Sieve of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime.This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes.
Sieve of Atkin
What is a factor?
Factors of a number are numbers that divide the given number exactly without leaving any remainder. That is, they are exact divisors of the given numbers. For example, the number 12 is divisible by 2, 3, 4, and 6.
What is HCF?
The greatest common factor (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides evenly all the given numbers with zero remainders. HCF stands for Highest Common Factor. HCF is also known as GCF (Greatest Common Factor) or GCD (Greatest Common Divisor).
The Highest Common Factor of two or more numbers is the highest number by which all the given numbers are divisible without leaving any remainders. Basically, it is the largest number that divides all the given numbers.
Let’s understand the concept of HCF with an example:
Consider two numbers: 12 and 18
Factors of 12: 1, 2, 3, 4, 6, and 12
Factors of 18: 1, 2, 3, 6, 9, and 18
Both 12 and 18 have 1, 2, 3, and 6 as common factors, of which 6 is the highest. So, 6 is the HCF of 12 and 18.
What is LCM?
LCM stands for Lowest or Least Common Multiple. The LCM of two or more numbers is the smallest positive integer that is divisible by all the given numbers.
Let’s understand the concept of LCM with an example:
Consider two numbers: 8 and 12.
The multiples of 8 are:
8 X 1 = 8,
8 X 2 = 16,
8 X 3 = 24,
8 X 4 = 32, and so on…
The multiples of 12 are:
12 X 1 = 12,
12 X 2 = 24,
12 X 3 = 36,
12 X 4 = 48, and so on…
Of all these multiples of 8 and 12, 24 is the lowest and common multiple of both. So, 24 is the LCM of 8 and 12.
LCM And HCF Formula
The relation between HCF and LCM for two numbers is defined by the HCF and LCM formula given by:
Product of two numbers = (HCF of the two numbers) x (LCM of the two numbers)
CAUCHY INTEGRAL,LIOUVILLE AND ROUCHE THEOREM
Cauchy Integral Theorem
If f(z) is analytic in some simply connected region R, then
\[\int f(z)\,dz\space =\space 0\]
for any closed contour gamma completely contained in R. Writing z as
\[z\space =\space x+iy\]
For a multiply connected region,
\[\int_{C_1} f(z)\,dz\space =\space \int_{C_2} f(z)\,dz\]
Liouville Theorem
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f(z)|<= M for all z in C is constant. Equivalently, non-constant holomorphic functions on C have unbounded images.
\[{\displaystyle |a_{k}|\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {|f(\zeta )|}{|\zeta |^{k+1}}}\,|d\zeta |\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {M}{r^{k+1}}}\,|d\zeta |={\frac {M}{2\pi r^{k+1}}}\oint _{C_{r}}|d\zeta |={\frac {M}{2\pi r^{k+1}}}2\pi r={\frac {M}{r^{k}}},}\]
A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if f and g are entire, and |f| ≤ |g| everywhere, then f = α·g for some complex number α. Consider that for g = 0 the theorem is trivial so we assume. Consider the function h = f/g. It is enough to prove that h can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of h is clear except at points in g−1(0). But since h is bounded and all the zeroes of g are isolated, any singularities must be removable. Thus h can be extended to an entire bounded function which by Liouville's theorem implies it is constant.
Rouche Theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K with closed contour K, if |g(z)| < |f(z)| on partial K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour K is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.
Let C be a closed, simple curve (i.e., not self-intersecting). Let h(z) = f(z) + g(z). If f and g are both holomorphic on the interior of C, then h must also be holomorphic on the interior of C. Then, with the conditions imposed above, the Rouche's theorem in its original (and not symmetric) form says that
If |f(z)| > |h(z) − f(z)|, for every z in C, then f and h have the same number of zeros in the interior of C.
Inequalities of Triangles
Hyperbolic trignometric ratios
Distance Formula: To Calculate Distance Between Two Points
\[Let \,the\, two \,points\, be \,A\, and \,B,\, having\, coordinates\, to\]
\[ be\, (x1,y1) \,and \,(x2,y2)\, respectively.\]
\[Thus,\, the \,distance\, between\, two\, points \,is\, given\, as-\]
\[\large d = \sqrt{(x_{2} -x_{1})^{2} + (y_{2} – y_{1})^{2}}\]
Midpoint Theorem: To Find Mid-point of a Line Connecting Two Points
\[Consider\, the\, same\, points \,A\, and \,B, \,having\, coordinates\, to\]
\[ be\, (x1,y1) \,and\, (x2,y2) \,respectively.\, Let\, M(x,y)\, be\, the\]
\[ midpoint\, of\, lying\, on \,the\, line\, connecting\, these\, two\]
\[ points \,A\, and\, B.\, The \,coordinates\, of\, the\, point\, M \,is\, given\, as-\]
\[\large M (x,y)= \left ( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right )\]
Angle Formula: To Find The Angle Between Two Lines
\[Consider\, two \,lines A\, and\, B,\, having \,their \,slopes\, to\, be\, m1\, and\, m2\, respectively.\]
\[Let\, “θ” \,be\, the\, angle\, between \,these \,two\, lines,\, then\, the\, angle \,between \,them \,can \,be\, represented\, as\]
\[\large \tan \theta = \frac{m_{1} – m_{2}}{1 + m_{1} m_{2}}\]
\[SPECIAL\, \, CASES:\]
\[i) \space \,Case \,1\,:\, When\, the\, two \,lines \,are\, parallel\, to \,each\, other,\]
\[\large m_{1} = m_{2}=m\]
\[Substituting\, the\, value \,in \,the\, equation\, above,\]
\[\large \tan \theta = \frac{m – m }{1 + m^{2}} = 0\]
\[\large \Rightarrow \theta = 0\]
\[ii) \space \,Case\, 2\,:\, When \,the\, two\, lines \,are\, perpendicular \,to \,each\, other,\]
\[m1 . m2 = -1\]
\[Substituting\, the\, value\, in \,the\, original \,equation,\]
\[\large \tan \theta = \frac{m_{1} – m_{2}}{1 + (-1)} = \frac{m_{1} – m_{2}}{0}\,\, which\, is\, undefined.\]
\[⇒ \theta = 90°\]
Section Formula: To Find a Point Which Divides a Line into m:n Ratio
\[Consider \, a \, line \, A \, and \, B \, having \, coordinates \, (x_1,y_1) \, and \, (x_2,y_2) \, respectively. \]
\[Let \,P\, be\, a\, point\, that \,which\, divides\, the \,line\, in \,the\, ratio\, m:n, \]
\[then \,the\, coordinates\,of\, the \,point\, P \,is \,given\, as-\]
\[i) \space When\, the\, ratio\, m:n \,is\, internal\,:\]
\[\large \left (\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n} \right )\]
\[ii) \space When \,the\, ratio \,m:n \,is\, external\,:\]
\[\large \left (\frac{mx_{2} – nx_{1}}{m – n}, \frac{my_{2} – ny_{1}}{m – n} \right )\]
Area of a Triangle in Cartesian Plane
\[The\, area \,of\, a\, triangle\, In \,coordinate\, geometry\,whose \,vertices\, are\, (x_1,y_1),\,(x_2,y_2) \,and\, (x_3,y_3)\, is\]
\[\frac{1}{2}|x_1 (y_2~ -~ y_3)~ + ~x_2(y_3~ – ~y_1)~ +~ x_3(y_1~ – ~y_2)|\]
Distance/MidPoint/Endpoint
3D Distance calculator
When Two Lines Meet
Distance between point and a line
Line Checker
Section Formula
Collinear Points Checker
Line and one Point
Distance Between Two Parallel Line
Perpendiculer Distance From Origin
Circle Calculator
Angle Bisector
Circle and a point
Two circle system
Common Tangent
2-D Shapes Inscribed
Largest triangle that can be inscribed in a semicircle
Regular polygon circumscribed to a circle
Minimum side of square embedded in Regular polygon with N sides
Smallest square formed with given rectangle
Area of a triangle with two vertices at midpoints of opposite sides of a square and the other vertex lying on vertex of a square
Area of largest semicircle that can be drawn inside a square
Number of rectangles in a circle
Incircle of a regular polygon
Largest rectangle that can be inscribed in a semicircle
Largest Hexagon that can be inscribed in a Square
Largest trapezoid that can be inscribed in a semicircle
Area of inner circle which passes through center of outer circle and touches its circumference
Area of circle inscribed in a Isosceles Trapezoid
Area of the largest triangle inscribed within a rectangle
Minimum area of square holding two identical rectangles
Count of distinct rectangles inscribed in an equilateral triangle
Largest square that can be inscribed within a hexagon which is inscribed within an equilateral triangle
Largest hexagon that can be inscribed within an equilateral triangle
Area of a Reuleaux Triangle inscirbed in a square
Area of a Reuleaux Triangle inscirbed in a square which is inscribed in a circle
Area of a Reuleaux Triangle inscribed in a Square which is inscribed in an equilateral triangle
Area of the largest Rhombus inscribed within a rectangle
Number of unique rectangles formed using N unit squares
Largest circle that can be inscribed in a semicircle
Area of a square inscribed in a circle which is inscribed in an equilateral triangle
Area of a Rectangle inscribed in an Equilateral Triangle
Area of a Reuleaux Triangle inscirbed in a square which is inscribed in a semicircle
Area of a Reuleaux Triangle inscirbed in a square which is inscribed in a ellipse
Area of a Reuleaux Triangle inscirbed in a square which is inscribed in a Right angle Triangle
Area of a Reuleaux Triangle inscribed in a Square which is inscribed in a Hexagon
Area of Equilateral triangle inscribed in a Circle of radius R
Area of a largest square fit in a right angle triangle
Area of a Largest square fit in a semicircle
Maximize a value for a semicircle of given radius
To check two Convex Regular Polygon have same center or not
Area of a circle inscribed in a rectangle which is inscribed in a semicircle
Largest ellipse that can be inscribed within a rectangle which in turn is inscribed within a semicircle
Radius of the biggest possible circle inscribed in rhombus which in turn is inscribed in a rectangle
Largest ellipse that can be inscribed within a rectangle which in turn is inscribed within a square
Area of largest isosceles triangle that can be inscribed in an Ellipse whose vertex coincides with one extremity of the major axis
Largest triangle that can be inscribed in an ellipse
Find the remaining vertices of a square from two given vertices
Division of an isosceles triangle in two parts with ratio of areas as n:m
Count of integral coordinates that lies inside a Square
Point that lies inside exactly K given squares
To check whether given Point (P) lies inside a Rectangle
To check whether given Point (P) lies inside a Sphere
To check whether given Point (P) lies inside a Parabola
Check if a given circle lies completely inside the ring formed by two concentric circles
To check whether given Point (P) lies inside a Hyperbola
To check whether given Point (P) lies inside a Ellipse
Check if a circle lies inside another circle or not
Square root calculator
Straight Line
\[1)\space Centroid\,with\,given\,three \,coordinates{(x_1,y_1),(x_2,y_2),(x_3,y_3)}:\,\]
\[\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\]
\[2)\space Relation\,between\,two\,lines\]
\[Let \,L1 \,and\, L2 \,be \,the \,two\, lines \,as\]
\[ L_1\, :\, a_1x + b_1y + c_1 = 0 \space \space \space \, \, \, L_2 \, :\, a_2x + b_2y + c_2 = 0\]
\[a)\space For\, Parallel\, Lines:\]
\[\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\]
\[b)\space For\, Intersecting\, Lines:\]
\[\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\]
\[c)\space For\, Coincident\, Lines:\]
\[\frac{a_1}{a_2}=\frac{b_1}{b_2} = \frac{c_1}{c_2}\]
\[3) \space Angle \,between\, Straight\, lines\]
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\[Let\,L_1\,≡y=m_1 x+c_1 \, \,and \,\,\, {{L}_{2}}\,\,\,\equiv \,\,\,y={{m}_{2}}x+{{c}_{2}}\] \[Angle = \theta ={{\tan }^{-1}}|( \frac{{{m}_{2}}-{{m}_{1}}}{1+{{m}_{1}}{{m}_{2}}})\] \[Special Cases:\] \[i) \space m_2=m_1\,→\,lines\,are\,parallel\] \[ii) \space m_1 m_2=−1\,→\,lines \,are\,perpendicular\,to\,each\,other\] |
\[4) \space Length\, of\, Perpendicular \,from \,a\, Point\, on\, a\, Line\]
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\[The\, length\, of\, the\, perpendicular \,from\] \[ P(x1, y1) \,on\, ax + by + c = 0 \,is\] \[\]\[l=\,|\frac{|ax_1+by_1+c}{\sqrt{a^2+b^2}}|\] \[B (x, y)\, is\, foot\, of\, perpendicular\, is\, given \,by:\] \[\frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{(ax_1+by_1+c)}{(a^2+b^2)}\] \[A’(h, k)\, is\, mirror\, image, \,given\, by:\] \[\frac{h-x_1}{a}=\frac{k-y_1}{b}=-2\,\frac{(ax_1+by_1+c)}{(a^2+b^2)}\] |
\[5) \space Angular\, Bisector\, of \,Straight \,lines\]
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\[An \,angle\, bisector \,has \,equal \,perpendicular\, distance\] \[ from \,the \,two\, given\, lines\] \[The\, equation\, of\, line\, L \,can\, be\, given\, as:\] \[\frac{(a_1 x+b_1 y+c_1)}{\sqrt{(a_1^2+b_1^2)}}\,=\,± \frac{(a_2 x+b_2 y+c_2)}{\sqrt{(a_2^2+b_2^2)}}\] |
\[6) \space Distance\, of\, a \,Point\, From\, a\, Line\]
\[Let\, the\, perpendicular\, distance (d)\, of \,a \,line \,Ax + By+ C = 0\, from \,a \,point (x1, y1)\, is\, defined \,by\]
\[d=\frac{\left | Ax_{1}+By_{1}+C \right |}{\sqrt{A^{2}+B^{2}}}\]
\[7) \space Distance \,Between \,Two\, Parallel\, Lines\]
\[The \,distance \,d\, between\, two \,parallel\, lines \,say, \,y= m x+ c1\, and\, y = m x + c2 \,is \,given\, by:\]
\[d=\frac{\left | C_{1}-C_{2} \right |}{\sqrt{1+m^{2}}}\]
\[Consider\, the \,general\, form\, of \,the\, line,\, i.e.,\, Ax + By + C1= 0 \,and\, Ax + By + C2=0\]
\[d=\frac{\left | c_{1}-c_{2} \right |}{\sqrt{A^{2}+B^{2}}}\]
\[8) \space Concurrency \,of \,Three \,Lines\]
\[Let \,the\, lines \,be:\]
\[{{L}_{1}}\equiv {{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}\]
