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<H2>Addition and subtraction formulas</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))}}} In this lesson you can learn how to prove these formulas. <TABLE cellspacing="10"> <TR> <TD> <H3>Proof of the addition formula for cosines</H3> In the unit circle consider the point P1 with the central angle {{{-alpha}}} (coordinates ({{{x[1]}}}, {{{y[1]}}}), see the <B>Figure 1a</B>). Consider also the point P2 with the central angle {{{beta}}} (coordinates ({{{x[2]}}}, {{{y[2]}}}), see the <B>Figure 1a</B>). Let P3 be the point with the central angle {{{alpha+beta}}} (coordinates ({{{x[3]}}}, {{{y[3]}}}), see the <B>Figure 1b</B>). We have {{{x[1] = cos(alpha)}}}, {{{y[1] = sin(-alpha)=-sin(alpha)}}}, (1) {{{x[2] = cos(beta)}}}, {{{y[2] = sin(beta)}}}, (2) {{{x[3] = cos(alpha+beta)}}}, {{{y[3] = sin(alpha+beta)}}}. (3) </TD> <TD> {{{drawing( 300, 300, -1.4, 1.4, -1.4, 1.4, circle (0, 0, 1.0), line (-1.4, 0, 1.4, 0), line ( 0, -1.4, 0, 1.4), locate (-0.08, 0.0, O), locate ( 1.06, 0.0, 1), locate (-1.2, 0.0, -1), locate (-0.2, -1, -1), locate (-0.08, 1.15, 1), locate ( 0.75, -0.5, P1(x1,y1)), line ( 0, 0, 0.866, -0.5), arc ( 0, 0, 0.4, 0.4, 0, 30) locate ( 0.7, 0.8, P2(x2,y2)), line ( 0, 0, 0.707, 0.707), arc ( 0, 0, 0.6, 0.6, 315, 0), arc ( 0, 0, 0.64, 0.64, 315, 0), red(line ( 0.866,-0.5, 0.707, 0.707)) )}}} <B>Figure 1a. Proof of the addition formula</B> <B> for cosines</B> </TD> <TD> {{{drawing( 300, 300, -1.4, 1.4, -1.4, 1.4, circle (0, 0, 1.0), line (-1.4, 0, 1.4, 0), line ( 0, -1.4, 0, 1.4), locate (-0.08, 0.0, O), locate ( 1.06, 0.0, 1), locate (-1.2, 0.0, -1), locate (-0.2, -1, -1), locate (-0.08, 1.15, 1), locate ( 0.24, 1.10, P3(x2,y2)), locate ( 1, 0.15, A(1,0)), line ( 0, 0, 0.259, 0.966), arc ( 0, 0, 0.8, 0.8, 285, 0), arc ( 0, 0, 0.84, 0.84, 285, 0), arc ( 0, 0, 0.88, 0.88, 285, 0), red(line ( 1, 0, 0.259, 0.966)) )}}} <B>Figure 1b. Proof of the addition formula</B> <B> for cosines</B> </TD> </TR> </TABLE>Since triangles <B>P1OP2</B> and <B>AOP3</B> are congruent, the segment <B>[P1,P2]</B> (<B>Figure 1a</B>) has the same length as the segment <B>[A,P3]</B> (<B>Figure 1b</B>), where <B>A</B> is the point with coordinates (1,0). This gives you the equation {{{(x[2]-x[1])^2 + (y[2]-y[1])^2 = (x[3]-1)^2 + y[3]^2}}}. Simplify this equation step by step. You get {{{x[2]^2-2x[1]x[2]+x[1]^2 + y[2]^2-2y[1]y[2]+y[1]^2 = x[3]^2-2x[3]+1 +y[3]^2}}}, (after opening the brackets), {{{-2x[1]x[2] - 2y[1]y[2] = -2x[3]}}}, (after using {{{x[1]^2+y[1]^2=1}}}, {{{x[2]^2+y[2]^2=1}}} and {{{x[3]^2+y[3]^2=1}}}), {{{x[1]x[2] + y[1]y[2] = x[3]}}}. (after dividing both sides by -2). Substituting expressions (1), (2) and (3) for {{{x[1]}}}, {{{y[1]}}}, {{{x[2]}}}, {{{y[2]}}}, {{{x[3]}}} and {{{y[3]}}}, you get exactly the addition formula {{{cos(alpha + beta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta)}}}. The proof is completed. <H3>Proof of the subtraction formula for cosines</H3> Now, when the addition formula for cosines is proved, the proof of the subtraction formula can be made in couple of lines. Simply introduce the angle {{{gamma = -beta}}} and then apply the