Section 7. Addition and Subtraction Identities 47 Section 7. Addition and Subtraction Identities In this section, we begin expanding our repertoire of trigonometric identities. Identities The sum and difference identities cos( α β cos( αcos( β + sin( αsin( β cos( α + β cos( αcos( β sin( αsin( β sin( α + β sin( αcos( β + cos( αsin( β sin( α β sin( αcos( β cos( αsin( β We will prove the difference of angles identity for cosine. The rest of the identities can be derived from this one. Proof of the difference of angles identity for cosine Consider two points on a unit circle: P at an angle of α from the positive x axis with coordinates ( cos( α,sin( α Q at an angle of β with coordinates cos( β,sin( β ( P C Q Notice the measure of angle POQ is α β. Label two more points: C at an angle of α β, with coordinates ( cos( α β,sin( α β, D at the point (, 0. α O α - β β D Notice that the distance from C to D is the same as the distance from P to Q because triangle COD is a rotation of triangle POQ. Using the distance formula to find the distance from P to Q yields ( cos( α cos( β + ( sin( α sin( β Expanding this cos ( α cos( αcos( β + cos ( β + sin ( α sin( αsin( β + sin Applying the Pythagorean Identity and simplifying cos( αcos( β sin( αsin( β ( β
48 Chapter 7 Similarly, using the distance formula to find the distance from C to D cos( α β + sin( α β 0 ( ( Expanding this cos ( α β cos( α β + + sin ( α β Applying the Pythagorean Identity and simplifying cos( α β + Since the two distances are the same we set these two formulas equal to each other and simplify cos( α cos( β sin( αsin( β cos( α β + cos( α cos( β sin( αsin( β cos( α β + cos( αcos( β + sin( αsin( β cos( α β Establishing the identity. Try it Now. By writing cos( α + β as cos ( α ( β, show the sum of angles identity for cosine follows from the difference of angles identity proven above. The sum and difference of angles identities are often used to rewrite expressions in other forms, or to rewrite an angle in terms of simpler angles. Example Find the exact value of cos( 75. Since 75 0 + 45, we can evaluate cos( 75 as cos( 75 cos(0 + 45 Apply the cosine sum of angles identity cos( 0 cos(45 sin(0 sin(45 Evaluate Simply 6 4 Try it Now. Find the exact value of sin π.
Section 7. Addition and Subtraction Identities 49 Example π Rewrite sin x in terms of sin(x and cos(x. 4 π sin x 4 Use the difference of angles identity for sine π π sin( x cos cos( x sin 4 4 Evaluate the cosine and sine and rearrange sin ( x cos( x Additionally, these identities can be used to simplify expressions or prove new identities Example sin( a + b Prove sin( a b tan( a + tan( b. tan( a tan( b As with any identity, we need to first decide which side to begin with. Since the left side involves sum and difference of angles, we might start there sin( a + b sin( a b sin( acos( b + cos( asin( b sin( acos( b cos( asin( b Apply the sum and difference of angle identities Since it is not immediately obvious how to proceed, we might start on the other side, and see if the path is more apparent. tan( a + tan( b Rewriting the tangents using the tangent identity tan( a tan( b sin( a sin( b + cos( a cos( b Multiplying the top and bottom by cos(acos(b sin( a sin( b cos( a cos( b sin( a sin( b + cos( acos( b cos( a cos( b Distributing and simplifying sin( a sin( b cos( acos( b cos( a cos( b
40 Chapter 7 sin( acos( a + sin( bcos( b sin( acos( a sin( bcos( b From above, we recognize this sin( a + b sin( a b Establishing the identity These identities can also be used to solve equations. Example 4 Solve sin( x sin(x + cos( xcos(x. By recognizing the left side of the equation as the result of the difference of angles identity for cosine, we can simplify the equation sin( x sin(x + cos( xcos(x Apply the difference of angles identity cos( x x cos( x Use the negative angle identity cos( x Since this is a special cosine value we recognize from the unit circle, we can quickly write the answers: π x + πk 6, where k is an integer π x + πk 6 Combining Waves of Equal Period A sinusoidal function of the form f ( x Asin( Bx + C can be rewritten using the sum of angles identity. Example 5 π Rewrite f ( x 4sin x + as a sum of sine and cosine.