\[ {{L}_{2}}\equiv {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}\]
\[ {{L}_{3}}\equiv {{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}\]
\[\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{vmatrix} = 0\]
Equation of a plane
Distance between two Points
Distance from point to plane calculator
Distance between perpendicular line and plane
Section Formula
Distance between two planes calculator
Angle between planes calculator
Centroid of a Tetrahedron
3-D Distance Calculator
3-D Geometry
\[1)\space Distance\, of\,P(x,y,z)\, from\, origin(0,0,0)= \sqrt {(x^2+y^2+z^2)}\]
\[2) \space Distance\, between \,2 \,points\, P\left( x_1,y_1,z_1 \right)\, and \,Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\]
\[=\sqrt{ (x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]
\[3) \space Let\, P\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)\, and \, Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\,be\, 2\, points.\]
\[ R\, divides\, the\, line \,segment\, PQ\, in\, ratio\, internally.\, Then\, R\, has\, coordinate:\]
\[(\frac {mx_2+nx_1}{m+n},\,\frac {my_2+ny_1}{m+n},\,\frac {mz_2+nz_1}{m+n})\]
\[4) \space Direction\, cosines\, and \,direction \,Ratios\, of\, a \,Line \,in \,Cartesian \,Plane\]
\[Cosines\, of \,the \,angles\, a \,line \,makes\, with \,the \,positive\, x,\, y\, and \, z \,axis\, respectively,\]
\[ are \,called \,direction \,cosines\, of \,that \,line.\]
\[So\, if\, those \,angles\, are\, \alpha ,\,\beta ,\,\gamma\, then\, \cos \alpha ,\,cos\beta ,\,cos\gamma\, are \,the\, direction\, cosines\, of\, the\]
\[ line. \,They \,are\, denoted\, by\, l,m,n.\]
\[l^2+m^2+n^2=1\]
\[Any\, 3 \,number\, a,\,b,\,c\,which\, are\, proportional\, to\, direction\, cosines\]
\[ are \,called \,direction\, ratios.\]
\[Hence \frac{l}{a}=\frac{m}{b}=\frac{n}{c}=\frac{\sqrt{{{l}^{2}}+{{m}^{2}}+{{n}^{2}}}}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}=\frac{1}{\sqrt{{{a}^{2}}{{b}^{2}}+{{c}^{2}}}} \]
\[5) \space Let \,P(x,y,z) \,and\, Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\, be \,2 \,points. \]
\[Then\, direction \,cosines\, will \,be\]
\[ l=\frac{{{x}_{2}}-{{x}_{1}}}{\left| PQ \right|},m=\frac{{{y}_{2}}-{{y}_{1}}}{\left| PQ \right|},n=\frac{{{z}_{2}}-{{z}_{1}}}{\left| PQ \right|}\]
\[6) \space Projection\, of\, PQ,\,(P(x_1,y_1,z_1),\, Q(x_2,y_2,z_2)) \,on \,a \,line\, whose\]
\[direction \,cosines\, are \,l,m,n is \,l\left( {{x}_{2}}-{{x}_{1}} \right)+m\left( {{y}_{2}}-{{y}_{1}} \right)+n\left( {{z}_{2}}-{{z}_{1}} \right)\]
\[7) \space 2\, lines \,having\, direction\, cosines\, \left( {{l}_{1}},{{m}_{1}},{{n}_{1}} \right)\, and\, \left( {{l}_{2}},{{m}_{2}},{{n}_{2}} \right).\]
\[ Then \,angle\, between\, them\, is\, \theta ={{\cos }^{-1}}\left( {{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}} \right)\]
\[8) \space Let\, \bar{A} \,be\, the\, vector\, area. \]
\[If\, its\, direction\, cosines \,are \, \cos \alpha ,\cos \beta ,\cos \gamma.\]
\[ Then\, projections \,are\, {{A}_{1}}=A\cos \alpha ,{{A}_{2}}=A\cos \beta ,{{A}_{3}}=A\cos \gamma .\]
\[∴ {{A}^{2}}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}\]
\[9) \space Area \,of\, a\, triangle:\]
\[\alpha = \frac 12\left| \begin{matrix} x_1 & y_1 & 1\cr x_2 & y_2 & 1 \cr x_3 & y_3 & 1 \cr \end{matrix} \right|\]
\[10) \space A \,first\, degree\, equation\, in\, x,\, y, \,z \, \, represents\, a\, plane\, in\, 3D\]
\[ ax+by+cz=0,{{z}^{2}}{{b}^{2}}+{{c}^{2}} \ne \,represents\, a \,plane.\]
\[11) \space Let\, P\, be\, the\, length\, of\, the\, normal\, from\, the \,origin\, to\, the\, plane\]
\[ and \,\, l,m,n \,be\, the \,direction\, cosines\, of\, that \,normal. \]
\[Then\, the\, equation \,of \,the\, plane\, is\, given\, by \, lx+my+nz=P.\]
\[12) \space Let\, a\, plane\, cuts\, length\, a,b,c \,from\, the\, coordinate\, axis.\]
\[ Then\, equation\, of \,the \,plane \,is \, \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1.\]
\[13) \space Plane\, passing\, through \, \left( {{x}_{1}}{{y}_{1}}{{z}_{1}} \right), \, \left( {{x}_{2}}{{y}_{2}}{{z}_{2}} \right), \, \left( {{x}_{3}}{{y}_{3}}{{z}_{3}} \right) \, is\]
\[ \left| \begin{matrix} x & y & z & 1 \\ {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1 \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1 \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1 \\ \end{matrix} \right|=0\]
\[14) \space Angle\, between\, 2\, planes\]
\[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0 \, and \, {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0 \, is \, given \, by:\]
\[ \cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]
\[15) \space Two\, sides \,of\, a\, plane\]
\[2\, points\, A\left( {{x}_{1}}{{y}_{1}}{{z}_{1}} \right) \,and\, B\left( {{x}_{2}}{{y}_{2}}{{z}_{2}} \right)\, lie\, on\, the\, same\, side\, or\]
\[ opposite \,sides\, of\, a\, plane\, ax+by+cz+d=0 \,accordingly\, as\, a{{x}_{1}}+b{{y}_{1}}+c{{z}_{1}}+d \]
\[and\, a{{x}_{2}}+b{{y}_{2}}+c{{z}_{2}}+d \,are\, of\, same\, sign \,or\, opposite\, sign.\]
\[16) \space Distance\, from\, a\, point\, to\, a \,plane\]
\[Distance \,of\,{{x}_{1}}{{y}_{1}}{{z}_{1}} \,from\, ax+by+cz+d\, is\]
\[ | \frac{a{{x}_{1}}+b{{y}_{1}}+c{{z}_{1}}+d}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\]
\[17) \space Equation\, of \,the\, planes\, bisecting\, the\, angle\, between \,2\, planes\]
\[Let\, {{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+d=0 \, and \, {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+d=0 \, be \, 2 \, planes. \]
\[The\, planes \,bisecting\, the\, angles \,between \,them \,are \]
\[ \frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\pm \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]
\[18) \space Two\, Intersecting\, plane\]
\[If\, U=0\, and\, V=0 \,be\, 2\, planes\, then \,the\, plane\]
\[ passing\, through \,the\, line \,of \,their\, intersection\, is\]
\[ U+\lambda V=0\lambda \,to\, be \,determined\, from\, given\, condition.\]
Probability Properties
General Formula
\[P(E)\,=\,\frac{Number\space of\space favourable\space outcomes}{Total\space Number\space of\space outcomes}\]
General form of Addition Theorem of Probability
\[P(A_1\,\cup\,A_2\,\cup\,....\,\cup\,A_n)\,=[\,\sum_{i=1}^{n} P(A_i) \,-\,\sum_{i < j} P(A_i \,\cap \, A_j) \, + \,\sum_{i < j < k} P(A_i \,\cap \, A_j \, \cap \, A_k) \, - \\ ...\,+\,(-1)^{n-1}\,P(A_1 \,\cap\, A_2\,\cap\,...\,\cap \, A_n) ] \]
Conditional Probability
\[The\,probability\,of\,occurrence\,of\,an\,event\,E_1\, , \,given\,that\,E_2\,has\,already\,occured\\is\,called\,the\,Conditional\,Probabilty\, , \,it\,is\,denoted\,by\,P(\frac{E_1}{E_2}) \]
\[P(\frac{E_1}{E_2})\,=\,\frac{P(E_1\,\cap\,E_2)}{P(E_2)}\,=\,\frac{n(E_1\,\cap\,E_2)}{n(E_2)}\]
Total Probability Theorem
\[Let\,E_1\, , \, E_2 \, ,\, ... \,E_n\,be\,n\,mutually\,exclusive\,and\,exhaustive\,events\,i.e.,\\E_i\, \cap \, E_j\,=\,\phi\,for\,i\,!= \,j\,and\,\bigcup_{i=1}^{n}\,E_i\,=\,S\,.\,Suppose\,that\, , \\ P(E_i)\,>\,0\, ,\,\forall\,1\,\leq\,i\,\leq\,n\,Then\,for\,any\,event\,E \]
\[P(E)\,=\,\sum_{i=1}^{n}\,P(E_i)\,.\,P(\frac{E}{E_i})\]
Baye's Theorem
\[If\,an\,event\, E \,can\, occur \,only\,with\,one\,of\,the\,n\,mutually\,exclusive\,and\,exhaustive\,events\\E_1\, ,\, E_2 \, , \, E_3 \, ... \, E_n\,and\,the\,probabilities\,P(E/E_1)\, ,\,P(E/E_2)\, ,\, ...\, , \, P(E/E_n)\,are\,known\, , \,then \]
\[P(\frac{E_k}{E})\,=\,\frac{P(E_k)\,.\,\frac{P(E)}{P(E_k)}}{\sum_{k=1}^{n}P(E_k)\,.\,\frac{P(E)}{P(E_k)}}\]
Binomial Theorem on Probability
\[Suppose\,a\,binomial\,experiment\,has\,probability\,of\,success\,p\,and\,that\,of\,failure\,q\,.\\If\,E\,be\,an\,event\,and\,let\,X\,=\,number\,of\,times\,event\,E\,occurs\,in\,n\,trials\\Then\,the\,probability\,of\,occurence\,of\,event\,E\,exactly\,r\,times\,in\,n\,trials\,is\,denoted\,by \]
\[P(r)\,=\,nC_r\,p^{r}\,q^{n-r}\\where\, , \,r\,=\,0\, ,\,1\, , \,2\, , \,3\, , \, ... \, ,\,n\, . \]
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Statistics Formula Sheet
| \[Mean\] | \[\bar{x}=\frac{\sum x}{n}\] |
\[x\,=\, Observations \,given\] \[n\, =\, Total\, number\, of\] \[observations\] |
|---|---|---|
| \[Median\] |
\[If\, n\, is\, odd,\, then\] \[M\,=(\frac{n+1}{2})\,term\] \[If\, n\, is\, even,\, then\] \[M \,=\,\frac{(\frac{n}{2})^{th}term+(\frac{n}{2}+1)^{th}term}{2}\] |
\[n \,= \,Total\, number \,of\] \[observations\] |
| \[Mode\] |
\[The\, value\, which\, occurs\]
\[most\, frequently\] |
\[-\] |
| \[Variance\] | \[\sigma ^{2}\,=\,\frac{\sum (x-\bar{x})^{2}}{n}\] |
\[x\, =\, Observations\, given\] \[x¯\, =\, Mean\] \[n \,= \,Total\, number \,of\] \[observations\] |
| \[Standard \,Deviation\] | \[S = \sigma = \sqrt{\frac{\sum (x-\bar{x})^{2}}{n}}\] |
\[x\, =\, Observations\, given\] \[x¯\, =\, Mean\] \[n \,= \,Total\, number \,of\] \[observations\] |
| \[Range\] | \[L-S\] |
\[L\, =\, Largest\, value\] \[S\, =\, Smallest\, value\] |
| \[Coeff.\,of\,Range\] | \[\frac{L-S}{L+S}\] |
\[L\, =\, Largest\, value\] \[S\, =\, Smallest\, value\] |
| \[Coeff.\,of\,Variation\] | \[\frac{\sigma}{\bar{x}}*100\] |
\[\sigma=Standard\, Deviation\] \[x¯\, =\, Mean\] |
| \[Combined \,Mean\] | \[\frac{m\bar{x}\,+\,n\bar{y}}{m+n}\] |
\[x¯\,/y¯\, =\, Mean\, of\, two\] \[distributions\] \[m\,/n\,=no.\,of\,elements\,in\,\] \[each\,distributions\] |
| \[Weighted\,Mean\] | \[\frac{\sum Wi\,xi}{\sum Wi}\] |
\[Wi=Weight \,of\, each\] \[distributions\] \[xi=ith\,observation\] |
|
\[Mean\,Deviation\] \[about\,mean\] \[\bar{X}\] |
\[\frac{\sum|xi-\bar{x}|}{n}\] |
\[n \,= \,Total\, number \,of\] \[observations\] \[x¯\, =\, Mean\] \[xi=ith\,observation\] |
|
\[Mean\,Deviation\] \[about\,median\] \[M\] |
\[\frac{\sum|xi-M|}{n}\] |
\[n \,= \,Total\, number \,of\] \[observations\] \[M\, =\, Median\] \[xi=ith\,observation\] |
|
\[Mean\,Deviation\] \[about\,mode\] \[Z\] |
\[\frac{\sum|xi-Z|}{n}\] |
\[n \,= \,Total\, number \,of\] \[observations\] \[Z\, =\, Mode\] \[xi=ith\,observation\] |
|
\[Mean\,Deviation\] \[about\,any\,no.\,'a'\] |
\[\frac{\sum|xi-a|}{n}\] |
\[n \,= \,Total\, number \,of\] \[observations\] \[xi=ith\,observation\] |
|
\[For \,symmetrical \] \[Distribution:\] |
\[Mode\,=\,3Median\,-\,2Mean\] | \[-\] |
n-th base root calculator
Identities:
\[\left(Z_{1} \space + \space Z_{2}\right)^2 \space = \space \left(Z_{1}\right)^2 \space + \space \left(Z_{2}\right)^2 \space + \space 2 \space Z_{1}Z_{2} \]
\[\left(Z_{1} \space - \space Z_{2}\right)^2 \space = \space \left(Z_{1}\right)^2 \space + \space \left(Z_{2}\right)^2 \space - \space 2 \space Z_{1}Z_{2} \]
\[\left(Z_{1}\right)^2 \space - \space \left(Z_{2}\right)^2 \space = \space \left(Z_{1} \space + \space Z_{2} \right)\left(Z_{1} \space - \space Z_{2} \right) \]
\[\left(Z_{1} \space + \space Z_{2}\right)^3 \space = \space \left(Z_{1}\right)^3 \space + \space 3\left(Z_{1}\right)^2Z_{2} \space + \space 3\left(Z_{2}\right)^2Z_{1} \space + \left(Z_{2}\right)^3 \]
\[\left(Z_{1} \space - \space Z_{2}\right)^3 \space = \space \left(Z_{1}\right)^3 \space - \space 3\left(Z_{1}\right)^2Z_{2} \space + \space 3\left(Z_{2}\right)^2Z_{1} \space - \left(Z_{2}\right)^3 \]
De Moivre's Theorem
For any real number x, we have
\[(cos(x)+isin(x))^{n}\space =\space cos(nx)\space +\space isin(nx)\]
Remark: Result can be shown true when n is a negative integer and even n is a rational number.
Cauchy-Reimaan Theorem
What are analytic functions?
A function f(z) is said to be analytic in a region R of the complex plane if f(z) has a derivative at each point of R and if f(z) is single valued.
Let f(x,y) = u(x,y) + iv(x,y) and the function should be analytic
Then,
\[\frac{∂v}{∂x}\space =\space -\frac{∂u}{∂y}\]
If, z = reiθ, then Cauchy-Reimaan Theorem become-
\[\frac{1}{r}\frac{∂u}{∂θ}\space =\space -\frac{∂v}{∂r}\]
Properties of Complex Numbers:
- 1. The addition of two conjugate complex numbers will result in a real number.
- 2. The multiplication of two conjugate complex number will also result in a real number.
- 3. If x and y are the real numbers and x+yi =0, then x =0 and y =0.
- 4. If p, q, r, and s are the real numbers and p+qi = r+si, then p = r, and q=s.
- 5. The
complex number obeys the commutative law of addition and multiplication.
\[Z_{1} \space + \space Z_{2} \space = \space Z_{2} \space + \space Z_{1} \\ Z_{1} \space * \space Z_{2} \space = \space Z_{2} \space * \space Z_{1} \]
- 6. The
complex number obeys the associative law of addition and multiplication.
\[\left(Z_{1} \space + \space Z_{2}\right) \space + Z_{3} \space = \space Z_{1} \space + \space \left(Z_{2} \space + \space Z_{3}\right) \\ \left(Z_{1} \space * \space Z_{2}\right) \space * Z_{3} \space = \space Z_{1} \space * \space \left(Z_{2} \space * \space Z_{3}\right) \]
- 7. The
complex number obeys the distributive law.
\[Z_{1} \space * \space \left(Z_{2} \space + \space Z_{3}\right) \space = \space Z_{1}*Z_{2} \space + \space Z_{1}*Z_{3} \]
- 8. If the sum of two complex number is real, and also the product of two complex number is also real, then these complex numbers are conjugate to each other.
- 9. For
any two complex numbers, say Z1 and Z2, then
\[\mid Z_{1} \space + \space Z_{2}\mid \space \le \space \mid Z_{1}\mid \space + \space \mid Z_{2}\mid \]
- 10. The result of the multiplication of two complex numbers and its conjugate value should result in a complex number and it should be a positive value.
