5.3 Sum and Difference Identities

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1 SECTION 5.3 Sum and Difference Identities Sum and Difference Identities Wat you ll learn about Cosine of a Difference Cosine of a Sum Sine of a Difference or Sum Tangent of a Difference or Sum Verifying a Sinusoid Algebraically... and wy Tese identities provide clear examples of ow different te algebra of functions can be from te algebra of real numbers. Cosine of a Difference Tere is a powerful instinct in all of us to believe tat all functions obey te following law of additivity: ƒ1u + v2 = ƒ1u) + ƒ1v2 In fact, very few do. If tere were a all of fame for algebraic blunders, te following would probably be te first two inductees: 1u + v2 2 = u 2 + v 2 1u + v = 1u + 1v So, before we derive te true sum formulas for sine and cosine, let us clear te air wit te following exploration. EXPLORATION 1 Getting Past te Obvious but Incorrect Formulas 1. Let u = p and v = p/2. Find sin 1u + v2. Find sin 1u2 + sin 1v2. Does sin 1u + v2 = sin 1u2 + sin 1v2? 2. Let u = 0 and v = 2p. Find cos 1u + v2. Find cos 1u2 + cos 1v2. Does cos 1u + v2 = cos 1u2 + cos 1v2? 3. Find your own values of u and v tat will confirm tat tan 1u + v2 Z tan 1u2 + tan 1v2. We could also sow easily tat cos 1u - v) Z cos 1u2 - cos 1v2 and sin 1u - v2 Z sin 1u2 - sin 1v2. As you migt expect, tere are formulas for sin 1u v2, cos 1u v2, and tan 1u v2, but Exploration 1 sows tat tey are not te ones our instincts would suggest. In a sense, tat makes tem all te more interesting. We will derive tem all, beginning wit te formula for cos 1u - v2. Figure 5.9a on te next page sows angles u and v in standard position on te unit circle, determining points A and B wit coordinates 1cos u, sin u2 and 1cos v, sin v 2, respectively. Figure 5.9b sows te triangle ABO rotated so tat te angle u = u - v is in standard position. Te angle u determines point C wit coordinates 1cos u, sin u2. Te cord opposite angle u as te same lengt in bot circles, even toug te coordinatization of te endpoints is different. Use te distance formula to find te lengt in eac case, and set te formulas equal to eac oter: AB = CD 21cos v - cos u sin v - sin u2 2 = 21cos u sin u Square bot sides to eliminate te radical and expand te binomials to get cos 2 u - 2 cos u cos v + cos 2 v + sin 2 u - 2 sin u sin v + sin 2 v = cos 2 u - 2 cos u sin 2 u 1cos 2 u + sin 2 u2 + 1cos 2 v + sin 2 v2-2 cos u cos v - 2 sin u sin v = 1cos 2 u + sin 2 u cos u 2-2 cos u cos v - 2 sin u sin v = 2-2 cos u cos u cos v + sin u sin v = cos u

2 22 CHAPTER 5 Analytic Trigonometry y Finally, since u = u - v, we can write cos 1u - v2 = cos u cos v + sin u sin v. B(cos v, sin v) A(cos u, sin u) y O (a) C(cos θ, sin θ) θ v u O D(1, 0) FIGURE 5.9 Angles u and v are in standard position in (a), wile angle u = u - v is in standard position in. Te cords sown in te two circles are equal in lengt. x x EXAMPLE 1 Using te Cosine-of-a-Difference Identity Find te exact value of cos 15 witout using a calculator. SOLUTION Te trick is to write cos 15 as cos ; ten we can use our knowledge of te special angles. cos 15 = cos = cos 5 cos 30 + sin 5 sin 30 Cosine difference identity = a 2 ba13 2 b + a 2 ba1 2 b 16 + = Now try Exercise 5. Cosine of a Sum Now tat we ave te formula for te cosine of a difference, we can get te formula for te cosine of a sum almost for free by using te odd-even identities. cos 1u + v2 = cos 1u - 1-v22 = cos u cos 1-v2 + sin u sin 1-v2 = cos u cos v + sin u 1-sin v2 = cos u cos v - sin u sin v Cosine difference identity Odd-even identities We can combine te sum and difference formulas for cosine as follows: Cosine of a Sum or Difference (Note te sign switc in eiter case.) cos 1u v2 = cos u cos v < sin u sin v We pointed out in Section 5.1 tat te cofunction identities would be easier to prove wit te results of Section 5.3. Here is wat we mean. EXAMPLE 2 Confirming Cofunction Identities Prove te identities (a) cos 11p/22 - x2 = sin x and sin 11p/22 - x2 = cos x. SOLUTION (a) cos a p 2 - xb = cos a p 2 b cos x + sin a p 2 b sin x = 0 # cos x + 1 # sin x = sin x sin a p 2 - xb = cos a p 2 - a p 2 - xbb = cos 10 + x2 = cos x Cosine sum identity sin u = cos 11p/22 - u2 by previous proof Now try Exercise 1.

