Lesson FORMULAS FOR TRIGONOMETRIC FUNCTIONS
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<H2>Formulas for Trigonometric functions</H2> This lesson is a collection of formulas that express properties of Trigonometric Functions. <H3>Simplest relations</H3> {{{sin(alpha)^2 + cos(alpha)^2 = 1}}}, {{{sin(-alpha) = -sin(alpha)}}}, {{{cos(-alpha) = cos(alpha)}}}, {{{tan(alpha) = sin(alpha)/cos(alpha)}}}, {{{cot(alpha) = cos(alpha)/sin(alpha)}}}, {{{tan(-alpha) = -tan(alpha)}}}, {{{cot(-alpha) = -cot(alpha)}}}, {{{1 + tan^2(alpha) = 1/cos^2(alpha)}}}, {{{1 + cot^2(alpha) = 1/sin^2(alpha)}}}. <H3>Reduction formulas</H3> {{{sin(alpha +- 2*pi*n) = sin(alpha)}}}, {{{cos(alpha +- 2*pi*n) = cos(alpha)}}}, {{{sin(alpha +- pi*n) = (-1)^n*sin(alpha)}}}, {{{cos(alpha +- pi*n) = (-1)^n*cos(alpha)}}}, {{{sin(alpha + pi/4) = (sqrt(2)/2)*(sin(alpha) + cos(alpha))}}}, {{{cos(alpha + pi/4) = (sqrt(2)/2)*(cos(alpha)- sin(alpha))}}}, {{{sin(alpha - pi/4) = (sqrt(2)/2)*(sin(alpha) - cos(alpha))}}}, {{{cos(alpha - pi/4) = (sqrt(2)/2)*(cos(alpha)+ sin(alpha))}}}, {{{tan(alpha +- pi*n) = tan(alpha)}}}, {{{cot(alpha +- pi*n) = cot(alpha)}}}. For the proof and examples see the lesson <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trig-identities.lesson>The Amazing Unit Circle: Trigonometric Identities</A> in this module. <H3>Addition and subtraction formulas</H3> {{{cos(alpha + beta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta)}}}, {{{cos(alpha - beta) = cos(alpha)*cos(beta) + sin(alpha)*sin(beta)}}}, {{{sin(alpha + beta) = sin(alpha)*cos(beta) + cos(alpha)*sin(beta)}}}, {{{sin(alpha - beta) = sin(alpha)*cos(beta) - cos(alpha)*sin(beta)}}}, {{{tan(alpha + beta) = (tan(alpha) + tan(beta))/(1 - tan(alpha)*tan(beta))}}}, {{{tan(alpha - beta) = (tan(alpha) - tan(beta))/(1 + tan(alpha)*tan(beta))}}}. For the proof and examples see the lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A> and <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas-Examples.lesson>Addition and subtraction formulas - Examples</A> in this module. <H3>Addition and subtraction of trigonometric functions</H3> {{{sin(alpha) + sin(beta) = 2*sin((alpha+beta)/2)*cos((alpha-beta)/2)}}}, {{{sin(alpha) - sin(beta) = 2*sin((alpha-beta)/2)*cos((alpha+beta)/2)}}}, {{{cos(alpha) + cos(beta) = 2*cos((alpha+beta)/2)*cos((alpha-beta)/2)}}}, {{{cos(alpha) - cos(beta) = -2*sin((alpha+beta)/2)*sin((alpha-beta)/2)}}}, {{{tan(alpha) +- tan(beta) = sin(alpha +- beta)/(cos(alpha)*cos(beta))}}}, {{{cot(alpha) +- cot(beta) = sin(alpha +- beta)/(sin(alpha)*sin(beta))}}}. For the proof and examples see the lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions.lesson>Addition and subtraction of trigonometric functions</A> and <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions-Examples.lesson>Addition and subtraction of trigonometric functions - Examples</A> in this module. <H3>Product of trigonometric functions</H3> {{{sin(alpha)*sin(beta) = (1/2)*(cos(alpha-beta) - cos(alpha+beta))}}}, {{{cos(alpha)*cos(beta) = (1/2)*(cos(alpha-beta) + cos(alpha+beta))}}}, {{{sin(alpha)*cos(beta) = (1/2)*(sin(alpha-beta) + sin(alpha+beta))}}}. For the proof and examples see the lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Product-of-trigonometric-functions.lesson>Product of trigonometric functions</A> and <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Product-of-trigonometric-functions-Examples.lesson>Product of trigonometric functions - Examples</A> in this module. <H3>Powers of trigonometric functions</H3> {{{cos^2(alpha) = (1/2)*cos(2alpha) + 1/2}}}, {{{sin^2(alpha) = -(1/2)*cos(2alpha) + 1/2}}}, {{{cos^3(alpha) = (1/4)*cos(3alpha) + (3/4)*cos(alpha)}}}, {{{sin^3(alpha) = -(1/4)*sin(3alpha) + (3/4)*sin(alpha)}}}. For the proof and examples see the lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of-trigonometric-functions.lesson>Powers of Trigonometric functions</A> and <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of=trigonometric-functions-Examples.lesson>Powers of Trigonometric functions - Examples</A> in this module. <H3>Trigonometric functions of multiply argument</H3> {{{cos(2alpha) = 2*cos^2(alpha) - 1}}}, {{{sin(2alpha) = 2*sin(alpha)*cos(alpha)}}}, {{{cos(3alpha) = 4*cos^3(alpha) - 3*cos(alpha)}}}, {{{sin(3alpha) = -4*sin^3(alpha) + 3*sin(alpha)}}}. For the proof and examples see the lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-multiply-argument.lesson>Trigonometric functions of multiply argument</A> and <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-multiply-argument-Examples.lesson>Trigonometric functions of multiply argument - Examples</A> in this module. <H3>Trigonometric functions of half argument</H3> {{{sin^2(alpha/2) = (1-cos(alpha))/2}}}, {{{cos^2(alpha/2) = (1+cos(alpha))/2}}}, {{{tan(alpha/2) = sin(alpha)/(1+cos(alpha)) = (1-cos(alpha))/sin(alpha)}}}, {{{sin(alpha) = 2*tan(alpha/2)/(1+tan^2(alpha/2))}}}, {{{cos(alpha) = (1-tan^2(alpha/2))/(1+tan^2(alpha/2))}}}, {{{tan(alpha) = 2*tan(alpha/2)/(1-tan^2(alpha/2))}}}. For the proof and examples see the lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A> and <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument-Examples.lesson>Trigonometric functions of half argument - Examples</A> in this module. <B>Miscellaneous Trigonometry problems</B> The lesson <A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Miscellaneous-Trigonometry-problems.lesson>Miscellaneous Trigonometry problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