Previous perfect square and cube number smaller than number N
Perfect cube greater than a given number
Perfect square greater than given number
Least number to be added to or subtracted from N to make it a Perfect Cube
Least number to be added to or subtracted from N to make it a Perfect Square
Count numbers upto N which are both perfect square and perfect cube
Find minimum number to be divided to make a number a perfect square
Minimum divisor of a number to make the number perfect cube
Cube Free Numbers smaller than n
Number of times the largest Perfect Cube can be subtracted from N
Find minimum number to be multiplied to make a number a perfect square
Find minimum number to be divided to make a number a perfect cube
Find minimum number to be multiplied to make a number a perfect cube
Check if number formed by joining two Numbers is Perfect Cube
Difference between Sum of Cubes and Sum of First N Natural Numbers
Difference between Sum of Cubes and Sum of Squares of First N Natural Numbers
Minimum digits to remove to make a number Perfect Square
Find square root of number upto given precision
Difference between Sum of Squares and Sum of First N Natural Numbers
Count square and non-square numbers before n
Check whether a number can be represented by sum of two squares
Check whether a number can be represented by difference of two squares
Check whether a number can be represented by product of two squares
Check whether a number can be represented by sum of two cubes
Check whether a number can be represented by difference of two cubes
Calculations for 1 complex number
Polar representaion of complex number
Find Euler Representaion of complex number
N-th Power of complex number
Calculations for 2 complex numbers
Roots of Unity Calculator
Volume, CSA/LSA and TSA of 3D solids
| \[Shape\] | \[Volume\] | \[C\,S\,A \space/\,L\,S\,A \] | \[T\,S\,A\] |
|---|---|---|---|
| \[Cube\] | \[a^{3}\] | \[4\,a^{2}\] | \[6\,a^{2}\] |
| \[Cuboid\] | \[l\,b\,h\] | \[2\,(\,l\,+\,b\,)\,h\] | \[2\,(\,lb\,+\,bh\,+\,lh\,)\] |
| \[Cylinder\] | \[ π\,r^{2}\,h\] | \[2\,π\,r\,h\] | \[2πr\, (\, r\, +\, h\, )\] |
| \[Hollow\space Cylinder\] | \[ πh\,(\, R \,+ \,r\, )\, (\, R\, -\, r\, )\] | \[2πh\space ( \,R\, +\, r\, )\] | \[2πh\, (\, R\, +\, r\, )\]
\[+\] \[2π\, (\, R\, +\, r\, )\, (\, R\, -\, r\, )\] |
| \[Cone\] | \[ \frac{1}{3} \, π\,r^{2} \, h\] | \[πrl\, =\, πr\,√( r^{2}\,+ \,h^{2} )\] | \[πr \,(\, l\, +\, r\, )\] |
| \[Frustum\space Of\space Cone\] | \[ ( \frac{rh}{3} )\, ( \,r^{2} \,+\, R^{2} \,+\, rR \,)\] | \[π(R+r)l\, =\, π(R+r)√(h^{2}+(R-r)^{2})\] | \[π(rl+Rl+r^{2}+R^{2})\] |
| \[Sphere\] | \[\frac{4}{3}\, π\,r^{3}\] | \[ 4\, π\,r^{2}\] | \[4\, π\,r^{2}\] |
| \[Hollow\space Sphere\] | \[\frac{4}{3}\, π\,(\,r_2^{3}\,-\,r_1^{3})\] |
\[4\, π\,(r_2^{2}-r_1^{2})\] |
\[4\, π\,(r_2^{2}-r_1^{2})\] \[TSA\,=\,CSA\] |
| \[Hemisphere\] | \[\frac{(2\, π\,r^{3}\,)}{3}\] | \[2\, π\,r^{2}\] | \[3\, π\,r^{2}\] |
| \[Prism\] | \[ Area \,of \,base× Height \] |
\[Perimeter\, of\, base\]
\[×\] \[Height\, of\, prism\] |
\[LSA + 2\,× Area \,of\, base\] |
| \[Rectangular\space Prism\] | \[(l\,b)\,×h\] | \[(\,2l\,+\,2b\,)\,×h\] | \[LSA +2×(\,lb\,)\] |
| \[Triangular\space Prism\] |
\[(\frac{1}{2}×apothem\space length\] \[×\] \[base\space length)\,×Height\] |
\[(3\,×base\,length)\] \[×\] \[Height\] |
\[LSA + 2×(\,\frac{1}{2}×apothem\space length\] \[×\] \[base\space length\,)\] |
| \[Pentagonal\space Prism\] |
\[(\frac{5}{2}×apothem\space length\] \[×\] \[base\space length)\,×Height\] |
\[(5\,×base\,length)\] \[×\] \[Height\] |
\[LSA + 2×(\,\frac{5}{2}×apothem\space length\] \[×\] \[base\space length\,)\] |
| \[Hexagonal\space Prism\] |
\[(3\,×apothem\space length\] \[×\] \[base\space length)\,×Height\] |
\[(6\,×base\,length)\] \[×\] \[Height\] |
\[LSA + 2×(\,3\,×apothem\space length\] \[×\] \[base\space length\,)\] |
| \[Pyramid\] | \[\frac{1}{3}×base\,area×Height\] |
\[ \frac{1}{2} × Perimeter\, of\, base\]
\[×\] \[Slant\, height\, Total\] |
\[LSA + Area\, of \,base\] |
| \[Square\space Pyramid\] | \[\frac{1}{3}(\,base\space length\,)^{2}\,×Height\] | \[\frac{1}{2} (4\,×base\, length)\]\[×\]\[slant\, height\] | \[LSA + (\,base\space length\,)^{2}\] |
| \[Triangular\space Pyramid\] |
\[\frac{1}{3}(\,\frac{1}{2}×apothem\space length\] \[×\] \[base\space length\,)\,×Height\] |
\[\frac{1}{2}(\,3×base\,length)\] \[×\] \[slant\space height\] |
\[LSA +(\, \frac{1}{2}×apothem\space length\] \[×\] \[base\space length\,)\] |
| \[Pentagonal\space Pyramid\] |
\[\frac{1}{3}(\,\frac{5}{2}×apothem\space length\] \[×\] \[base\space length\,)\,×Height\] |
\[\frac{1}{2}(\,5×base\space length)\] \[×\] \[slant\space height\] |
\[LSA +(\, \frac{5}{2}×apothem\space length\] \[×\] \[base\space length\,)\] |
| \[Hexagonal\space Pyramid\] |
\[\frac{1}{3}(\,3×\,apothem\space length\] \[×\] \[base\space length\,)\,×Height\] |
\[\frac{1}{2}(\, 6×base\space length)\] \[×\] \[slant\space height\] |
\[LSA +(\, 3×apothem\space length\] \[×\] \[base\space length\,)\] |
| \[Frustum\space Of\space Pyramid\] |
\[\frac{1}{3}×(\,A_1+A_2+\,\sqrt{A_1A_2})\,×Height\] \[^{*}A_1\,-Area\, of\, Upper\, Base\] \[^{*}A_2\,-Area\, of\, Lower\, Base\] |
\[\frac{1}{2}×(P_1+P_2)\,×slant\,height\]\[^{*}P_1\,-Perimeter\,Of\,Upper\,Base\]\[^{*}P_2\,-Perimeter\,Of\,Lower\,Base\] | \[LSA\,+\,A_1\,+\,A_2\] |
| \[Torus\] | \[(\,2π\,R\,)\,(\,π\,r^{2}\,)\]\[^{*}R\,-\,Major\,Radius\]\[^{*}r\,-\,Minor\,Radius\] | \[(\,2π\,R\,)\,(\,2π\,r\,)\] | \[(\,2π\,R\,)\,(\,2π\,r\,)\] |
Volumetric Weight Calculator
Percentage change in Volume of a Sphere
Percentage change in Volume of a Cube
Percentage change in Volume of a Cuboid
| \[Angles→\] \[Ratios↓\] |
\[0^{\circ}\] | \[30^{\circ}\] | \[45^{\circ}\] | \[60^{\circ}\] | \[90^{\circ}\] |
|---|---|---|---|---|---|
| \[\sin\theta\] | \[0\] | \[\frac{1}{2}\] | \[\frac{1}{\sqrt{2}}\] | \[\frac{\sqrt{3}}{2}\] | \[1\] |
| \[\cos\theta\] | \[1\] | \[\frac{\sqrt{3}}{2}\] | \[\frac{1}{\sqrt{2}}\] | \[\frac{1}{2}\] | \[0\] |
| \[\tan\theta\] | \[0\] | \[\frac{1}{\sqrt{3}}\] | \[1\] | \[\sqrt{3}\] | \[Not \space Defined\] |
| \[\cosec\theta\] | \[Not \space Defined\] | \[2\] | \[\sqrt{2}\] | \[\frac{2}{\sqrt{3}}\] | \[1\] |
| \[\sec\theta\] | \[1\] | \[\frac{2}{\sqrt{3}}\] | \[\sqrt{2}\] | \[2\] | \[Not \space Defined\] |
| \[\cot\theta\] | \[Not \space Defined\] | \[\sqrt{3}\] | \[1\] | \[\frac{1}{\sqrt{3}}\] | \[0\] |
\[Values \space of \space some \space T-Ratios \space for \space many \space angles \]
\[1) \space sin(7.5^{\circ})= \frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}{2}= cos(82.5^{\circ})= sin \frac{\pi}{24}\]
\[2) \space cos(7.5^{\circ})= \frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}= sin(82.5^{\circ})= cos \frac{\pi}{24}\]
\[3) \space tan(7.5^{\circ})= \sqrt{6}-\sqrt{3}+\sqrt{2}-2=(\sqrt{2}-1)(\sqrt{3}-\sqrt{2})= cot(82.5^{\circ})= tan\frac{\pi}{24}\]
\[4) \space cot(7.5^{\circ})= \sqrt{6}+\sqrt{3}+\sqrt{2}+2=(\sqrt{2}+1)(\sqrt{3}+\sqrt{2})= tan(82.5^{\circ})= cot\frac{\pi}{24}\]
\[5) \space sin15^{\circ}= \frac{\sqrt{3}-1}{2\sqrt{2}}= cos75^{\circ}= sin \frac{\pi}{12}\]
\[6) \space cos15^{\circ}= \frac{\sqrt{3}+1}{2\sqrt{2}}= sin75^{\circ}= cos \frac{\pi}{12}\]
\[7) \space tan15^{\circ}= 2-\sqrt{3}= cot75^{\circ}= tan\frac{\pi}{12}\]
\[8) \space cot15^{\circ}= 2+\sqrt{3}= tan75^{\circ}= cot\frac{\pi}{12}\]
\[9) \space sin18^{\circ}= \frac{\sqrt{5}-1}{4}= \sqrt{\frac{3-\sqrt{5}}{8}} = cos72^{\circ}= sin\frac{\pi}{10} \]
\[10) \space cos18^{\circ}= \frac{\sqrt{10+2\sqrt{5}}}{4}= \sqrt{\frac{5+\sqrt{5}}{8}} = sin72^{\circ}= cos\frac{\pi}{10} \]
\[11) \space tan18^{\circ}= \sqrt{1-\frac{2\sqrt{5}}{5}}= cot72^{\circ}= tan\frac{\pi}{10}\]
\[12) \space cot18^{\circ}= \sqrt{5+2\sqrt{5}}= tan72^{\circ}= cot\frac{\pi}{10}\]
\[13) \space sin(22.5^{\circ})= \frac{\sqrt{2-\sqrt{2}}}{2}= \sqrt{\frac{4-\sqrt{8}}{8}} = cos(67.5^{\circ})= sin\frac{\pi}{8}\]
\[14) \space cos(22.5^{\circ})= \frac{\sqrt{2+\sqrt{2}}}{2}= \sqrt{\frac{4+\sqrt{8}}{8}} = sin(67.5^{\circ})= cos\frac{\pi}{8}\]
\[15) \space tan(22.5^{\circ})=\sqrt{2}-1=cot(67.5^{\circ})= tan\frac{\pi}{8}\]
\[16) \space cot(22.5^{\circ})=1+\sqrt{2}=tan(67.5^{\circ})= cot\frac{\pi}{8}\]
\[17) \space sin36^{\circ}= \frac{\sqrt{10-2\sqrt{5}}}{4}= \sqrt{\frac{5-\sqrt{5}}{8}}= cos54^{\circ}= sin\frac{\pi}{5}\]
\[18) \space cos36^{\circ}= \frac{\sqrt{5}+1}{4}= \sqrt{\frac{3+\sqrt{5}}{8}}= sin54^{\circ}=cos \frac{\pi}{5}\]
\[19) \space tan36^{\circ}= \sqrt{5-2\sqrt{5}}= cot54^{\circ}= tan \frac{\pi}{5}\]
\[20) \space cot36^{\circ}= \sqrt{\frac{5+2\sqrt{5}}{5}}= tan54^{\circ}= cot \frac{\pi}{5}\]
\[21) \space sin(37.5^{\circ})= \frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}{2}= cos(52.5^{\circ})= sin \frac{5\pi}{24}\]
\[22) \space cos(37.5^{\circ})= \frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}{2}= sin(52.5^{\circ})= cos \frac{5\pi}{24}\]
\[23) \space tan(37.5^{\circ})= \sqrt{6}+\sqrt{3}-\sqrt{2}-2=(\sqrt{2}+1)(\sqrt{3}-\sqrt{2})= cot(52.5^{\circ})= tan\frac{5\pi}{24}\]
\[24) \space cot(37.5^{\circ})= \sqrt{6}-\sqrt{3}-\sqrt{2}+2=(\sqrt{2}-1)(\sqrt{3}+\sqrt{2})= tan(52.5^{\circ})= cot\frac{5\pi}{24} \]
Quotient Identities
\[tan \space \theta=\frac{sin\space\theta}{cos\space\theta}\]
\[cot \space \theta=\frac{cos\space\theta}{sin\space\theta}\]
Reciprocal Identities
\[sin\space\theta=\frac{1}{cosec\space\theta}\]
\[cos\space\theta=\frac{1}{sec\space\theta}\]
\[tan\space\theta=\frac{1}{cot\space\theta}\]
\[cosec\space\theta=\frac{1}{sin\space\theta}\]
\[sec\space\theta=\frac{1}{cos\space\theta}\]
\[cot\space\theta=\frac{1}{tan\space\theta}\]
Pythagorean Identities
\[sin^2\space\theta\space+\space cos^2\space\theta=1\]
\[sec^2\space\theta\space-\space tan^2\space\theta=1\]
\[cosec^2\space\theta\space-\space cot^2\space\theta=1\]
Even/Odd Identities
\[sin\space (-\theta )=\space -sin\space\theta\]
\[tan\space (-\theta )=\space -tan\space\theta\]
\[cosec\space (-\theta )=\space -cosec\space\theta\]
\[cos\space (-\theta )=\space cos\space\theta\]
\[cot\space (-\theta )=\space -cot\space\theta\]
\[sec\space (-\theta )=\space sec\space\theta\]
Double angle Identities
\[sin\space 2 \theta=2 \space sin\space\theta\space cos\space\theta\]
\[cos\space 2 \theta= cos^2 \space\theta - sin^2 \space\theta\]
\[cos\space 2 \theta = 2\space cos^2\space\theta - 1\]
\[cos\space 2 \theta= 1- 2\space sin^2 \space\theta\]
\[tan\space 2 \theta=\frac{2\space tan \theta}{1 - tan^2 \theta}\]
Cofunction Identities
\[sin \space (\frac{π}{2} - \theta ) = cos\space\theta\]
\[tan \space (\frac{π}{2} - \theta ) = cot\space\theta\]
\[cosec \space (\frac{π}{2} - \theta ) = sec\space\theta\]
\[cos \space (\frac{π}{2} - \theta ) = sin\space\theta\]
\[cot \space (\frac{π}{2} - \theta ) = tan\space\theta\]
\[sec \space (\frac{π}{2} - \theta ) = cosec\space\theta\]
Half angle Identities
\[sin^2\space\theta = \frac{1 - cos 2\theta}{2}\]
\[cos^2\space\theta = \frac{1 + cos 2\theta}{2}\]
\[tan^2\space\theta = \frac{1 - cos 2\theta}{1 + cos 2\theta}\]
Sum/Difference Identities
\[sin (\theta ± \phi ) = sin\space\theta cos\space\phi ± cos\space\theta sin\space\phi\]
\[cos (\theta ± \phi ) = cos\space\theta cos\space\phi ∓ sin\space\theta sin\space\phi\]
\[tan (\theta ± \phi ) = \frac{tan \space\theta ± tan \space\phi}{1 ∓ tan \space\theta tan\space\phi}\]
Product-to-Sum Identities
\[2 cos \space \theta \cos \space \varphi = {{\cos(\theta - \varphi) + \cos(\theta + \varphi)}}\]
\[2 sin \space \theta \sin \space \varphi = {{\cos(\theta - \varphi) - \cos(\theta + \varphi)}}\]
\[2 sin \space \theta \cos \space \varphi = {{\sin(\theta + \varphi) + \sin(\theta - \varphi)}}\]
\[2 cos \space \theta \sin \space \varphi = {{\sin(\theta + \varphi) - \sin(\theta - \varphi)}}\]
\[2 tan \space \theta \tan \space \varphi =\frac{\cos(\theta-\varphi)-\cos(\theta+\varphi)}{\cos(\theta-\varphi)+\cos(\theta+\varphi)}\]
Sum-to-Product Identities
\[sin \space \theta \pm \sin \space \varphi = 2\sin\left( \frac{\theta \pm \varphi}{2} \right) \cos\left( \frac{\theta \mp \varphi}{2} \right)\]
\[cos \space \theta + \cos \space \varphi = 2\cos\left( \frac{\theta + \varphi} {2} \right) \cos\left( \frac{\theta - \varphi}{2} \right)\]
\[cos \space \theta - \cos \space \varphi = -2\sin\left( \frac{\theta + \varphi}{2}\right) \sin\left(\frac{\theta - \varphi}{2}\right)\]
Triple Angle Identities
\[sin\space 3 \theta=3 \space sin\space\theta - 4 \space sin^3 \space\theta\]
\[cos\space 3 \theta=4 \space cos^3\space\theta - 3 \space cos \space\theta\]
\[tan\space 3 \theta=\frac{3 \space tan\space\theta - tan^3 \space\theta}{1 - 3 \space tan^2 \space \theta}\]
Conditional Identities
\[Given:\space A\,+\,B\,+\,C\,=\,π\]
\[sin\space 2A\space+\space sin\space 2B\space+\space sin\space 2C\space=4\,sin\,A\,sin\,B\,sin\,C\]
\[sin\space A\space+\space sin\space B\space+\space sin\space C\space=4\,cos\,\frac{A}{2}\,cos\,\frac{B}{2}\,cos\,\frac{C}{2}\]
\[cos\space 2A\space+\space cos\space 2B\space+\space cos\space 2C\space=-\,1\,-\,4\,cos\,A\,cos\,B\,cos\,C\]
\[cos\space A\space+\space cos\space B\space+\space cos\space C\space=1\,+\,4\,sin\,\frac{A}{2}\,sin\,\frac{B}{2}\,sin\,\frac{C}{2}\]
\[tan\space A\space+\space tan\space B\space+\space tan\space C\space=\,tan\,A\,tan\,B\,tan\,C\]
\[tan\space \frac{A}{2}\space tan\space \frac{B}{2}\space+\space tan\space \frac{B}{2}\space tan\space \frac{C}{2}\space+\space tan\space \frac{C}{2}\space tan\space \frac{A}{2}\space=\,1\]
\[cot\space \frac{A}{2}\space+\space cot\space \frac{B}{2}\space+\space cot\space \frac{C}{2}\space=\,cot\,\frac{A}{2}\,cot\,\frac{B}{2}\,cot\,\frac{C}{2}\]
\[cot\space A\space cot\space B\space+\space cot\space B\space cot\space C\space+\space cot\space C\space cot\space A\space=\,1\]
Trigonometric Identities involving three different angles
\[1) \space sin\space (A+B+C) \space = sinAcosBcosC \space + \space sinBcosAcosC \space + \space sinCcosAcosB \space - \space sinAsinBsinC \]\[=cosAcosBcosC \space [tanA +tanB+tanc-tanAtanBtanc]\]
\[2) \space cos\space (A+B+C) \space = cosAcosBcosC \space - \space sinAsinBcosC \space \space sinAcosBsinC \space - \space cosAsinBsinC \]\[=cosAcosBcosC \space [1-tanAtanB - tanBtanC-tanCtanA]\]
\[3) \space tan\space (A+B+C) \space = \frac{tanA \space + \space tanB \space + \space tanC \space - \space tanAtanBtanC}{1-tanAtanB-tanBtanC-tanCtanA}\]
Sine and Cosine Series
\[sin\alpha + sin(\alpha + \beta) + sin(\alpha +2\beta) +...+sin(\alpha+ \overline{n-1}\beta)\]\[=\frac{sin( \alpha+(\frac{n-1}{2})\beta) sin(\frac{n\beta}{2})}{sin(\frac{\beta}{2})}\]
\[cos\alpha + cos(\alpha + \beta) + cos(\alpha +2\beta) +...+cos(\alpha+ \overline{n-1}\beta)\]\[=\frac{cos( \alpha+(\frac{n-1}{2})\beta) sin(\frac{n\beta}{2})}{sin(\frac{\beta}{2})}\]
IMPORTANT RESULTS
\[1) \space sin\theta \space sin(60^{\circ}-\theta) \space sin(60^{\circ}+\theta) \space = \space \frac{1}{4}sin3\theta\]
\[2) \space cos\theta \space cos(60^{\circ}-\theta) \space cos(60^{\circ}+\theta) \space = \space \frac{1}{4}cos3\theta\]
\[3) \space tan\theta \space tan(60^{\circ}-\theta) \space tan(60^{\circ}+\theta) \space = \space tan3\theta\]
\[4) \space cot\theta \space cot(60^{\circ}-\theta) \space cot(60^{\circ}+\theta) \space = \space cot3\theta\]
\[5) \space (i) sin^2\theta \space + \space sin^2(60^{\circ}+\theta) \space + \space sin^2(60^{\circ}-\theta) \space = \frac{3}{2}\]
\[\space \space \space \space (ii) cos^2\theta \space + \space cos^2(60^{\circ}+\theta) \space + \space cos^2(60^{\circ}-\theta) \space = \frac{3}{2}\]
\[6) \space If \space tanA+tanB+tanC \space = \space tanAtanBtanC,\]\[then \space A+B+C \space = \space n\pi, \space n \in I\]
\[7) \space If \space tanAtanB+tanBtanC+tanCtanA \space = \space 1,\]\[\space \space then \space A+B+C \space = \space (2n+1)\frac{\pi}{2}, \space n \in I\]
\[8) \space cos\theta \space cos2\theta \space cos4\theta ..... cos(2^{n-1}\theta) \space = \space \frac{sin(2^n\theta)}{2^nsin\theta}\]
\[9) \space cotA \space - \space tanA \space = \space 2cot2A\]
Inverse Hyperbolic Trigonometric Identities
The hyperbolic sine function is a one-to-one function and thus has an inverse. As usual, the graph of the inverse hyperbolic sine function sinh-1(x) also denoted by arcsinh(x) by reflecting the graph of sinh(x) about the line y=x For all inverse hyperbolic functions but the inverse hyperbolic cotangent and the inverse hyperbolic cosecant, the domain of the real function is connected.