Section 7. Addition and Subtraction Identities 4 Using the sum of angles identity π 4sin x + π π 4 sin( x cos + cos( x sin Evaluate the sine and cosine 4 sin x + cos x Distribute and simplify ( ( ( x cos( x sin + Notice that the result is a stretch of the sine added to a different stretch of the cosine, but both have the same horizontal compression, which results in the same period. We might ask now whether this process can be reversed can a combination of a sine and cosine of the same period be written as a single sinusoidal function? To explore this, we will look in general at the procedure used in the example above. f ( x Asin( Bx + C Use the sum of angles identity A ( sin( Bxcos( C + cos( Bxsin( C Distribute the A A sin( Bxcos( C + Acos( Bxsin( C Rearrange the terms a bit A cos( Csin( Bx + Asin( Ccos( Bx Based on this result, if we have an expression of the form m sin( Bx + ncos( Bx, we could rewrite it as a single sinusoidal function if we can find values A and C so that m sin( Bx + ncos( Bx A cos( Csin( Bx + Asin( Ccos( Bx, which will require that: m cos( C m Acos( C which can be rewritten as A n Asin( C n sin( C A To find A, m + n ( Acos( C + ( Asin( C A cos ( C + A sin ( C A ( cos ( C + sin ( C Apply the Pythagorean Identity and simplify A Rewriting a Sum of Sine and Cosine as a Single Sine To rewrite m sin( Bx + n cos( Bx as A sin( Bx + C m n A m + n, cos( C, and sin( C A A
4 Chapter 7 We can use either of the last two equations to solve for possible values of C. Since there will usually be two possible solutions, we will need to look at both to determine which quadrant C is in and determine which solution for C satisfies both equations. Example 6 Rewrite 4 sin(x 4cos(x as a single sinusoidal function. Using the formulas above, A ( 4 + ( 4 6 + 6 64, so A 8. Solving for C, 4 π π cos( C, so C or C. 8 6 6 4 π However, since sin( C, the angle that works for both is C 8 6 Combining these results gives us the expression π 8sin x + 6 Try it Now. Rewrite sin(5x + cos(5x as a single sinusoidal function. Rewriting a combination of sine and cosine of equal periods as a single sinusoidal function provides an approach for solving some equations. Example 7 Solve sin(x + 4cos(x to find two positive solutions. To approach this, since the sine and cosine have the same period, we can rewrite them as a single sinusoidal function. A ( + ( 4 5, so A 5 cos( C, so cos C 0. 97 or C π 0.97 5. 56 5 5 4 Since sin( C, a positive value, we need the angle in the first quadrant, C 0.97. 5 Using this, our equation becomes 5 sin( x + 0.97 Divide by 5 sin ( x + 0.97 Make the substitution u x + 0.97 5
Section 7. Addition and Subtraction Identities 4 sin ( u 5 The inverse gives a first solution sin u 0.0 5 By symmetry, the second solution is u π 0.0.940 A third solution is u π + 0.0 6.485 Undoing the substitution, we can find two positive solutions for x. x + 0.97 0.0 or x + 0.97. 940 or x + 0.97 6. 485 x 0.76 x. 0 x 5. 