# Lesson Addition and subtraction of trigonometric functions - Examples

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Addition and subtraction of trigonometric functions - Examples

The addition and subtraction trigonometric functions formulas are: {{{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))}}} The proofs of these formulas are presented in the lesson Addition and subtraction of trigonometric functions in this module. Below are examples of application of these formulas. Example 1 Find sin(75°) + sin(15°). Solution Use the addition formula for sines: {{{sin(alpha) + sin(beta) = 2*sin((alpha+beta)/2)*cos((alpha-beta)/2)}}}. You have sin(75°) + sin(15°) = 2*sin((75°+15°)/2)*cos((75°-15°)/2) = 2*sin(45°)*cos(30°) = {{{2*sqrt(2)/2*sqrt(3)/2 = sqrt(6)/2}}}. Example 2 Prove that {{{(sin(alpha)+sin(3alpha))/(cos(alpha)+cos(3alpha)) = tan(2alpha)}}}. Solution Use the addition formula for sines and cosines: {{{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)}}}. You have {{{(sin(alpha)+sin(3alpha))/(cos(alpha)+cos(3alpha)) = 2*sin((alpha+3alpha)/2)*cos((alpha-3alpha)/2)/(2*cos((alpha+3alpha)/2)*cos((alpha-3alpha)/2)))}}} = {{{2*sin(2alpha)*cos(-2alpha)/(2*cos(2alpha)*cos(-2alpha)) = sin(2alpha)/cos(2alpha) = tan(2alpha)}}}. The proof is completed. Example 3 Prove that {{{1 + cos(2alpha) + cos(4alpha) + cos(6alpha) = 4*cos(alpha)*cos(2alpha)*cos(3alpha)}}}. Solution Using the addition formula for cosines you have {{{1 + cos(2alpha) = cos(0) + cos(2alpha) = 2*cos((0+2alpha)/2)*cos((0-2alpha)/2) = 2*cos^2(alpha)}}}, {{{cos(4alpha) + cos(6alpha) = 2*cos((4alpha+6alpha)/2)*cos((4alpha-6alpha)/2) = 2*cos(5alpha)*cos(-alpha) = 2*cos(alpha)*cos(5alpha)}}}. By summing the left and the right sides of these two equalities, you get {{{1 + cos(2alpha) + cos(4alpha) + cos(6alpha) = 2*cos^2(alpha) + 2*cos(alpha)*cos(5alpha)}}}. Furthermore, you can transform the right side as follows: {{{2*cos^2(alpha) + 2*cos(alpha)*cos(5alpha) = 2*cos(alpha)*(cos(alpha) + cos(5alpha))}}} = {{{2*cos(alpha)*(cos(alpha+5alpha)/2)*(cos(alpha-5alpha)/2)}}} = {{{2*cos(alpha)*cos(3alpha)*cos(-2alpha) = 2*cos(alpha)*cos(2alpha)*cos(3alpha)}}}. The proof is completed. Example 4 Prove yourself that {{{1 - cos(2alpha) + cos(4alpha) - cos(6alpha) = 4*sin(alpha)*cos(2alpha)*sin(3alpha)}}}. Solution The proof is similar to that of the Example 3. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For your convenience, below is the list of my lessons on Trigonometry in this site in the logical order. They all are under the current topic Trigonometry in the section Algebra II. Addition and subtraction formulas
{{{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))}}}.     The lessons Addition and subtraction formulas and                      Addition and subtraction formulas - Examples
Addition and subtraction of trigonometric functions
{{{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))}}}.     The lessons Addition and subtraction of trigonometric functions and                      Addition and subtraction of trigonometric functions - Examples (this lesson)
Product of trigonometric functions
{{{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))}}}.                                  The lessons Product of trigonometric functions and                                                    Product of trigonometric functions - Examples
Powers of trigonometric functions
{{{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)}}}.                                           The lessons Powers of Trigonometric functions and                                                             Powers of Trigonometric functions - Examples
Trigonometric functions of multiply argument
{{{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)}}}.                                                 The lessons Trigonometric functions of multiply argument and                                                                 Trigonometric functions of multiply argument - Examples
Trigonometric functions of half argument
{{{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))}}}. The lessons Trigonometric functions of half argument and                   Trigonometric functions of half argument - Examples
Miscellaneous Trigonometry problems The lesson Miscellaneous Trigonometry problems