\[sinh^{-1}(x)\space =\space ln(x+\sqrt{x^{2}+1})\]
\[cosh^{-1}(x)\space =\space ln(x+\sqrt{x^{2}-1})\]
\[tanh^{-1}(x)\space =\space \frac{1}{2}\space ln(\frac{1+x}{1-x})\]
\[coth^{-1}(x)\space =\space \frac{1}{2}\space ln(\frac{x+1}{x-1})\]
\[sech^{-1}(x)\space =\space ln(\frac{1}{x}+\sqrt{\frac{1}{x^{2}}-1})\]
\[cosech^{-1}(x)\space =\space ln(\frac{1}{x}+\sqrt{\frac{1}{x^{2}}+1})\]
Differentiation Formulaes
\[\frac{d}{dx}sinh^{-1}(x)\space =\space \frac{1}{\sqrt{x^{2}+1}}\]
\[\frac{d}{dx}cosh^{-1}(x)\space =\space \frac{1}{\sqrt{x^{2}-1}}\]
\[\frac{d}{dx}tanh^{-1}(x)\space =\space \frac{1}{1+x^{2}}\]
\[\frac{d}{dx}coth^{-1}(x)\space =\space \frac{1}{1-x^{2}}\]
\[\frac{d}{dx}sech^{-1}(x)\space =\space \frac{1}{x\sqrt{1-x^{2}}}\]
\[\frac{d}{dx}cosech^{-1}(x)\space =\space \frac{1}{x\sqrt{1+x^{2}}}\]
Unit Circle
A unit circle is a circle with a radius of 1 (unit radius). This tool will help determine the coordinates of any point on the unit circle just by entering the angle.
Hyperbolic Trigonometric Identities
The hyperbolic functions are analogs of the circular function or the trigonometric functions. The hyperbolic function occurs in the solutions of linear differential equations, calculation of distance and angles in the hyperbolic geometry, Laplace’s equations in the cartesian coordinates. Generally, the hyperbolic function takes place in the real argument called the hyperbolic angle. The basic hyperbolic functions are:
- 1 Hyperbolic Sine(sinh)
- 2 Hyperbolic Cosine(cosh)
- 3 Hyperbolic Tangent(tanh)
From these three basic functions, the other functions such as hyperbolic cosecant (cosech), hyperbolic secant(sech) and hyperbolic cotangent (coth) functions are derived.
\[sinh(x)\space =\space \frac{e^{x}-e^{-x}}{2}\]
\[cosh(x)\space =\space \frac{e^{x}+e^{-x}}{2}\]
\[tanh(x)\space =\space \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\]
Properties of Hyperbolic Functions
\[sinh(-x)\space =\space -sinh(x)\]
\[cosh(-x)\space =\space cosh(x)\]
\[sinh(2x)\space =\space 2sinh(x)cosh(x)\]
\[cosh(2x)\space =\space cosh^{2}x + sinh^{2}x\]
\[sinh(x)\space =\space -isinh(ix)\]
\[cosh(2x)\space –\space sinh(2x)\space =\space 1\]
\[tanh(2x)\space +\space sech(2x)\space =\space 1\]
\[coth(2x)\space –\space cosech(2x)\space =\space 1\]
Inverse Trigonometric Identities
Negative Formulaes
\[ sin^{-1}(-x) = -sin^{-1}(x)\]
\[ cos^{-1}(-x) = π-cos^{-1}(x)\]
\[ tan^{-1}(-x) = -tan^{-1}(x)\]
\[ cot^{-1}(-x) = π-cot^{-1}(x)\]
\[ sec^{-1}(-x) = π-sec^{-1}(x)\]
\[ cosec^{-1}(-x) = -cosec^{-1}(x)\]
Addition of two functions
\[sin^{-1}x + cos^{-1}x = π/2 \space,\space x ∈ [-1, 1] \]
\[tan^{-1}x + cot^{-1}x = π/2 , x ∈ R\]
\[cosec^{-1}x + sec^{-1}x = π/2 \]
Reciprocal Formulaes
\[sin^{-1}\frac{1}{x} = cosec^{-1}x \]
\[cos^{-1}\frac{1}{x} = sec^{-1}x \]
\[tan^{-1}\frac{1}{x} = cot^{-1}x, x > 0\]
Identity Formulaes
\[sin(sin^{-1}x) = x, -1≤ x ≤1\]
\[cos(cos^{-1}x) = x, -1≤ x ≤1\]
\[tan(tan^{-1}x) = x, – ∞ < x < ∞ \]
\[cot(cot^{-1}x) = x, – ∞ < x < ∞ \]
\[sec(sec^{-1}x) = x, - ∞ < x ≤ 1\space or\space 1 ≤ x < ∞\]
\[cosec(cosec^{-1}x) = x, - ∞ < x ≤ 1\space or\space 1 ≤ x < ∞\]
Addition of Same functions
\[sin^{-1}x+sin^{-1}y = sin^{-1}(x\sqrt{1-y^2} + y\sqrt{1-x^2})\]
\[sin^{-1}x-sin^{-1}y = sin^{-1}(x\sqrt{1-y^2} - y\sqrt{1-x^2})\]
\[cos^{-1}x+cos{-1}y = cos^{-1}(xy-\sqrt{(1-x^2)(1-y^2)}\]
\[cos^{-1}x-cos{-1}y = cos^{-1}(xy+\sqrt{(1-x^2)(1-y^2)}\]
\[tan^{-1}x+tan^{-1}y = tan^{-1}\frac{x+y}{1-xy}\]
\[tan^{-1}x-tan^{-1}y = tan^{-1}\frac{x-y}{1+xy}\]
Simplified Inverse Trigonometric Functions
| \[Equation\] | \[Conditions\] | \[Graph\] |
|---|---|---|
| \[y=sin^{-1}(\frac{2x}{1+x^2})\] | \[\space \space \space \space \space \space \space \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space |x| \leqslant 1\] \[ \pi \space - \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x>1\]\[-(\pi + 2tan^{-1}x) \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space \space x<-1\] | |
| \[y=cos^{-1}(\frac{1-x^2}{1+x^2})\] | \[2tan^{-1}x \space \space \space \space \space \space \space \space \space if \space \space \space \space \space x \geqslant 0\] \[ -2tan^{-1}x \space \space \space \space \space \space if \space \space \space \space \space x<0\] | |
| \[y=tan^{-1}(\frac{2x}{1-x^2})\] | \[\space \space \space \space \space \space \space \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space |x|<1\] \[ \pi \space + \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x<-1\]\[-(\pi - 2tan^{-1}x) \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x>1\] | |
| \[y=sin^{-1}(3x-4x^3)\] | \[-(\pi+3sin^{-1}x) \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x \leqslant -\frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 3sin^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space - \frac{1}{2} \leqslant x \leqslant \frac{1}{2}\]\[\pi - 3sin^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \space \frac{1}{2} \leqslant x \leqslant 1\] | |
| \[y=cos^{-1}(4x^3-3x)\] | \[3cos^{-1}x-2\pi \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x \leqslant -\frac{1}{2}\]\[2\pi - 3cos^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space -\frac{1}{2} \leqslant x \leqslant \frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 3cos^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \frac{1}{2} \leqslant x \leqslant 1\] | |
| \[y=sin^{-1}(2x \sqrt{1-x^2})\] | \[-(\pi+2sin^{-1}x) \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x\leqslant -\frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 2sin^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space -\frac{1}{\sqrt{2}} \leqslant x \leqslant \frac{1}{\sqrt{2}}\]\[\pi - 2sin^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \frac{1}{\sqrt{2}} \leqslant x \leqslant 1\] | |
| \[y=cos^{-1}(2x^2-1)\] | \[\space \space \space \space 2cos^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \space 0 \leqslant x \leqslant 1\]\[2\pi-2cos^{-1}x \space \space \space \space \space \space if \space \space \space \space\space -1 \leqslant x \leqslant 0\] |
Domain,Range and Graph of Inverse Functions
| \[Function\] | \[Domain\] |
\[Range \,of\, an\]
\[ Inverse\, Function\] |
\[Graphs\] |
|---|---|---|---|
|
\[sin^{-1}x\] \[(arcsine\,x)\] |
\[-1≤ x ≤1\] | \[-\frac{π}{2}≤y≤ \frac{π}{2}\] | |
|
\[cos^{-1}x\] \[(arccosine\,x)\] |
\[-1≤ x ≤1\] | \[0≤y ≤π\] | |
|
\[tan^{-1}x\] \[(arctangent\,x)\] |
\[– ∞ < x < ∞\] | \[-\frac{π}{2}<y<\frac{π}{2}\] | |
|
\[cot^{-1}x\] \[(arccotangent\,x)\] |
\[– ∞ < x < ∞\] | \[0<y<π\] | |
|
\[sec^{-1}x\] \[(arcsecant\,x)\] |
\[– ∞ ≤ x ≤-1 \]
\[or\] \[ 1≤x≤ ∞\] |
\[0≤y≤\pi,\] \[y \neq \frac{\pi}{2}\]
|
|
|
\[cosec^{-1}x\] \[(arccosecant\,x)\] |
\[– ∞ ≤ x ≤-1\, or\, 1≤x≤ ∞\] | \[-\frac{π}{2}≤y≤\frac{π}{2},y \neq 0\] |
The Ackermann function is multi-variable function from the natural numbers to the natural numbers with very fast rate of growth.
Domain,Range,Period and Graph of Trigonometric Functions
| \[Function\] | \[Domain\] |
\[Range \,and\,Period\,of\, a\]
\[ Trigonometric\, Function\] |
\[Graphs\] |
|---|---|---|---|
|
\[sin\,x\] |
\[R\] |
\[-1≤ y ≤1\] \[2π\] |
|
|
\[cos\,x\] |
\[R\] |
\[-1≤ y ≤1\] \[2π\] |
|
|
\[tan\,x\] |
\[R\,-\,(\,2n+1\,)\,\frac{π}{2}\,,\,n\,\in\,I\] |
\[– ∞ < y < ∞\] \[π\] |
|
|
\[cot\,x\] |
\[R\,-\,nπ\,,\,n\,\in\,I\] |
\[– ∞ < y < ∞\] \[π\] |
|
|
\[sec\,x\] |
\[R\,-\,(\,2n+1\,)\,\frac{π}{2}\,,\,n\,\in\,I\] |
\[– ∞ < y ≤-1\, or\, 1≤y < ∞\] \[2π\,,\,sec^2x\,,\,|\,secx\,|\,\in\,[\,1\,,\, ∞\,)\] |
|
|
\[cosec\,x\] |
\[R\,-\,nπ\,,\,n\,\in\,I\] |
\[– ∞ < y ≤-1\, or\, 1≤y< ∞\] \[2π\,,\,cosec^2x\,,\,|\,cosecx\,|\,\in\,[\,1\,,\, ∞\,)\] |
General Solution of Some Trigonometric Equations
\[1)\space If \space sin\theta=0, \space then \space \theta=n\pi, \space n \in I(set \space of \space integers)\]
\[2)\space If \space cos\theta=0, \space then \space \theta=(2n+1)\frac{\pi}{2}, \space n \in I\]
\[3)\space If \space tan\theta=0, \space then \space \theta=n\pi, \space n \in I\]
\[4)\space If \space sin\theta=sin\alpha, \space then \space \theta=n\pi +(-1)^n\alpha, \space where \space \alpha \in [\frac{-\pi}{2},\frac{\pi}{2}], \space n\in I\]
\[5)\space If \space cos\theta=cos\alpha, \space then \space \theta=2n\pi ± \alpha, \space n \in I \space \alpha \in [0,\pi]\]
\[6)\space If \space tan\theta=tan\alpha, \space then \space \theta=n\pi + \alpha, \space n \in I \space \alpha \in (\frac{-\pi}{2},\frac{\pi}{2})\]
\[7)\space If \space sin\theta=1, \space \theta=2n\pi+ \frac{\pi}{2}=(4n+1)\frac{\pi}{2}, \space n \in I\]
\[8)\space If \space cos\theta=1 \space then \space \theta=2n\pi, \space n \in I\]
\[9)\space If sin^2\theta \space =sin^2\alpha \space \space or \space cos^2\theta \space = cos^2\alpha \space \space or \space tan^2\theta \space = tan^2\alpha,\]\[then \space \theta=n\pi \space ± \space \alpha, \space n \in I\]
\[10)\space For \space n \in I, \space sinn\pi=0 \space and \space cosn\pi \space =(-1)^n, \space n \in I \]\[sin(n\pi + \theta) \space = (-1)^nsin\theta\]\[cos(n\pi+ \theta) \space =(-1)^ncos\theta\]
\[11)\space cos \space n\pi \space = (-1)^n, \space n \in I\]
\[12)\space If \space n \space is \space an \space odd \space integer \space then \space sin\frac{n\pi}{2}=(-1)^{\frac{n-1}{2}}, \space cos\frac{n\pi}{2}=0\]
\[13)\space sin(\frac{n\pi}{2}+\theta) \space = (-1)^{\frac{n-1}{2}} \space cos\theta, \space cos(\frac{n\pi}{2}+\theta) \space = (-1)^{\frac{n+1}{2}}sin\theta\]
Negative Binomial Distribution Calculator
The negative binomial distribution calculator, otherwise called the Pascal distribution calculator, helps in determining what is the probability of requiring n trials to achieve a fixed number of successes.