558 x 0.6 x. 007 x. 779 Since the first of these is negative, we eliminate it and keep the two positive solutions, x.007 and x. 779. The Product-to-Sum and Sum-to-Product Identities Identities The Product-to-Sum Identities sin( αcos( β ( sin( α + β + sin( α β sin( αsin( β ( cos( α β cos( α + β cos( αcos( β cos( α + β + cos( α β ( We will prove the first of these, using the sum and difference of angles identities from the beginning of the section. The proofs of the other two identities are similar and are left as an exercise. Proof of the product-to-sum identity for sin(αcos(β Recall the sum and difference of angles identities from earlier sin( α + β sin( αcos( β + cos( αsin( β sin( α β sin( αcos( β cos( αsin( β Adding these two equations, we obtain sin( α + β + sin( α β sin( αcos( β Dividing by, we establish the identity sin( αcos( β sin( α + β + sin( α β (
44 Chapter 7 Example 8 Write sin( t sin(4t as a sum or difference. Using the product-to-sum identity for a product of sines sin( t sin(4t ( cos(t 4t cos(t + 4t ( cos( t cos(6t If desired, apply the negative angle identity ( cos(t cos(6t Distribute cos(t cos(6t Try it Now π π 4. Evaluate cos. Identities The Sum-to-Product Identities u + v u v sin( u + sin( v sin u v u + v sin( u sin( v sin cos u + cos v u + v u v cos cos u cos v u + v u v sin sin ( ( ( ( We will again prove one of these and leave the rest as an exercise. Proof of the sum-to-product identity for sine functions We begin with the product-to-sum identity sin( αcos( β ( sin( α + β + sin( α β We define two new variables: u α + β v α β
Section 7. Addition and Subtraction Identities 45 Adding these equations yields u + v α, giving Subtracting the equations yields u v β, or u + v α u v β Substituting these expressions into the product-to-sum identity above, u + v u v sin ( sin( u + sin( v Multiply by on both sides u + v u v sin sin( u + sin( v Establishing the identity Example 9 Evaluate cos( 5 cos(75. Using the sum-to-product identity for the difference of cosines, cos( 5 cos(75 5 + 75 5 75 sin sin Simplify ( 45 sin( sin 0 Evaluate Example 0 cos(4t cos(t Prove the identity tan( t. sin(4t + sin(t Since the left side seems more complicated, we can start there and simplify. cos(4t cos(t Using the sum-to-product identities sin(4t + sin(t 4t + t 4t t sin sin Simplify 4t + t 4t t sin sin( t sin( t Simplify further sin( t cos( t sin( t Rewrite as a tangent cos( t tan(t Establishing the identity
46 Chapter 7 Try it Now 5. Notice that, using the negative angle identity, sin( u sin( v sin( u + sin( v. Use this along with the sum of sines identity to prove the sum-to-product identity for sin u sin v. ( ( Example sin πt + sin πt cos( πt for all solutions with 0 t <. Solve ( ( In an equation like this, it is not immediately obvious how to proceed. One option would be to combine the two sine functions on the left side of the equation. Another would be to move the cosine to the left side of the equation, and combine it with one of the sines. For no particularly good reason, we ll begin by combining the sines on the left side of the equation and see how things work out. sin ( πt + sin ( πt cos( πt Apply the sum to product identity on the left πt+ πt πt πt sin cos( π t Simplify sin πt cos πt cos( πt Apply the negative angle identity ( ( ( ( ( ( ( t ( t sin πt cos πt cos( πt Rearrange the equation to be 0 on one side sin πt cos πt cos( πt 0 Factor out the cosine ( cos π sin π 0 Using the Zero Product Theorem we know that at least one of the two factors must be π cos π t, has period P, so the solution interval of π zero. The first factor, ( 0 t < represents one full cycle of this function. cos( π t 0 Substitute u π t cos ( u 0 On one cycle, this has solutions π π u or u Undo the substitution π π t, so t π π t, so t π sin π, has period of P, so the solution interval π 0 t < contains two complete cycles of this function. The second factor, ( t
