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<H2>Addition and subtraction formulas - Examples</H2> The addition and subtraction Trigonometry formulas are: {{{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 proofs of these formulas are presented in the lesson <A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson> Addition and subtraction formulas</A> in this module. Below are examples of application of these formulas. <H3>Example 1</H3>Find cos(75°), sin(75°) and tan(75°). <B>Solution</B> Note that 75° = 30° + 45°. Calculate cos(75°) by applying the addition formula: cos(75°) = cos(30°)*cos(45°) - sin(30°)*sin(45°) = {{{sqrt(3)/2*sqrt(2)/2 - 1/2*sqrt(2)/2 = (sqrt(6) - sqrt(2))/4}}}. Calculate sin(75°) by applying the addition formula: sin(75°) = sin(30°)*cos(45°) + cos(30°)*sin(45°) = {{{1/2*sqrt(2)/2 + sqrt(3)/2*sqrt(2)/2 = (sqrt(2) + sqrt(6))/4}}}. Now, calculate tan(75°) as the fraction sin(75°)/cos(75°): tan(75°) = sin(75°)/cos(75°) = {{{(sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))}}}, as it follows from the lines above. Simplify it: {{{(sqrt(6)+sqrt(2))/(sqrt(6)-sqrt(2)) = ((sqrt(6)+sqrt(2))*(sqrt(6)+sqrt(2)))/((sqrt(6)-sqrt(2))*(sqrt(6)+sqrt(2))) = (sqrt(6)+sqrt(2))^2/(6-2) = (6+2sqrt(12)+2)/4 = (8+4sqrt(3))/4 = 2+sqrt(3)}}}. Or, you can calculate tan(75°) by applying the addition formula for tangents: tan(75°) = (tan(30°) + tan(45°))/(1 - tan(30°)*tan(45°)) = {{{(sqrt(3)/3 + 1)/(1 - sqrt(3)/3) = (3+sqrt(3))/(3-sqrt(3))}}}. Simplify it: {{{(3+sqrt(3))/(3-sqrt(3)) = ((3+sqrt(3))*(3+sqrt(3)))/((3-sqrt(3))*(3+sqrt(3))) = (3+sqrt(3))^2/(9-3) = (9+6sqrt(3)+3)/6 = (12+6sqrt(3))/6 = 2+sqrt(3)}}}. Thus, both calculations produce the same result for tan(75°), namely, {{{2+sqrt(3)}}}. <H3>Example 2</H3>Find cos(15°), sin(15°) and tan(15°). <B>Solution</B> Note that 15° = 45° - 30°. Calculate cos(15°) by applying the subtraction formula: cos(15°) = cos(45°)*cos(30°) + sin(45°)*sin(30°) = {{{sqrt(2)/2*sqrt(3)/2 + sqrt(2)/2*1/2 = (sqrt(6) + sqrt(2))/4}}}. Calculate sin(15°) by applying the subtraction formula: sin(15°) = sin(45°)*cos(30°) - cos(45°)*sin(30°) = {{{sqrt(2)/2*sqrt(3)/2 - sqrt(2)/2*1/2 = (sqrt(6) - sqrt(2))/4}}}. Now, calculate tan(15°) as the fraction sin(15°)/cos(15°): tan(15°) = sin(15°)/cos(15°) = {{{(sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))}}}, as it follows from the lines above. Simplify it: {{{(sqrt(6)-sqrt(2))/(sqrt(6)+sqrt(2)) = ((sqrt(6)-sqrt(2))*(sqrt(6)-sqrt(2)))/((sqrt(6)-sqrt(2))*(sqrt(6)+sqrt(2))) = (sqrt(6)-sqrt(2))^2/(6-2) = (6-2sqrt(12)+2)/4 = (8-4sqrt(3))/4 = 2-sqrt(3)}}}. Or, you can calculate tan(15°) by applying the subtraction formula for tangents: tan(15°) = (tan(45°) - tan(30°))/(1 + tan(45°)*tan(30°)) = {{{(1 - sqrt(3)/3)/(1 + sqrt(3)/3) = (3-sqrt(3))/(3+sqrt(3))}}}. Simplify it: {{{(3-sqrt(3))/(3+sqrt(3)) = ((3-sqrt(3))*(3-sqrt(3)))/((3+sqrt(3))*(3-sqrt(3))) = (3-sqrt(3))^2/(9-3) = (9-6sqrt(3)+3)/6 = (12-6sqrt(3))/6 = 2-sqrt(3)}}}. Thus, both calculations produce the same result for tan(15°), namely, {{{2-sqrt(3)}}}. Another way to solve the <B>Example 2</B> is to note that 15° = 90° - 75° and then to apply the formulas