Formula: P(Y=n) = (n-1)C(r-1) * pr * (1-p)(n-r)
where,
n is the total number of trials
r is the number of successes
p is the probability of one success
(n-1)C(r-1) is the number of combinations
P(Y=n) is the probability of the exact number of trials n needed to achieve r successes.
PPV Calculator
NPV Calculator
Relative Risk Calculator
Relative risk compares the probability of a disease being developed in a group exposed to a particular factor to the probability of the same disease being developed in a control group
where,
a is the number of members of the exposed group who developed the disease;
b is the number of members of the exposed group who didn't develop the disease;
c is the number of members of the control group who developed the disease;
d is the number of members of the control group who didn't develop the disease;
Expansion of Trigonometric Functions
\[sine\space =\space x\space - \frac{x^3}{3!}\space + \frac{x^5}{5!} - \frac{x^7}{7!}.....\]
\[cosine\space =\space 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}.....\]
\[tan\space =\space x+\frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315}.....\]
\[cot\space = \space x^{-1} - \frac{1}{3x} -\frac{1}{45}x^3 - \frac{2}{945}x^5.....\]
\[sec\space = \space 1 + \frac{1}{2x^2} + \frac{5}{24}x^4 + \frac{61}{720}x^6......\]
\[cosec\space =\space x^{-1}+\frac{1}{6x+7}360x^3 + \frac{31}{15120}x^5.....\]
Expansion of Inverse Trigonometric Functions
\[sin^{-1}\space =\space x + \frac{1}{6}x^3 +\frac{3}{40}x^5 +\frac{5}{112}x^7.....\]
\[cos^{-1}\space =\space \frac{π}{2} - (expansion\space of\space sine)\]
\[tan^{-1}\space =\space x - \frac{1}{3}x^3 + \frac{1}{5}x^5.....\]
\[cot^{-1}\space =\space \frac{π}{2} - (expansion\space of\space tan)\]
The rest can be found out using Identities
Solution of Triangles
Law of Sines Calculator
Law of Cosines Calculator
\[1) \space Sine\space rule : \frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}\]
\[2) \space Cosine\space rule : cosA = \frac{(b^2+c^2-a^2)}{2bc}\]
\[3) \space Projection\space Formula : a = b cos C + c cos B \]
\[4) \space Napier's\space Analogy : tan\frac{B-C}{2} = (\frac{b-c}{b+c}) cot\frac{A}{2}\]
\[5) \space Half\space Angle\space Formula: sin\frac{A}{2} = \sqrt\frac{(s-b)(s-c)}{bc}\]
\[cos\frac{A}{2} = \sqrt\frac{(s)(s-a)}{bc}\]
\[tan\frac{A}{2} = \sqrt\frac{(s-b)(s-c)}{(s)(s-a)}\]
\[6) \space Area\space of\space \triangle : \frac{1}{2}absinC = \frac{1}{2}bcsinA = \frac{1}{2}acsinB \]
\[8) \space m-n \space Theorem \space : \space If\space D\space be\space the\space point\space on\space the\space side\space BC\space of a triangle\space ABC\space which\space divides\space the\space side\space BC\space in\space the\space ratio\space m: n, \\If\space BD:DC = m:n, then\\ (m+n) cot θ = m cot α- n cot β = n cot B – m cot C\]
\[9) \space Radius\space of\space CircumCircle : R = \frac{a}{2sinA} = \frac{b}{2sinB} = \frac{c}{sinC} = \frac{abc}{4\triangle}\]
\[10) \space Radius\space of\space InCircle : r = \frac{\triangle}{s} \space = (s-a)tan\frac{A}{2} \space = 4R sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}\]
\[11) \space Radius\space of\space ExCircle : r_1 = \frac{\triangle}{s-a} \space = (s)tan\frac{A}{2} \space = 4R sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}\]
\[12) \space Orthocentre\space and\space Pedal\space triangle : \]
\[ The\space triangle\space formed\space by\space joining\space the\space feet\space of\space the\space altitudes\space is\space called\space the\space Pedal\space Triangle.\] \[a) \space The \space sides\space are\space a cos A = R sin 2A , \space a cos B = R sin 2B \space a cos C = R sin 2C\]
\[b) \space The \space distances \space of \space the \space orthocenter \space from \space the \space angular \space points \space of \space the \triangle ABC \space are \space \]\[2RcosA, \space 2RcosB, \space and \space 2RcosC.\]
\[c) \space The \space distance \space of \space orthocenter \space from \space sides \space are\]\[ 2RcosB \space cosC, \space 2RcosCcosA \space and 2RcosCcosA.\]
\[d) \space Area \space of \space Pedal \space Triangle \space = \space 2 \triangle cosAcosBcosC \space = \frac{1}{2}R^2sin2Asin2Bsin2C \]
\[e) \space Circumradii \space of \space pedal \space triangle \space = \space \frac{R}{2}\]
\[13) \space Important Results\]
\[a)\space Circumcentre (O) : O_A\space =\space R\space\space and\space\space O_A\space =\space R cos A\]
\[b)\space Orthocentre: H_A = 2R cos A and H_A = 2R cos B cos C\]
\[c)\space Excentre (I_1): I_1A\space =\space r_1\space cosec\frac{A}{2}\space=\,4Rcos \frac{A}{2}\]
\[d)\space Centroid(G) :G_A =\sqrt[3]{2b^2+2c^2-a^2}\]
\[e)\space Incentre(I) :I =r\,cosec\frac{A}{2}\]
\[f)\space Length\,of\,Angle\,Bisector\,from\,angle\,A\,=\frac{2bc\,cos\frac{A}{2}}{b+c}\]
\[g)\space Length\, of\, Median \,from \,angle \,A\, =\frac{1}{2}\sqrt{2b^2+2c^2-a^2}\]
\[h)\space Length \,of\, Altitude\, from \,Angle\, A\,=\,\frac{2\triangle}{a}\,=c\,sinB\]
\[i)\space Relation\, between\, area \,of\, Inscribed(A)\, and \,Excribed(A_1,A_2,A_3)\, Circle\, of \,\triangle ABC:\]
\[\frac{1}{\sqrt{A}}\,=\,\frac{1}{\sqrt{A_1}}\,+\,\frac{1}{\sqrt{A_2}}\,+\,\frac{1}{\sqrt{A_3}}\]
\[j)\space Relation \,between\, median \,and \,sides \,of\, \triangle ABC:\]
\[(Length\, of\, median\, from\, angle\, A)^2\,+\,(Length\, of\, median\, from\, angle\, B)^2\,+\]
\[(Length\, of\, median\, from\, angle\, C)^2\,=\,\frac{3}{4}(a^2+b^2+c^2)\]
\[k) \space Relation\,between\, angle\, of \triangle ABC,\, Inradius,\,Circumradius:\]
\[ cosA\,+\,cosB\,+\,cosC\,=\,1+\frac{r}{R}\]
\[l)\space Incentre \,and \,Centroid \,always\,lie\,inside\, the\, \triangle\]
\[m)\space Circumcentre\, lies\, inside\, acute\, angled\, \triangle,\,outside\, obtuse\, angled\, \triangle\]
\[and\, on \,hypotenuse\, for\, right\, angled\, \triangle\]
\[n)\space Orthocentre\, lies\, inside\, acute\, angled\, \triangle,\,outside\, obtuse\, angled\, \triangle\]
\[and\, at \,B\, for\, right\, angled\, \triangle ABC \, right \,angled \,at \,B\]
Sum And Difference Of Trignometric Function
Here you can calculate Sum and Difference of Trignometric Functions using sin, cos and tan.
Circum Area and External Radius
Area of External Triangle
\[8R^2cosA/2cosB/2cosC/2\]
Circum radius of External Triangle
\[2R\]
Lancent Arc Calculator
\[h = √ r * b - b² / 4\]
\[b = 2 (r - √ r² - h² )\]
\[r = ( 4h² + b² ) / (4b)\]
\[p = 2r * arccos [ 1 - b / ( 2 * r ) ] + b\]
\[A = r * ( u - b ) / 2 - h * ( r - b/2 )\]
Distance between Incenter and Excenter of a Triangle
\[4RsinA/2\]
Distance between excentre and circumcentre
For OI1
\[R√1+(8*sin(A/2)*sin(B/2)*sin(C/2))\]
For OI2 and OI3
\[R√1+(8*cos(A/2)*cos(B/2)*cos(C/2))\]
Sensitivity and Specificity
Sensitivity
Sensitivity is the ability of a test to correctly classify an individual as ′diseased′
Sensitivity = a / a+c = a (true positive) / a+c (true positive + false negative) = Probability of being test positive when disease present.
Specificity
The ability of a test to correctly classify an individual as disease- free is called the test′s specificity.
Specificity = d / b+d = d (true negative) / b+d (true negative + false positive) = Probability of being test negative when disease absent.
STANDARD DEVIATION
What is Standard Deviation
Standard deviation is the measure of dispersion of a set of data from its mean. It measures the absolute variability of a distribution; the higher the dispersion or variability, the greater is the standard deviation and greater will be the magnitude of the deviation of the value from their mean.
- Select Population if data contains all the measurable values or all of the values you are interested in.
- Select Sample if the data is sample of large or unlimited population an you wish to make a statement about entire population.
Algebraic Equations Formulas
\[1)\space a^2-b^2\space =(a-b)(a+b)\]
\[2)\space (a+b)^2\space =a^2+2ab+b^2\]
\[3)\space a^2+b^2\space =(a+b)^2-2ab\]
\[4)\space (a-b)^2\space =a^2-2ab+b^2\]
\[5)\space (a+b+c)^2\space =a^2+b^2+c^2+2ab+2bc+2ca\]
\[6)\space (a-b-c)^2\space =a^2+b^2+c^2-2ab+2bc-2ca\]
\[7)\space (a+b)^3\space =a^3+b^3+3a^2b+3ab^2\]
\[8)\space (a-b)^3\space =a^3-b^3-3a^2b+3ab^2\]
\[9)\space a^3-b^3\space =(a-b)(a^2+ab+b^2)\]
\[10)\space a^3+b^3\space =(a+b)(a^2-ab+b^2)\]
\[11)\space a^3+b^3+c^3-3abc\space=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\]
\[if\space a+b+c=0 \space then \space a^3+b^3+c^3\space=\space3abc\]
\[12)\space (a+b)^4\space =a^4+4a^3b+6a^2b^2+4ab^3+b^4\]
\[13)\space (a-b)^4\space =a^4-4a^3b+6a^2b^2-4ab^3+b^4\]
\[14)\space a^4-b^4\space =(a-b)(a+b)(a^2+b^2)\]
\[15)\space a^5-b^5\space =(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)\]
\[16)\space (a+b+c+...)^2\space =a^2+b^2+c^2+....+2(ab+bc+ca+...)\]
\[17)\space Considering\space n\space as\space a\space natural\space number:\] \[a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b+…+ b^{n-2}a + b^{n-1})\]
\[18)\space Considering\space n\space as\space an\space even\space number:\] \[a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b+…+ b^{n-2}a - b^{n-1})\]
\[18)\space Considering\space n\space as\space an\space odd\space number:\] \[a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b+ a^{n-3}b^2…- b^{n-2}a + b^{n-1})\]
\[19)\space Exponential\space laws:\] \[(a^m)(a^n) = a^{m+n}\] \[(ab)^m = a^mb^m \] \[(a^m)^n = a^{mn} \] \[{a^m\over a^n} = a^{m-n} \] \[a^m = {1\over a^{-m}} \] \[a^{-m} = {1\over a^m} \]
Location of Roots
\[Let\space f(x) = ax^2+bx+c = 0 where\space a>0\space then,\]
1.If both the roots are positive i.e. they lie in (0, ¥), then the sum of the roots as well as the product of the roots must be positive.
\[ ie. \space a+b = \frac{-b}{a} > 0 \space and\space ab=\frac{c}{a}>0\]
2.Similarly, if both the roots are negative i.e. they lie in (–¥, 0) then the sum of the roots will be negative and the product of the roots must be positive.
\[ie.\space a+b = \frac{-b}{a} < 0 \space and\space ab>0\]
3.Both the roots are greater than a given number k if the following three conditions are satisfied,
\[D>=0\space \frac{-b}{2a} > k\space and\space a.f(k)>0\]
4.Both the roots will lie in the given interval (k1, k2) if the following conditions are satisfied
\[D>=0\space k_1 < \frac{-b}{2a} < k_2\space and\space a.f(k_1)\space and\space a.f(k_2) are>0\]
5.Exactly one of the roots lies in the given interval (k1, k2) if,
\[f(k_1).f(k_2)\space >\space 0\]
6.A given number k will lie between the roots if a.f(k) < 0.
In particular, the roots of the equation will be of opposite signs if 0 lies between the roots a.f(0) < 0.
\[f(x)=ax^2+bx+c\]
\[a=\]
\[b=\]
\[c=\]
Nature of the roots of the Quadratic Equation ax2 + bx + c, a ≠ 0, a, b, c ∈ R
To find out the nature of the roots of a quadratic equation,
- We find the value of the discriminant D. If D < 0, we immediately say that equation has no solution in R.
- If D ≥ 0, then we examine whether it is a perfect
square of a rational number or not.
- If D > 0 and D is not a perfect square, then roots are real and distinct.
- If D > 0 and it is a perfect square of a rational number and a, b, c ∈ Q, then the roots are real and rational. Roots are also distinct.
- If D = 0 then the roots are real and equal and if a, b, c ∈ Q the roots are equal rational numbers.
Pythagorean Triplets
The integer solutions to the Pythagorean Theorem, a^2 + b^2 = c^2 are called Pythagorean Triples which contains three positive integers a, b, and c.
Enter exactly one number
Pythagorean Triplets Checker
a :
b :
c :
Greatest Integer Function and Fractional Part
Quartiles
The quartile measures the
spread of values above and below the mean by dividing the distribution into four groups.
A quartile divides data into three points a lower quartile, median, and upper quartile to form four
groups of the dataset. Where N is number of dataset points
Q1 = (N+1)(1/4)
Q2 = (N+1)(2/4)
Q3 = (N+1)(3/4)
Theory of Quartile
In statistics, a quartile is a type of quantile which divides the number of data points into four parts, or quarters, of more-or-less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are a form of order statistic. The three main quartiles are as follows:
The first quartile (Q1) is defined as the middle number between the smallest number (minimum) and the median of the data set. It is also known as the lower or 25th empirical quartile, as 25% of the data is below this point.
The second quartile (Q2) is the median of a data set; thus 50% of the data lies below this point.
The third quartile (Q3) is the middle value between the median and the highest value (maximum) of the data set. It is known as the upper or 75th empirical quartile, as 75% of the data lies below this point.[
Applications of Quartiles
Some examples of the use of range in the real world are: temperature ranges for the day as reported on a weather report, min/max levels of water in a reservoir. However, in the presence of outliers, the range values can be significantly influenced by them. Hence, we prefer to use the IQR instead as we “ignore” the bottom 25% of the data points and the top 25% of the data points. IQR statistic is more robust with respect to outliers. In the presence of outliers, IQR is a better representation of the amount of spread in the data rather than the range. Some companies use the quartiles to benchmark other companies. For example, the median company pay for a given position is set at the first quartile of the top 20 companies in that region. The quartiles and IQR information is typically used when you create a box-plot of your data set.
Deciles
In descriptive statistics, the term “decile” refers
to the nine values that split the population data into ten equal fragments such that each fragment
is representative of 1/10th of the population.
In other words, each successive decile corresponds to an increase of 10% points such that the
1st decile or D1 has 10% of the observations below it, then 2nd
decile or D2 has 20% of the observations below it, and so on so forth. N is the number of
data points in list.
Di = i * (n + 1) / 10 th data
Correlaton Coefficient Calculator
Correlation coefficient is used to find how strong a relationship is between data. The formulas return a value between -1 and 1, where:
Pearson Formula for Correlation Coefficient:
\[r_{x,y}=\frac{\sum_{i=1}^{n}(X_{i}-M_{x})(Y_{i}-M_{y})}{\sqrt{\sum_{i=1}^{n}(X_{i}-M_{y})^2 \sum_{i=1}^{n}(Y_{i}-M_{y})^2}}\]
R-Squared Calculator
R Squared Calculator is used for data analysis programmed to predict the future outcome with respect
to the proportion of variability in the other data set.
The coefficient of equation R2 as an overall summary of the effectiveness of a least
squares equation.
T - Test
The t-test is any statistical hypothesis test in which the test statistic follows a Student’s t-distribution under the null hypothesis. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. Also number of values in each set should not exceed 30. T-test uses means and standard deviations of two samples to make a comparison.
Formula for t-value:
Regression Coefficient Calculator
The Regression Coefficient is the constant ‘b’ in the regression equation that tells about the change in the value of dependent variable corresponding to the unit change in the independent variable.
AB - Test
An AB test is a statistical significance test in which two populations are compared to see if they differ significantly on a single characteristic. A sample size of 30 or more is recommended.
Relative Risk Calculator
Relative risk compares the probability of a disease being developed in a group exposed to a particular factor to the probability of the same disease being developed in a control group
Formula: [a / (a + b)] / [c / (c + d)]
where,
a is the number of members of the exposed group who developed the disease;
b is the number of members of the exposed group who didn't develop the disease;
c is the number of members of the control group who developed the disease;
d is the number of members of the control group who didn't develop the disease;
F-Test
A Statistical F Test uses an F Statistic to compare two variances, s1 and s2, by dividing them.