Section 7. Addition and Subtraction Identities 47 ( t sin π 0 Isolate the sine sin ( π t u π t sin( u On one cycle, this has solutions π 5π u or u 6 6 On the second cycle, the solutions are π π u π + or 6 6 5π 7π u π + Undo the substitution 6 6 π π t, so t 6 5π 5 π t, so t 6 π π t, so t 6 7π 7 π t, so t 6 Altogether, we found six solutions on 0 t <, which we can confirm by looking at the graph. 5 7 t,,,,, Important Topics of This Section The sum and difference identities Combining waves of equal periods Product-to-sum identities Sum-to-product identities Completing proofs Try it Now Answers cos( α + β cos( α ( β cos( αcos( β + sin( αsin( β. cos( αcos( β + sin( α( sin( β cos( αcos( β sin( αsin( β
48 Chapter 7. 6 4. π 6sin 5x + 4 4. 4 5. sin( u sin( v Use negative angle identity for sine sin( u + sin( v Use sum-to-product identity for sine u + ( v u ( v sin Eliminate the parenthesis u v u + v sin Establishing the identity
Section 7. Addition and Subtraction Identities 49 Section 7. Exercises Find an exact value for each of the following. sin 95. cos(65 4. cos(45. sin ( 75. ( 7π π 5. cos 6. cos 7. 5π sin 8. π sin Rewrite in terms of sin ( x and cos( x. 9. π sin x + 6 0. π sin x 4. 5π cos x 6. π cos x + Simplify each expression. π. csc t π 4. sec w π 5. cot x π 6. tan x Rewrite the product as a sum. 6sin 6 sin 7. ( x ( x 8. 0 cos( 6t cos( 6t 9. sin ( 5x cos( x 0. 0cos( 5x sin ( 0x Rewrite the sum as a product. cos 6t cos 4t. ( + (. cos( 6u + cos( 4u. sin ( x + sin ( 7x 4. sin ( h + sin ( h sin a 5. Given sin ( a and cos( π b, with a and b both in the interval, π 4 : cos a b a. Find ( + b b. Find ( 4 5 sin a 6. Given sin ( a and cos( π b, with a and b both in the interval 0, : cos a+ b a. Find ( b b. Find ( Solve each equation for all solutions. sin x cos 6x cos x sin 6 x 0.9 7. ( ( ( ( 8. ( x ( x ( x ( x 9. cos( x cos( x + sin ( x sin ( x sin 6 cos cos 6 sin 0. 0. cos( 5x cos( x sin ( 5x sin ( x
40 Chapter 7 Solve each equation for all solutions. cos 5x cos x. ( (. sin ( 5x sin ( x. cos( 6θ cos( θ sin ( 4θ 4. cos( 8θ cos( θ sin ( 5θ Rewrite as a single function of the form Asin( Bx + C. 4sin x 6cos x sin x 5. ( ( 6. ( 5cos( x 7. 5sin ( x + cos( x 8. sin ( 5x + 4cos( 5x Solve for the first two positive solutions. 5sin x cos x 9. ( + ( 40. sin ( x + cos( x 4. sin ( x 5cos( x 4. ( x ( x sin 4 cos 4 Simplify. sin 7t + sin 5t 4. cos 7t + cos 5t ( ( ( ( 44. ( t ( t ( t + ( t sin 9 sin cos 9 cos Prove the identity. π tan ( x + 44. tan x + 4 tan x ( π tan ( t 45. tan t 4 + tan ( t 46. cos( a+ b + cos( a b cos ( a cos ( b cos( a+ b tan ( a tan ( b 47. cos( a b + tan ( a tan ( b tan ( a+ b sin ( a cos( a + sin ( b cos( b 48. tan ( a b sin ( a cos( a sin ( b cos( b 49. sin ( a+ b sin ( a b cos( b cos( a sin ( x + sin ( y 50. tan ( x+ y cos( x + cos( y cos( a+ b 5. tan ( a tan ( b cos( a cos( b cos x+ y cos x y cos x sin y 5. ( (