for the complementary angle and to use results of the <B>Example 1</B>: cos(15°) = sin(75°) = {{{(sqrt(2) + sqrt(6))/4}}}, sin(15°) = cos(75°) = {{{(sqrt(6) - sqrt(2))/4}}}, tan(15°) = cot(75°) = 1/tan(75°) = {{{1/((sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) = (sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2)) = 1/(2+sqrt(3)) = 2-sqrt(3)}}}. <H3>Example 3</H3>Find {{{cos(alpha+beta)}}}, {{{sin(alpha+beta)}}} and {{{tan(alpha+beta)}}}, if {{{cos(alpha) = 4/5}}}, {{{sin(beta) = 15/17}}} and {{{alpha}}} and {{{beta}}} are the first quadrant angles. <B>Solution</B> Since {{{cos(alpha) = 4/5}}} and {{{alpha}}} is the first quadrant angle, we have {{{sin(alpha) = sqrt(1-cos^2(alpha)) = sqrt(1 - (4/5)^2) = sqrt(1-16/25) = sqrt(9/25) = 3/5}}}. Since {{{sin(beta) = 15/17}}} and {{{beta}}} is the first quadrant angle, we have {{{cos(beta) = sqrt(1-sin^2(beta)) = sqrt(1 - (15/17)^2) = sqrt(1-225/289) = sqrt(64/289) = 8/17}}}. Now apply the addition formulas: {{{cos(alpha+beta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta) = 4/5*8/17-3/5*15/17 = 32/85 - 45/85 = (32 - 45)/85 = -13/85}}}, {{{sin(alpha+beta) = sin(alpha)*cos(beta) + cos(alpha)*sin(beta) = 3/5*8/17+4/5*15/17 = 24/85 + 60/85 = (24 + 60)/85 = 84/85}}}. {{{tan(alpha+beta) = sin(alpha+beta)/cos(alpha+beta) = -(84/85)/(13/85) = -84/13}}}, as it follows from the previous two lines, OR {{{tan(alpha+beta) = (tan(alpha) + tan(beta))/(1 - tan(alpha)*tan(beta)) = (3/4 + 15/8)/(1 - (3/4)*(15/8)) = (21/8)/(1-45/32) = -(21/8)/(13/32) = -84/13}}} from the addition formula for tangents. Both calculated results for tangents are identical. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 <B>Trigonometry</B> in the section <B>Algebra II</B>. <B>Addition and subtraction formulas</B> <TABLE> <TR> <TD>{{{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))}}}. </TD> <TD> The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A> and <B>Addition and subtraction formulas - Examples</B> (this lesson) </TD> </TR> </TABLE> <B>Addition and subtraction of trigonometric functions</B> <TABLE> <TR> <TD>{{{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))}}}. </TD> <TD> 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> </TD> </TR> </TABLE> <B>Product of trigonometric functions</B> <TABLE> <TR> <TD>{{{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))}}}. </TD> <TD> 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> </TD> </TR> </TABLE> <B>Powers of trigonometric functions</B> <TABLE> <TR> <TD>{{{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)}}}. </TD> <TD> 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> </TD> </TR> </TABLE> <B>Trigonometric functions of multiply argument</B> <TABLE> <TR> <TD>{{{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)}}}. </TD> <TD> 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> </TD> </TR> </TABLE> <B>Trigonometric functions of half argument</B> <TABLE> <TR> <TD>{{{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))}}}. </TD> <TD> 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> </TD> </TR> </TABLE> <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.