The result is always a positive number (because variances are always positive).
The equation for comparing two variances with the f-test is:
F = s1^2 / s2^2 ( if s1^2> s2^2).
Any statistical test that uses F-distribution can be called an F-test. It is used when the sample
size is small i.e. n < 30.
Formula for f-value:
(Larger Sample Variance)/(Smaller Sample Variance)
Z - Test
What is Z-Test
Z-test is a statistical test to determine whether two population means are different when the variances are known and the sample size is large. Z-test is a hypothesis test in which the z-statistic follows a normal distribution. A z-statistic, or z-score, is a number representing the result from the z-test.
Ho : μ = μo (Null Hypothesis)
Ha : μ ≠ μo (Alternative Hypothesis)
C chart Calculator
A c-chart is an attributes control chart used with data collected in subgroups that are the same size. C-charts show how the process, measured by the number of nonconformities per item or group of items, changes over time. Nonconformities are defects or occurrences found in the sampled subgroup.
p chart Calculator
A p-chart is an attributes control chart used with data collected in subgroups of varying sizes. Because the subgroup size can vary, it shows a proportion on nonconforming items rather than the actual count. Control limits of p-chart determines whether the process is in control or not.
R chart Calculator
R chart is a type of scheme, popularly known as control chart, used to monitor range of a normally distributed variables simultaneously, when samples are collected at regular intervals from a business or industrial process.
Sheppard's correction for moments Calculator
Sheppard’s Correction for Grouping Errors are adjustments to calculated sample moments for grouped data (i.e. data that has been binned). The correction is named after W.F. Shepard, who noted than when moments are calculated for continuous frequency distributions, it is assumed that the data is centered around the class interval midpoints. This erroneous assumption introduces “grouping errors” in calculation of moments. Grouping errors only happen in even moments (e.g. second and fourth moments), so the correction isn’t needed for odd-numbered moments.
np chart Calculator
NP charts are used to monitor the number of nonconforming units of a process based on samples taken from the process at given times (hours, shifts, days, weeks, months, etc.) .We plot a graph of No defective items vs Sample No. If all the points of the graph lie within the control limits, then the process is under statistial control.
X̄ chart Calculator
The X-bar chart is a type of Shewhart control chart that is used to monitor the arithmetic means of successive samples of constant size, n. This type of control chart is used for characteristics that can be measured on a continuous scale, such as weight, temperature, thickness etc.
Confidence Interval
What is the confidence interval?
In statistics, a confidence interval is a range of values that is determined through use of observed data, calculated at a desired confidence level, that may contain the true value of the parameter being studied. The confidence level, for example, a 95% confidence level, relates to how reliable the estimation procedure is, not the degree of certainty that the computed confidence interval contains the true value of the parameter being studied. The desired confidence level is chosen prior to the computation of the confidence interval and indicates the proportion of confidence intervals, that when constructed given the chosen confidence level over an infinite number of independent trials, will contain the true value of the parameter.
Confidence intervals are typically written as (some value) ± (a range). The range can be written as
an actual value or a percentage. It can also be written as simply the range of values. For example,
the following are all equivalent confidence intervals:
20.6 ±0.887
or
20.6 ±4.3%
or
[19.713 – 21.487]
\[ CI = X̄ ± Z * \frac{s}{\sqrt{n}}\]
Odds Calculator
Chi-Square Test
(Single Row Data)
A chi-square (χ2) statistic is a test that measures how a model compares to actual observed data. The data used in calculating a chi-square statistic must be random, raw, mutually exclusive, drawn from independent variables, and drawn from a large enough sample.
Lead Time Calculator
Lead Time is the delay that occurs between the initiation and completion of a process. Lead Time is used in manufacturing, supply chain management, and project management.
What is the unknown value:
Takt Time Calculator
Takt time is the rate at which you need to complete a product to meet customer demand. Takt time is your sell rate and can easily be categorized as the heartbeat of your work process. It allows you to optimize your capacity in the most appropriate way to meet demand without keeping too much inventory in reserve.
Throughput Time Calculator
Throughput time or cycle time is the amount of time it takes for a manufacturer to make a product including process time, inspection time, move time, and wait time.Notice the process in the throughput time calculation that is considered value-added time is the process time. This is the only step that actually produces a product. The rest of the time is administrative and logistical non-value-added time.
Cycle Time Calculator
Cycle Time is the time it takes you to complete the production process of one product, from start to finish.It’s one of the most crucial flow metrics that’ll help you run a project smoothly efficiently.
Direct And Indirect Proportion
Direct Proportion Calculator
X1/Y1 = X2/Fourth Value
Indirect Proportion Calculator
X1.Y1 = X2.(Fourth Value)
Cross Multiplication Calculator
\[\frac{A}{B}=\frac{C}{D}\]
Enter any 3 fields
Diamond Problem Calculator
Enter any 2 fields
Work And Time Calculator
Work, Time, Men
Work = Number of days (Time)(T or D) X Number of men(M)
Derived Formula from above equation: W1/W2 = (T1 X N1) / (T2 X N2)
What is the unknown value:
Profit Loss calculations
Profit/Loss calculations over Discount
From
To
Simple and Compund Interest
Midrange Calculator
To calculate Height when Angle of Elevation and Distance is given
To calculate Height when Angle of Depression and Distance is given
To calculate Distance when Angle of Elevation and Height is given
To calculate Distance when Angle of Depression and Height is given
To calculate Angle of Elevation
To calculate Angle of Depression
Quadratic equation whose roots are K times the roots of given equation
Quadratic equation whose roots are reciprocal to the roots of given equation
Generate Quadratic Equation having given sum and product of roots
Common Roots
Form the Cubic equation from the given roots
Maximum and Minimum value of a quadratic function
Check whether one root of the Quadratic Equation is twice of other or not
Intercept form of Plane
\[ax+by+cz+d=0\]
PLOT BAR GRAPH
Instructions :
Seperate Each data with ',' for both the axis
- Example :
for Y axis enter ----> a,b,c,d :
for Y axis enter ----> 10,5,6,4 :
Properties of Parallel Lines
What are parallel lines?
Parallel lines are equidistant lines (lines having equal distance from each other) that will never meet.
Conditions for Lines to be parallel
- 1. the pair of alternate angles is equal, then two straight lines are parallel to each other.
- 2. the pair of interior angles are on the same side of traversals is supplementary, then the two straight lines are parallel.
- 3. the pair of corresponding angles is equal, then the two straight lines are parallel to each other.
- 4. Two lines cut by a transversal line are parallel when the sum of the consecutive exterior angles is 180 °
- 5. The sum of angles together on a line is 180 °
Coordinate Systems
1. Cartesian Coordinate System
It uses the concept of mutually perpendicular lines to denote the coordinate of a point. To locate the position of a point in a plane using two perpendicular lines, we use the cartesian coordinate system. Points are represented in the form of coordinates (x, y) in two-dimension with respect to x- and y- axes. The x-coordinate of a point is its perpendicular distance from the y-axis measured along the x-axis and it is known as Abscissa. The y-coordinate of a point is its perpendicular distance from the x-axis measured along the y-axis and it is known as Ordinate.
2. Polar Coordinate System
Here, a point is chosen as the pole and a ray from this point is taken as the polar axis. Basically, we have two parameters namely angle and radius. The angle Ɵ with the polar axis has a single line through the pole measured anti-clockwise from the axis to the line.
The point will have a unique distance from the origin (r). Thus, a point in Polar coordinate system is represented as a pair of coordinates (r, Ɵ). The pole is represented by (0, Ɵ) for any value of Ɵ, where r = 0.
The distance from the pole is called the radial coordinate, radial distance or simply radius and the angular coordinate, polar angle or azimuth.
Consider the figure below that depicts the relationship between polar and
cartesian coordinates.
X = r cos Ɵ and y = r sin Ɵ
\[r =\space \sqrt{x^2 + y^2}\space and\space tan Ɵ\space =\space \frac{y}{x}\]
3. Spherical Coordinate System
The coordinate ρ is the distance from P to the origin. If the point Q is the projection of P to the xy-plane, then θ is the angle between the positive x-axis and the line segment from the origin to Q. Lastly, ϕ is the angle between the positive z-axis and the line segment from the origin to P.
x = ρsinϕcosθ
y = ρsinϕsinθ
z = ρcosϕ
4. Cylindrical Coordinate System
x=rcosθ
y=rsinθ
z=z
Its just Polar coordinates with z as height forming a cylinder
Important Graph Calculators
FInding angle between chord and a tangent
Length of the chord of the circle whose radius and the angle subtended at the center by the chord is given
Angle subtended by the chord when the angle subtended by another chord of same length is given
Angle subtended by the chord to center of the circle when the angle subtended by the another equal chord of a congruent circle is given
Angle on circumference subtended by the chord when the central angle subtended by the chord is given
Possibility to create a polygon with a given angle
Angle of rotational symmetry
Minimum side of square embedded in Regular polygon with N sides
Length of Diagonal of a n-sided regular polygon
Number of triangles formed by joining vertices of n-sided polygon with two common sides and no common sides
Number of rectangles in N*M grid
Number of occurrences of a given angle formed using 3 vertices of a n-sided regular polygon
Number of ways a convex polygon of n+2 sides can split into triangles by connecting vertices
Check if given polygon is a convex polygon or not
Find number of diagonals in n sided convex polygon
Polygon with maximum sides that can be inscribed in an N-sided regular polygon
Area of largest Circle inscribe in N-sided Regular polygon
Area of largest Triangle inscribe in N-sided Regular polygon
Area of a n-sided regular polygon with given side length
Number of triangles formed by joining vertices of n-sided polygon with one side common
Count of nested polygons that can be drawn by joining vertices internally
Regular polygons inscribed to a circle
Ratio of area of two nested polygons formed by connecting midpoints of sides of a regular N-sided polygon
Number of cycles formed by joining vertices of n sided polygon at the center
Apothem of a n-sided regular polygon
Central angle of a N sided Regular Polygon
Angle between 3 vertices in a N sided Regular Polygon
Side of a regular n-sided polygon circumscribed in a circle
Number of cycles in a Polygon with lines from Centroid to Vertices
Sum of sides of largest and smallest child polygons possible from a given polygon
Vertex diagonally opposite to the vertex M from an N-sided polygon
Ceva and Thales Theorem
Ceva's Theorem
Ceva’s theorem is a theorem regarding triangles in Euclidean Plane Geometry.
Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.
Ceva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O, to meet opposite sides at D, E and F respectively.
\[{\frac {AF}{FB}}\cdot {\frac {BD}{DC}}\cdot {\frac {CE}{EA}}=1\]
Another form of the theorem is that three concurrent lines from the polygon vertices of a triangle divide the opposite sides in such fashion that the product of three nonadjacent segments equals the product of the other three
Thales' Theorem
Basic Proportionality theorem was introduced by a famous Greek Mathematician, Thales, hence it is also called Thales Theorem. According to him, for any two equiangular triangles, the ratio of any two corresponding sides is always the same. Based on this concept, he gave theorem of basic proportionality (BPT). This concept has been introduced in similar triangles. If two triangles are similar to each other then,
1. Corresponding angles of both the triangles are equal
2. Corresponding sides of both the triangles are in proportion to each other
Thales Theorem Statement
If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.
\[\frac{AP}{PB}\space =\space \frac{AQ}{QC}\]
The MidPoint theorem is a special case of the basic proportionality theorem. According to mid-point theorem, a line drawn joining the midpoints of the two sides of a triangle is parallel to the third side.
Converse
According to this theorem, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Properties of Circles
- 1. The circles are said to be congruent if they have equal radii
- 2. The diameter of a circle is the longest chord of a circle
- 3. Equal chords subtend equal angles at the circumference
- 4. The radius drawn perpendicular to the chord bisects the chord
- 5. Circles having different radius are similar
- 6. A circle can circumscribe a rectangle, trapezium, triangle, square, kite
- 7. A circle can be inscribed inside a square, triangle and kite
- 8. The chords that are equidistant from the centre are equal in length
- 9. The distance from the centre of the circle to the longest chord (diameter) is zero
- 10. The perpendicular distance from the centre of the circle decreases when the length of the chord increases
- 11. If the tangents are drawn at the end of the diameter, they are parallel to each other
- 12. An isosceles triangle is formed when the radii joining the ends of a chord to the centre of a circle
Properties of Quadrilaterals
Different types of Quadrilaterals Are:
| Parallelogram |
| Rectangle |
| Square |
| Rhombus |
| Kite |
| Trapezium |
| Summary Of Quadrilaterals |
Faulhaber's Formula
Faulhaber formula is given by
Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described below
| \[Property\] | \[Parallelogram\] | \[Rectangle\] | \[Square\] | \[Rhombus\] | \[Kite\] | \[Trapezium\] | |
|---|---|---|---|---|---|---|---|
| Sides | All sides are equal | ❌ | ❌ | ✔ | ✔ | ❌ | ❌ |
| Opposite sides are equal | ✔ | ✔ | ✔ | ✔ | ❌ | ❌ | |
| Opposite sides are parallel | ✔ | ✔ | ✔ | ✔ | ❌ | ✔ | |
| Angles | All angles are equal | ❌ | ✔ | ✔ | ❌ | ❌ | ❌ |
| Opposite angles are equal | ✔ | ✔ | ✔ | ✔ | ❌(only 1 pair equal) | ❌ | |
| Sum of two adjacent angles is 180 | ✔ | ✔ | ✔ | ✔ | ❌ | ❌ | |
| Diagonals | Bisect each other | ✔ | ✔ | ✔ | ✔ | ❌(only 1 diagonal bisects) | ❌ |
| Bisect perpendicularly | ❌ | ❌ | ✔ | ✔ | ❌ | ❌ | |
| Intersect Perpendicularly | ❌ | ❌ | ✔ | ✔ | ✔ | ❌ | |
| Formulas | Area | l x h | l x b | a2 | 1⁄ 2 x d1 x d2 | 1⁄ 2 x d1 x d2 | 1⁄ 2 x Sum of parallel sides x h |
| Perimeter | 2 x (l + b) | 2 x (l + b) | 4a | 4a | 2 x (l + b) | Sum of length of sides |
Cartesian Coordinates to Polar Coordinate Calculator
\[r = √(x^2 + y^2)\]
\[θ = tan-1 ( y / x )\]
Pentope Calculator
Cylindrical to Cartesian Coordinate Calculator
\[x = pcosθ\]
\[y = psinθ\]
\[z = z\]
Curve Tracing
For Cartesian Curves
\[1.\space Symmetry\space - \\ If\space even\space powers\space of\space x,\space then\space symmetrical\space about\space y\space axis \\ If\space even\space powers\space of\space y,\space then\space symmetrical\space about\space x\space axis \\ If\space replacing\space y\space by\space x\space no\space change\space in\space curve,\space symmetrical\space about\space y=x\]
\[2.\space Origin\space - \\ If\space curve\space has\space no\space constant\space term,\space Equate\space to\space 0\space the\space lowest\space degree\space term\space to\space get\space equation\space of\space tangent\space at\space origin\]
\[3.\space Asymptotes\space - \\ For\space Asymptotes\space parallel\space to\space x\space axis,\space equate\space highest\space degree's\space coefficient\space of \space x\space to\space 0 \\ For\space Asymptotes\space parallel\space to\space y\space axis,\space equate\space highest\space degree's\space coefficient\space of \space y\space to\space 0 \]
\[4.\space Region\space - \\ Using\space logic\space find\space region\space in\space which\space the\space curve\space will\space lie \\ Use\space wavy\space curve\space method\space to\space find\space the\space region(best\space way)\]
\[5.\space Point\space of\space Intersection\space - \\ Put\space certain\space values\space of\space x\space to\space get\space corresponding\space values\space of\space y\space and\space plot\space them\space to\space get\space final\space answer\]
For Polar Curves
\[1.\space Symmetry\space -\\ To\space check\space for\space any\space angle\space \theta\space for\space symmetry\space check\space for\space (2\alpha - \theta),\space If\space curve\space remains\space same\space then\space symmetrical\]
\[2.\space Pole\space -\\ Put\space r=0\space to\space get\space equation\space of\space tangent\space at\space pole\]
\[3.\space Asymptotes\space -\\ Put\space r = \infty\space to\space get\space Asymptotes\]
\[4.\space Region\space -\\ Put\space extreme\space values\space of\space cosine/sine\space to\space get\space region\space in\space which\space curve\space lie\]
\[5.\space Point\space of\space Intersection\space \\ Put\space values\space of\space \theta\space to\space get\space specific\space values\space of\space r,\space then\space plot\]
Calculator for 1-D Shapes
| Line |
| Koch Curve |
| Circular Segment |
Calculator for Shapes
Congruence Properties of Triangles
1. SSS Property (Side-Side-Side) - When all the three corresponding sides of 2 triangles are of equal size then triangles are congruent
2. SAS Property (Side-Angle-Side) - If any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by SAS rule.
3. ASA Property (Angle-Side-Angle) - If any two angles and the side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule.
4. AAS Property (Angle-Angle-Side) - If any two angles and a non included side of one triangle are equivalent to the corresponding two angles and the non included side of the second triangle, then the two triangles are said to be congruent by AAS rule.