addition formula for cosines. Use {{{cos(gamma) = cos(beta)}}}, {{{sin(gamma) = -sin(beta)}}}: {{{cos(alpha - gamma) = cos(alpha + beta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta)}}} = {{{cos(alpha)*cos(gamma) - sin(alpha)*(-sin(gamma)) = cos(alpha)*cos(gamma) + sin(alpha)*sin(gamma))}}}. The proof is completed. <H3>Proof of the addition formula for sines</H3> You can easy get the addition formula for sines from the subtraction formula for cosines, which is already proved. Simply use the reduction formulas {{{sin(alpha) = cos(pi/2 - alpha)}}}, {{{cos(alpha) = sin(pi/2 - alpha)}}} (see, for example, the lesson <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trig-identities.lesson>The Amazing Unit Circle: Trigonometric Identities</A> of this module). You have {{{sin(alpha + beta) = cos(pi/2 - (alpha+beta)) = cos((pi/2 - alpha)-beta)) }}} = {{{cos(pi/2 - alpha)*cos(beta) + sin(pi/2 - alpha)*sin(beta)}}} = {{{sin(alpha)*cos(beta) + cos(alpha)*sin(beta)}}}. The proof is completed. <H3>Proof of the subtraction formula for sines</H3> Similarly, you can easy get the subtraction formula for sines from the addition formula for cosines, which is already proved. Simply use the same reduction formula as in the previous proof. {{{sin(alpha - beta) = cos(pi/2 - (alpha-beta)) = cos((pi/2 - alpha)+beta)) }}} = {{{cos(pi/2 - alpha)*cos(beta) - sin(pi/2 - alpha)*sin(beta)}}} = {{{sin(alpha)*cos(beta) - cos(alpha)*sin(beta)}}}. The proof is completed. <H3>Proof of the addition and subtraction formulas for tangents</H3> Now, when the addition and subtraction formulas for cosines and sines are proved, the proof of the addition and subtraction formulas for tangents is straightforward. For addition you have {{{tan(alpha + beta) = sin(alpha + beta)/cos(alpha + beta)}}} = {{{(sin(alpha)*cos(beta) + cos(alpha)*sin(beta))/(cos(alpha)*cos(beta) - sin(alpha)*sin(beta))}}} = {{{((sin(alpha)*cos(beta))/(cos(alpha)*cos(beta)) + (cos(alpha)*sin(beta))/(cos(alpha)*cos(beta)))/((cos(alpha)*cos(beta))/(cos(alpha)*cos(beta)) - (sin(alpha)*sin(beta))/(cos(alpha)*cos(beta)))}}} (after dividing both numerator and denominator by {{{cos(alpha)*cos(beta)}}}) = {{{(tan(alpha) + tan(beta))/(1 - tan(alpha)*tan(beta))}}}. The proof is completed. For subtraction you have {{{tan(alpha - beta) = sin(alpha - beta)/cos(alpha - beta)}}} = {{{(sin(alpha)*cos(beta) - cos(alpha)*sin(beta))/(cos(alpha)*cos(beta) + sin(alpha)*sin(beta))}}} = {{{((sin(alpha)*cos(beta))/(cos(alpha)*cos(beta)) - (cos(alpha)*sin(beta))/(cos(alpha)*cos(beta)))/((cos(alpha)*cos(beta))/(cos(alpha)*cos(beta)) + (sin(alpha)*sin(beta))/(cos(alpha)*cos(beta)))}}} (after dividing both numerator and denominator by {{{cos(alpha)*cos(beta)}}}) = {{{(tan(alpha) - tan(beta))/(1 + tan(alpha)*tan(beta))}}}. The proof is completed. For examples see the lesson <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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 <B>Addition and subtraction formulas</B> (this lesson) and <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas-Examples.lesson>Addition and subtraction formulas - Examples</A> </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.