5. RHS Property (Right-angle - Hypotenuse - Side) - If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right triangles are said to be congruent by RHS rule.
Similarity of Triangles
1. AA(or AAA) Similarity - If any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other.
2. SAS Similarity - If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar.
3. SSS Similarity - If all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar.
Fundamental Theorem of Proportionality
If a line parallel to one of the sides of a triangle intersects the other two sides in distinct points, then the segments of the other two sides in one halfplane are proportional to the segments in the other halfplane.
Using the fundamental theorem on proportionality, we can obtain some results:
- If a line parallel to one
side of a triangle
intersects the other two sides of the triangle in distinct points, the segments of the
other sides of the triangle in the same halfplane of the line are proportional to the
corresponding sides of the triangle.
- If three (or more than
three) parallel lines
are intercepted by two transversals, the segments cut off on the transversals between
the same parallel lines are proportional.
- In a triangle the
bisector of an angle divides
the side opposite to the angle in the segments whose lengths are in the ratio of their
corresponding sides.
Converse of the fundamental theorem of proportionality
If a line intersects two sides of a triangle such that segments of the sides in each half-plane are proportional then the line is parallel to the third side.
3-D Shapes Inscribed
Largest cone that can be inscribed within a cube
Largest right circular cone that can be inscribed within a sphere which is inscribed within a cube
Largest cube that can be inscribed within the sphere
Largest sphere that can be inscribed within a cube which is in turn inscribed within a right circular cone
Largest Cube that can be inscribed within a Right Circular Cylinder
Largest cube that can be inscribed within the cone
Largest cylinder that can be inscribed within a cube
Largest cylinder that can be inscribed within a sphere
Largest cube that can be inscribed within a cylinder
Minimum volume of cone that can be circumscribed about a sphere of radius R
Largest sphere that can be inscribed in a right circular cylinder inscribed in a frustum
Largest right circular cylinder that can be inscribed within a cone which is in turn inscribed within a cube
Largest right circular cone that can be inscribed within a sphere
Volume of Largest right circular cylinder that can be inscribed within a sphere
Volume of Largest right pyramid that can be inscribed within a sphere
Largest sphere that can be inscribed inside a cube
Largest Square that can be inscribed within a hexagon
Largest right circular cylinder within a frustum
Surface area of the square pyramid
Find slant height of cone and pyramid
Calculator for 3-D Shapes
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Cube
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Catalan Solids
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Cuboid
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Cupola
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Polyhedrons
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Prisms
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Pyramids
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| Triquetra |
Calculator for 4-D Shapes
| Glome |
| Tesseract |
| Hypersphere |
Names of n-faced polyhedra
| \[No.\, Of\, Faces\] | \[Name\,Of\,Shape\] |
|---|---|
| \[1\] | \[Monohedron\] |
| \[2\] | \[Dihedron\] |
| \[3\] | \[Trihedron\] |
| \[4\] | \[Tetrahedron\] |
| \[5\] | \[Pentahedron\] |
| \[6\] | \[Hexahedron\] |
| \[7\] | \[Heptahedron\] |
| \[8\] | \[Octohedron\] |
| \[9\] | \[Enneahedron\] |
| \[10\] | \[Decahedron\] |
| \[11\] | \[Hendecahedron\] |
| \[12\] | \[Dodecahedron\] |
| \[13\] | \[Tridecahedron\] |
| \[14\] | \[Tetradecahedron\] |
| \[15\] | \[Pentadecahedron\] |
| \[16\] | \[Hexadecahedron\] |
| \[17\] | \[Heptadecahedron\] |
| \[18\] | \[Octadecahedron\] |
| \[19\] | \[Enneadecahedron\] |
| \[20\] | \[Icosahedron\] |
| \[21\] | \[Henicosahedron\] |
| \[22\] | \[Doicosahedron\] |
| \[23\] | \[Triaicosahedron\] |
| \[24\] | \[Tetraicosahedron\] |
| \[25\] | \[Pentaicosahedron\] |
| \[26\] | \[Hexaicosahedron\] |
| \[27\] | \[Heptaicosahedron\] |
| \[28\] | \[Octaicosahedron\] |
| \[29\] | \[Enneaicosahedron\] |
| \[30\] | \[Triacontahedron\] |
| \[31\] | \[Hentriacontahedron\] |
| \[32\] | \[Dotriacontahedron\] |
| \[33\] | \[Tritriaconthedron\] |
| \[34\] | \[Tetratriacontahedron\] |
| \[35\] | \[Pentatriacontahedron\] |
| \[36\] | \[Hexatriacontahedron\] |
| \[37\] | \[Heptatriacontahedron\] |
| \[38\] | \[Octatriacontahedron\] |
| \[39\] | \[Enneatriacontahedron\] |
| \[40\] | \[Tetracontahedron\] | \[50\] | \[Pentacontahedron\] |
| \[60\] | \[Hexacontahedron\] |
| \[70\] | \[Heptacontahedron\] |
| \[80\] | \[Octacontahedron\] |
| \[90\] | \[Enneacontahedron\] |
| \[100\] | \[Hectohedron\] |
| \[200\] | \[Dihectohedron\] |
| \[300\] | \[Trihectohedron\] |
| \[400\] | \[Tetrahectohedron\] |
| \[500\] | \[Pentahectohedron\] |
| \[600\] | \[Hexahectohedron\] |
| \[700\] | \[Heptahectohedron\] |
| \[800\] | \[Octahectohedron\] |
| \[900\] | \[Enneahectohedron\] |
| \[1000\] | \[Chiliahedron\] |
| \[10000\] | \[Myriahedron\] |
| \[1000000\] | \[Megahedron\] |
| \[Infinite\] | \[Sphere\] |
Names of n-sided polygon
| \[No.\, Of\, Sides\] | \[Name\,Of\,Shape\] |
|---|---|
| \[1\] | \[Monogon\,(impossible\, figure\] \[in\, Euclidean\, geometry)\] |
| \[2\] | \[Digon\] |
| \[3\] | \[Trigon(or\,Triangle)\] |
| \[4\] | \[Tetragon(or\,Quadrilateral)\] |
| \[5\] | \[Pentagon\] |
| \[6\] | \[Hexagon\] |
| \[7\] | \[Heptagon\] |
| \[8\] | \[Octogon\] |
| \[9\] | \[Enneagon\] |
| \[10\] | \[Decagon\] |
| \[11\] | \[Hendecagon\] |
| \[12\] | \[Dodecagon\] |
| \[13\] | \[Tridecagon\] |
| \[14\] | \[Tetradecagon\] |
| \[15\] | \[Pentadecagon\] |
| \[16\] | \[Hexadecagon\] |
| \[17\] | \[Heptadecagon\] |
| \[18\] | \[Octadecagon\] |
| \[19\] | \[Enneadecagon\] |
| \[20\] | \[Icosagon\] |
| \[21\] | \[Henicosagon\] |
| \[22\] | \[Doicosagon\] |
| \[23\] | \[Triaicosagon\] |
| \[24\] | \[Tetraicosagon\] |
| \[25\] | \[Pentaicosagon\] |
| \[26\] | \[Hexaicosagon\] |
| \[27\] | \[Heptaicosagon\] |
| \[28\] | \[Octaicosagon\] |
| \[29\] | \[Enneaicosagon\] |
| \[30\] | \[Triacontagon\] |
| \[31\] | \[Hentriacontagon\] |
| \[32\] | \[Dotriacontagon\] |
| \[33\] | \[Tritriacontagon\] |
| \[34\] | \[Tetratriacontagon\] |
| \[35\] | \[Pentatriacontagon\] |
| \[36\] | \[Hexatriacontagon\] |
| \[37\] | \[Heptatriacontagon\] |
| \[38\] | \[Octatriacontagon\] |
| \[39\] | \[Enneatriacontagon\] |
| \[40\] | \[Tetracontagon\] |
| \[50\] | \[Pentacontagon\] |
| \[60\] | \[Hexacontagon\] |
| \[70\] | \[Heptacontagon\] |
| \[80\] | \[Octacontagon\] |
| \[90\] | \[Enneacontagon\] |
| \[100\] | \[Hectogon\] |
| \[200\] | \[Dihectogon\] |
| \[300\] | \[Trihectogon\] |
| \[400\] | \[Tetrahectogon\] |
| \[500\] | \[Pentahectogon\] |
| \[600\] | \[Hexahectogon\] |
| \[700\] | \[Heptahectogon\] |
| \[800\] | \[Octahectogon\] |
| \[900\] | \[Enneahectogon\] |
| \[1000\] | \[Chiliagon\] |
| \[10000\] | \[Myriagon\] |
| \[1000000\] | \[Megagon\] |
| \[Infinite\] | \[Circle\] |
Date
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Different Centers of Triangle
Using Edges
Using Vertex
Centroid Calculator
Circumcentre Calculator
Orthocentre Calculator
Incentre Calculator
Excentre Calculator
Area of Triangle
Area of Circumcircle of an Equilateral Triangle
Median of an Equilateral Triangle (M) =
Median of an Equilateral Triangle (M) =
To check whether given Point (P) lies inside a Triangle
Oblique Triangle Calculator
Radius of Escribed Circles of a Triangle Calculator
Median of Triangle Calculator
Circumradius of Triangle Calculator
Inradius of Triangle Calculator
Distance between Incenter and Circumcenter of a Triangle
Calculator
Radius (r) =
CircumRadius (R) =
Radius (r) =
CircumRadius (R) =
EFFECTIVE INTEREST RATE CALCULATOR
It also reveals the real percentage rate owed in interest on a loan, a credit card, or any other debt.
It is also called the effective interest rate, the effective rate, or the annual equivalent rate.
Set Calculator
Result
Unit Convertor
SPI/CGPA Convertor
1. Percentage(%) = SPI/CGPA * 9.5
2. SPI/CGPA =
Percentage(%) / 9.5
3. SPI/CGPA scaled out of 10.
\[x^n \space = \space y\]
Percentage Calculator
Cost and Selling Prices
Cost Price:
The price, at which an article is purchased, is called its cost price, abbreviated as C.P.
Selling Price:
The price, at which an article is sold, is called its selling prices, abbreviated as S.P.
Profit or Gain:
If S.P. is greater than C.P., the seller is said to have a profit or gain.
Loss:
If S.P. is less than C.P., the seller is said to have incurred a loss.
IMPORTANT FORMULAE
1) Selling Price: (S.P.): SP = [(100 + Gain %) x C.P] / 100
2) Selling Price: (S.P.): Sp = [(100 - Loss %) x C.P.] / 100
3) Cost Price: (C.P.): C.P. = (100 x S.P.)/(100 + Gain %)
4) Cost Price: (C.P.): C.P. = (100 x S.P.)/(100 - Loss %)
Exponential Growth Calculator
Formula Used:\[x(t)=x_{0}(1+\frac{r}{100})^t\]
Euclid's GCD
The algorithm is based on the below facts.
If we subtract a smaller number from a larger (we reduce a
larger number), GCD doesn’t change. So if we keep subtracting repeatedly the larger of two, we end up
with GCD.
Now instead of subtraction, if we divide the smaller number, the algorithm stops when we find remainder
0.
Sum of GCD of all numbers upto N with N itself
Moments Calculator
Moments are a statistical tools ,used in statistical investigations .The moments of a distribution are the arithmetic means of the various powers of the deviations of items from some given number.
Mean Absolute Deviation Calculator
Adam Number Checker
B.M.I Calculator
Enter Height:
Enter Weight:
B.M.I Calculator
Enter Height:
Enter Weight:
Hyperperfect Number Checker
A number n is called k-hyperperfect if: n = 1 + k ∑ i d i where all di are the proper divisors of n.
Kaprekar Number Checker
A Kaprekar number is a number whose square when divided into two parts and such that sum of parts is equal to the original number and none of the parts has value 0.
Abundant Number
A number n is said to be Abundant Number if sum of all the proper divisors of the number denoted by sum(n) is greater than the value of the number n.
Wagstaff Number
A Wagstaff prime is a prime number of the form \[{{2^p+1}\over 3}\] , where p is an odd prime.
Woodall Number
A Woodall number (Wn) is any natural number of the form \[W_n = n \cdot 2^n - 1\] for some natural number n.
Doubling Time
The doubling time calculator, is a simple tool that lets you calculate how long it will take an amount to double given a constant growth rate
Antilog Calculator
Log base Calculator
Percentage change
Percentage Off
Modulo Calculator
x mod y = r
Hypergeometric Distribution Statistics
Exponential Distribution Calculator
\[P.M.F = f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0, \\ 0 & x < 0 \end{cases}\]
where,λ = rate parameter
x = random variable
Gamma Distribution Calculator
\[P.M.F = f(x;\alpha,\beta) = \frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)}\]
where,α = shape parameter
x = random variable
β = scale parameter
Γ(x) = the gamma function
Hypergeometric Distribution PMF Calculator
The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of k successes in n draws with replacement.
In the below formula:
K = the number of successes in the population
k = the number of observed successes
N = the population size
n = the number of draws
\[Hypergeometric \space distribution \space P.M.F = P(X = k) = \frac{\binom{K}{k} \binom{N -
K}{n-k}}{\binom{N}{n}}\]
Poisson Distribution Statistic
ORDER OF MAGNITUDE CALCULATOR
MULTIPLICATIVE INVERSE
ADDITIVE INVERSE
MANHATTAN DISTANCE CALCULATOR
Cartesian Coordinate Calculator
Polar Coordinate Calculator
Spherical Coordinate Calculator
Cylindrical to Spherical
Cylindrical Coordinate Calculator
Cartesian to Cylindrical
Spherical to Cylindrical
Polar to Cylindrical
BILINEAR INTERPOLATION CALCULATOR
Bilinear interpolation calculator is used to estimate function's value at any point (x, y) inside the rectangle.
UNIT RATE CALCULATOR
MARGIN OF ERROR CALCULATOR
LINEAR REGRESSION CALCULATOR
Linear regression is a model of the relationship between a dependent variable y and independent variables x
by
linear prediction function y=a+bx.
Linear functions are used to model the data in linear regression and the unknown model parameters are
estimated from the data.
Such method of modeling data is known as linear models. For more two or more variables, this modeling is
called multiple linear regression.
Linear regression models are often fitted using the least squares regression line.
COVARIANCE CALCULATOR
Covariance indicates whether
two variables X and Y are related by measuring how the variables change in relation to each other.
It tells us if there is a relationship between two variables and which direction that relationship is
in.
A positive covariance means that the two variables are positively related, and they have the same direction.
A negative covariance means that the variables are negatively related, and they have the opposite
directions.
PERCENTILE RANK CALCULATOR
PR = L / N * 100
where:
PR is the percentile rank - It can take values from 0 to 100;
L is the number of values from the set A that are lower than or equal to your data value xₘ; and
N is the total number of values in the set A.
Half Life Calculator
In radioactivity, the interval of time required for one-half of the atomic nuclei of a radioactive sample to decay (change spontaneously into other nuclear species by emitting particles and energy), or, equivalently, the time interval required for the number of disintegrations per second of a radioactive material to decrease by one-half is called Half-life.
Percentage to Ratio
SKEWNESS CALCULATOR
How to Calculate Skewness:
1. Calculate the mean and standard deviation
2. Subtract the mean from each raw score
3. Raise each of these deviations from the mean to the third power and sum
4. Calculate skewness, which is the sum of the deviations from the mean, raise to the third power,
divided by number of cases minus 1, times the standard deviation raised to the third power.
KURTOSIS CALCULATOR
Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values. Kurtosis determines the heaviness of the distribution tails.
CLOCK ANGLE CALCULATOR
Weighted Mean Calculator
The Weighted mean for given
set of non negative data {x1, x2, x3, ...xn} with non negative weights {w1, w2, w3, ...wn}
can be derived from the formula:
where
x is the repeating value
w is the number of occurrences of x (weight)
x̄ is the weighted mean
Coefficient of Variation Calculator
CV= μ/σ
where:
σ= standard deviation
μ= mean
The coefficient of variation (CV) is a statistical measure of the dispersion of data points in a data series
around the mean.
The coefficient of variation represents the ratio of the standard deviation to the mean,
and it is a useful statistic for comparing the degree of variation from one data series to another,
even if the means are drastically different from one another.
Confidence Interval Calculator
Root Mean Square Calculator
In mathematics, the root
mean square is also known as the quadratic mean.
It is represented by RMS or rms. The root mean square is defined as the square root of the mean square.
We can also say that it is a generalised mean with the exponent value as 2.
P VALUE CALCULATOR
What is P value ?
The p-value is the probability that the test statistic will produce values at least as extreme as the value
it produced for your sample.
It is crucial to remember that this probability is calculated under the assumption that the null hypothesis
is true!
Z SCORE CALCULATOR
The z-score, also referred to as standard
score, z-value, and normal score, among other things, is a dimensionless quantity that is used to indicate
the signed, fractional, number of standard deviations by which an event is above the mean value being
measured.
Values above the mean have positive z-scores, while values below the mean have negative z-scores.
The
z-score can be calculated by subtracting the population mean from the raw score, or data point in question
(a test score, height, age, etc.), then dividing the difference by the population standard deviation
Supplementary Angle Calculator
Verify if two angles are supplementary
Complementary Angle Calculator
Verify if two angles are complementary
Coterminal Angle Calculator
Verify if two angles are coterminal
FACTORIZATION CALCULATOR
ERROR PERCENTAGE CALCULATOR
When measuring data, the result often varies from the true value.
Error can arise due to many different reasons that are often related to human error, but can also be due to estimations and limitations of devices used in measurement.
Regardless, in cases such as these, it can be valuable to calculate the percentage error.
The computation of percentage error involves the use of the absolute error, which is simply the difference between the observed and the true value. The absolute error is then divided by the true value, resulting in the relative error, which is multiplied by 100 to obtain the percentage error.
SSS Triangle's Angle Calculator
To solve an SSS triangle:
Use The Law of Cosines first to calculate one of the angles
Use The Law of Cosines again to find another angle
Finally use angles of a triangle add to 180° to find the last ang
IMPROPER TO MIXED FRACTION CONVERTER
Sum of Square Calculator
For three numbers : x2 + y2 +z2 = (x+y+z)2-2xy-2yz-2xz ; x,y and z are real numbers
For n Natural Numbers : Σn2 = [n(n+1)(2n+1)]/6
For first n Even Numbers : Σ(2n)2 =[2n(n+1)(2n+1)]/3
For first n Odd Numbers : Σ(2n-1)2 = [n(2n+1)(2n-1)]/3
Expansion Calculator
Sum of the Tan(x) expansion upto N terms
Sum of the ArcSin(x) expansion upto N terms
Sum of the cos(x) expansion upto N terms
Sum of the sin(x) expansion upto N terms
Number of Perfect Cubes in a Range
Sum of First N Natural Numbers
Sum of First N Even Natural Numbers
Sum of First N Odd Natural Numbers
Sum of Squares of N Natural Numbers
Sum of Cubes of N Natural Numbers
Sum of Power Four of N Natural Numbers
Sum of Power Five of N Natural Numbers
Squares in a Range
Cubes in a Range
Area of Segment Calculator
Area of a Segment in Radians: A = (½) × r x r x (θ – Sin θ)
Area of a Segment in Degrees: A = (½) × r x r x [(π/180) θ – sin θ]
ARC LENGTH CALCULATOR
L / θ = C / 2π
As circumference C = 2πr,
L / θ = 2πr / 2π
L / θ = r
We find out the arc length formula when multiplying this equation by θ:
L = r * θ
AREA OF CRESCENT AND LUNE
Platonic Solids
A Platonic Solid is a 3D shape where:
There are only five platonic solids.
1. Tetrahedron
2. Cube
3. Octahedron
4. Dodecahedron
5. Icosahedron
Archimedean Solids
A Archimedean Solids is a 3D shape where:
There are only 13 archimedean solids.
1. Truncated tetrahedron
2. Cuboctahedron
3. Truncated cube
4. Truncated Octahedron
5. Rhombicuboctahedron
6. Truncated cuboctahedron
7. Snub cube
8. Icosidodecahedron
9. Truncated dodecahedron
10. Truncated icosahedron
11. Rhombicosidodecahedron
12. Truncated icosidodecahedron
13. Snub dodecahedron
Sum of Square Calculator
For two numbers : x2 + y2 = (x + y)2– 2ab ; x and y are real numbers
For three numbers : x2 + y2+z2 = (x+y+z)2-2xy-2yz-2xz ; x,y and z are real numbers
For n Natural Numbers : Σn2 = [n(n+1)(2n+1)]/6]
For first n Even Numbers : Σ(2n)2 =[2n(n+1)(2n+1)]/3
For first n Odd Numbers : Σ(2n-1)2 = [n(2n+1)(2n-1)]/3
Binomial Distribution Calculator
\[P.M.F = \, \frac{n!}{(n-r)!r!} p^x (1-p)^y\]
where,n= no. of trials
r= no. of successful events
x= probability of success
y= porobability of not being successful(1-x)
PMF of Poisson Distribution Calculator
It classifies the experiment into two different categories, such as success and failure. Generally, the Poisson random variable “x” defines the probability of the success of the experiment. From the average rate of success, the Poisson distribution probability can be easily calculated.
e = is Euler's number (e = 2.71828...)
k = the number of occurrences (k=0,1,2...)
λ = parameter such that λ>0
\[P.M.F = P(X{=}k)= \frac{\lambda^k e^{-\lambda}}{k!}\]
Boolean Algebra
Laws of Boolean Algebra
\[1.\,Annulment \space\, Law\\\,(\,i\,)\,A \centerdot 0\,=\,0\\\,(\,ii\,)\,A\,+\,1\,=\,1\]
\[2.\,Identity \space\, Law\\\,(\,i\,)\,A\,+\,0\,=\,A\\\,(\,ii\,)\,A \centerdot 1\,=\,A\]
\[3.\,Idempotent \space\, Law\\\,(\,i\,)\,A\,+\,A\,=\,A\\\,(\,ii\,)\,A \centerdot A\,=\,A\]
\[4.\,Complement \space\, Law\\\,(\,i\,)\,A \centerdot A\,'\,=\,0\\\,(\,ii\,)\,A\,+\,A\,'\,=1\]
\[5.\,Demorgan's \space\,Law\\ \,(\,i\,)\, ( \,A\,+\,B\,) \,'\,=\,A\,' \centerdot B\,'\\\,(\,ii\,)\, (\,A \centerdot B \,)\,'\,= \, A\,'\,+\,B\,'\]
\[6.\,Distributive \space \, Law \\\,(\,i\,)\, A \, (\,B\,+\,C\,)\, = \,A \centerdot B \,+\,A \centerdot C \\\,(\,ii\,)\,A \,+ \,B \centerdot C\,=\,(\,A\,+\,B\,)\centerdot (\,A\,+\,C\,)\]
\[7.\,Absorption \,/\,Redundancy \space \, Law\\\,(\,i\,)\, A \, +\,A \centerdot B\,= \,A\\\,(\,ii\,)\,A\,(\,A\,+\,B\,)\, = \,A\]
\[8.\,Commutative \space Law \\\,(\,i\,)\, A \,+\,B\,=\,B\,+\,A\\\,(\,ii\,)\, A \centerdot B\,=\,B \centerdot A\]
\[9.\,Associative \space Law \\ \,(\,i\,)\,A \,+\,(\,B\,+\,C\,)\,=\,(\,A\,+\,B\,)\,+\,C \\\,(\,ii\,)\, A \centerdot (\, B \centerdot C \,)\, = \, (\, A \centerdot B \,) \centerdot C\]
\[10.\,Double \space Negation \space Law \\\,(\,i\,)\, A \,'\,'\, = \, A \\\,(\,ii\,)\, B\,'\,'\, = \, B\]
Result Of Subtraction
Choose your numbers system
First number
Operation
Second number
Result
From Base
To Base
Residue at Pole
Let z=a be a pole of order m of a one valued section f(z) and gamma be any circle of radius r with centre at z=a which doesnt have any singularities except at z=a, then f(z) is analytic within the annulus
Hence it can be written in form of Laurent Series as
\[f(z)\space =\space \sum_{n=0}^{\infty} a_n (z-a)^{n}\space +\space \sum_{n=0}^{\infty} b_n (z-a)^{-n}\]
\[a_n \space =\space \frac{1}{2\pi i} \int \frac{f(z)}{(z-a)^{n+1}}\,dz\]
\[b_n\space =\space \frac{1}{2\pi i} \int \frac{f(z)}{(z-a)^{-n+1}}\,dz\]
\[ Particularly,\space b_1 \space =\space \frac{1}{2\pi i} \int f(z)\,dz\]
The coefficient of b1 is called residue of f(z) at z=a
Cauchy Residue Theorem
In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem.
Let U be a simply connected open subset of the complex plane containing a finite list of points a1, ..., an, U0 = U \ {a1, ..., an}, and a function f defined and holomorphic on U0. Let γ be a closed rectifiable curve in U0, and denote the winding number of γ around ak by I(γ, ak). The line integral of f around γ is equal to 2πi times the sum of residues of f at the points, each counted as many times as γ winds around the point:
\[{\displaystyle \oint _{\gamma }f(z)\,dz=2\pi i\sum _{k=1}^{n}\operatorname {I} (\gamma ,a_{k})\operatorname {Res} (f,a_{k}).}\]
If γ is a positively oriented simple closed curve, I(γ, ak) = 1 if ak is in the interior of γ, and 0 if not, therefore
\[{\displaystyle \oint _{\gamma }f(z)\,dz=2\pi i\sum \operatorname {Res} (f,a_{k})}\]
The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve γ must first be reduced to a set of simple closed curves {γi} whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of f dz along a Jordan curve γi with interior V. The requirement that f be holomorphic on U0 = U \ {ak} is equivalent to the statement that the exterior derivative d(f dz) = 0 on U0. Thus if two planar regions V and W of U enclose the same subset {aj} of {ak}, the regions V \ W and W \ V lie entirely in U0, and hence
\[{\displaystyle \int _{V\smallsetminus W}d(f\,dz)-\int _{W\smallsetminus V}d(f\,dz)}\]
Contour Integration
Contour integration is the process of calculating the values of a contour integral around a given contour in the complex plane. As a result of a truly amazing property of holomorphic functions, such integrals can be computed easily simply by summing the values of the complex residues inside the contour.
Contour integration methods include:
1. direct integration of a complex-valued function along a curve in the complex plane (a contour);
2. application of the Cauchy integral formula; and
3. application of the residue theorem.
Contours
Contours are the class of curves on which we define contour integration.  This requires that the sequence of curves γ1, …, γn be such that the terminal point of γi coincides with the initial point of γi+1, ∀ i, 1 ≤ i < n. This includes all directed smooth curves. Also, a single point in the complex plane is considered a contour. The symbol + is often used to denote the piecing of curves together to form a new curve.
\[\gamma = y_1\space +\space y_2\space +\space y_3\space ....+\space y_n \]
The contour integral of a complex function f : C → C is a generalization of the integral for real-valued functions. For continuous functions in the complex plane, the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the partition of an interval and the Riemann integral. In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour.
Milne Thomson Method
With the help of this method, we can directly construct f(z) in terms of z without finding v when u is given or u when v is given
\[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]
Case -1 : WHen only real part of u(x,y) is given
\[1.\space Find\space \frac{\delta u}{\delta x}\]
\[2.\space Write\space it\space as\space equal\space to\space \phi _1 (x,y)\]
\[3.\space Find\space \frac{\delta u}{\delta y}\]
f(z) is obtained by the formula -
\[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]
Case -2 : When only imaginary part of v(x,y) is given
\[1.\space Find \frac{\delta v}{\delta y}\]
\[2.\space Write it equal to \psi _1(x,y)\]
\[3.\space Find \frac{\delta v}{\delta x}\space and\space equate\space to\space \psi _2(x,y)\]
Case -3 : When u-v is given
\[1.\space f(z)\space =\space u+iv\]
\[2.\space if(z)\space =\space iu-v\]
\[3.\space F(z)\space =\space f(z)+if(z)\]
\[4.\space Find\space \frac{\delta U}{\delta x}\space and\space \frac{\delta U}{\delta y}\]
\[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]
Case -4 : When u+v is given
\[1.\space f(z)\space =\space u+iv\]
\[2.\space if(z)\space =\space iu-v\]
\[3.\space F(z)\space =\space f(z)+if(z)\]
\[4.\space Find\space \frac{\delta V}{\delta x}\space and\space \frac{\delta V}{\delta y}\]
\[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]
Representation by Power Series
Every complex number which is analytic in domain D can be represented by a power series about a point z0 inside D. This is called Taylor Series. If f(z) is not analytic at point z0, we can still expand in both negative and positive side of z-z0. This is called Laurent's Series
Taylor Series
If f(z) is analytic inside a circle C with point a as centre, then for all z inside C,
\[f(z)\space =\space f(a)+f(z-a)f^{'}(a) + \frac{(z-a)^{2}}{2}f^{''}(a)+.......\frac{(z-a)^{n}}{n}f^{n}(a)\]
\[f(z)\space =\space \sum_{n=0}^{\infty}a_n (z_a)^{n},\space where\space a_n\space =\space \frac{f^{n}a}{n!}\]
If a = 0, the series becomes,
\[f(z)\space =\space f(0) + zf^{'}(0) + \frac{z^{2}}{2}f^{''}(0)+......+ \frac{z^{n}}{n!}f^{n}(0)\]
This series is called Maclaurin Series
Laurent's Series
If f(z) is analytic inside and on the boundary of annular(ring-shaped) region R bounded by two concentric circled C1 and C2 of radii r1 and r2 having centre at a then for all z in R,
\[f(z)\space =\space \sum_{n=0}^{\infty}a_n (z-a)^{n}\space +\space \sum_{n=1}^{\infty}b_n (z-a)^{-n}\]
\[a_n\space =\space \frac{1}{2\pi i} \int\frac{f(w)}{(w-a)^{-n+1}}\,dw\]
\[b_n\space =\space \frac{1}{2\pi i} \int\frac{f(w)}{(w-a)^{-n+1}}\]
In case f(z) is anaytic inside C1 then bn=0 Laurent Seris reduces to Taylor Series
BCD Code Converter
From
To
BCD Code Addition
+
Decimal/BCD(8421) Table
| Decimal Equivalent | BCD Code (8421) |
|---|---|
|
0 |
0 0 0 0 |
|
1 |
0 0 0 1 |
|
2 |
0 0 1 0 |
|
3 |
0 0 1 1 |
|
4 |
0 1 0 0 |
|
5 |
0 1 0 1 |
|
6 |
0 1 1 0 |
|
7 |
0 1 1 1 |
|
8 |
1 0 0 0 |
|
9 |
1 0 0 1 |
Excess-3 Code Converter
From
To
<< /p>
Decimal/Excess-3 Table
| Decimal Equivalent | Excess-3 Code |
|---|---|
|
0 |
0 0 1 1 |
|
1 |
0 1 0 0 |
|
2 |
0 1 0 1 |
|
3 |
0 1 1 0 |
|
4 |
0 1 1 1 |
|
5 |
1 0 0 0 |
|
6 |
1 0 0 1 |
|
7 |
1 0 1 0 |
|
8 |
1 0 1 1 |
|
9 |
1 1 0 0 |
From
To
Decimal/2421 Table
| Decimal Equivalent | 2421 Code |
|---|---|
|
0 |
0 0 0 0 |
|
1 |
0 0 0 1 |
|
2 |
0 0 1 0 |
|
3 |
0 0 1 1 |
|
4 |
0 1 0 0 |
|
5 |
1 0 1 1 |
|
6 |
1 1 0 0 |
|
7 |
1 1 0 1 |
|
8 |
1 1 1 0 |
|
9 |
1 1 1 1 |
If all the letters of the word '_____' are arranged in all possible ways and written out in alphabetical(dictionary) order, then find the rank of the given word
Word in english characters
Rank in Dictionary
Bin/Dec/Oct/Hex Converter
From
To
Bin/Dec/Oct/Hex Table
| Decimal Numbers | Binary Numbers | Octal Numbers | Hexa Decimal Numbers |
|---|---|---|---|
|
1 |
1 |
1 |
1 |
|
2 |
10 |
2 |
2 |
|
3 |
11 |
3 |
3 |
|
4 |
100 |
4 |
4 |
|
5 |
101 |
5 |
5 |
|
6 |
110 |
6 |
6 |
|
7 |
111 |
7 |
7 |
|
8 |
1000 |
10 |
8 |
|
9 |
1001 |
11 |
9 |
|
10 |
1010 |
12 |
A |
|
11 |
1011 |
13 |
B |
|
12 |
1100 |
14 |
C |
|
13 |
1101 |
15 |
D |
|
14 |
1110 |
16 |
E |
|
15 |
1111 |
17 |
F |
Empirical Probability
Geometric Probability Distribution
\[P.M.F = \, p(1-p)^x\]
where,p= probability of success
x= the number of failures until the first success
Conditional Probability
Mean and Variance of Random variable
Joint Probability
The probability that two events will both occur
Formula for Joint Probability
Joint Probability = P(A ⋂ B) = P (A) x P(B)
Where:
P(A ⋂ B) is the notation for the joint probability of event “A” and “B”.
P(A) is the probability of event “A” occurring.
P(B) is the probability of event “B” occurring.
Independent and Dependent event Probability
Independent Event Probability
The outcome of one event does not affect the outcome of the other.
If A and B are independent events then probability of both occuring is
P(A and B) = P(A) X P(B)
Dependent Event Probability
The outcome of one event affect the outcome of the other.
If A and B are dependent events then probability of both occuring is
P(A and B) = P(A) X P(B|A)
Bayes' Theorem
Bayes’ theorem describes the probability of an event based on prior knowledge of the conditions that might be relevant to the event.
\[ P \, \left( \frac{A}{B}\right)\space = \space \frac{P \left( \frac{B}{A} \right) \centerdot P(A)}{P(B)} \]
P(A|B) = probability of A given B is true
P(B|A) = probability of B given A is true
P(A), P(B) = the independent probabilities of A and B
Enter number of favourable outcomes in case of Event A
Enter number of possible outcomes in case of Event A
Enter number of favourable outcomes in case of Event B
Enter number of possible outcomes in case of Event B
| C | More | ( | ) | |
| e | e^ | 00 | ^ | |
| 7 | 8 | 9 | ÷ | |
| 4 | 5 | 6 | x | % |
| 1 | 2 | 3 | - | Deci. |
| 0 | . | √ | + | = |