3 SECTION 5.3 Sum and Difference Identities 23 Sine of a Difference or Sum We can use te cofunction identities in Example 2 to get te formula for te sine of a sum from te formula for te cosine of a difference. sin 1u + v2 = cos a p 2-1u + v2b Cofunction identity = cos aa p 2 - ub - vb A little algebra = cos a p 2 - ub cos v + sin a p 2 = sin u cos v + cos u sin v - ub sin v Cosine difference identity Cofunction identities Ten we can use te odd-even identities to get te formula for te sine of a difference from te formula for te sine of a sum. sin 1u - v2 = sin 1u + 1-v22 = sin u cos 1-v2 + cos u sin 1-v2 = sin u cos v + cos u 1-sin v2 = sin u cos v - cos u sin v A little algebra Sine sum identity Odd-even identities We can combine te sum and difference formulas for sine as follows: Sine of a Sum or Difference sin 1u v2 = sin u cos v cos u sin v (Note tat te sign does not switc in eiter case.) EXAMPLE 3 Using te Sum/Difference Formulas Write eac of te following expressions as te sine or cosine of an angle. (a) sin 22 cos 13 + cos 22 sin 13 cos p 3 cos p + sin p 3 sin p (c) sin x sin 2x - cos x cos 2x SOLUTION Te key in eac case is recognizing wic formula applies. (Indeed, te real purpose of suc exercises is to elp you remember te formulas.) (a) sin 22 cos 13 + cos 22 sin 13 = sin = sin 35 cos p 3 cos p + sin p 3 sin p Recognizing sine of sum formula Recognizing cosine of difference formula = cos a p 3 - p b (c) = cos p sin x sin 2x - cos x cos 2x = -1cos x cos 2x - sin x sin 2x2 = -cos1x + 2x) = -cos 3x Recognizing opposite of cos sum formula Applying formula Now try Exercise 19.

4 2 CHAPTER 5 Analytic Trigonometry If one of te angles in a sum or difference is a quadrantal angle (tat is, a multiple of 90 or of p/2 radians), ten te sum-difference identities yield single-termed expressions. Since te effect is to reduce te complexity, te resulting identity is called a reduction formula. EXAMPLE Proving Reduction Formulas Prove te reduction formulas: (a) sin1x + p2 = -sin x cos ax + 3p 2 b = sin x SOLUTION (a) sin 1x + p2 = sin x cos p + cos x sin p = sin x # 1- + cos x # 0 = -sin x cos ax + 3p 2 b = cos x cos 3p 2 - sin x sin 3p 2 = cos x # 0 - sin x # 1- = sin x Now try Exercise 23. Tangent of a Difference or Sum We can derive a formula for tan 1u v2 directly from te corresponding formulas for sine and cosine, as follows: tan 1u v2 = sin 1u v2 cos 1u v2 = sin u cos v cos u sin v cos u cos v sin u sin v Tere is also a formula for tan 1u v2 tat is written entirely in terms of tangent functions: tan u tan v tan 1u v2 = 1 tan u tan v We will leave te proof of te all-tangent formula to te exercises. EXAMPLE 5 Proving a Tangent Reduction Formula Prove te reduction formula: tan 1u - 13p/222 = -cot u. SOLUTION We can t use te all-tangent formula (Do you see wy?), so we convert to sines and cosines. tan au - 3p 2 b = sin 1u - 13p/222 cos 1u - 13p/222 = sin u # 0 - cos u # 1- cos u # 0 + sin u # 1- = -cot u Verifying a Sinusoid Algebraically = sin u cos 13p/22 - cos u sin 13p/22 cos u cos 13p/22 + sin u sin 13p/22 Now try Exercise 39. Example 7 of Section.6 asked us to verify tat te function ƒ1x2 = 2 sin x + 5 cos x is a sinusoid. We solved grapically, concluding tat ƒ1x2 L 5.39 sin 1x

5 SECTION 5.3 Sum and Difference Identities 25 We now ave a way of solving tis kind of problem algebraically, wit exact values for te amplitude and pase sift. Example 6 illustrates te tecnique. EXAMPLE 6 Expressing a Sum of Sinusoids as a Sinusoid Express ƒ1x2 = 2 sin x + 5 cos x as a sinusoid in te form ƒ1x2 = a sin 1bx + c2. SOLUTION Since a sin 1bx + c2 = a 1sin bx cos c + cos bx sin c2, we ave Comparing coefficients, we see tat b = 1 and tat a cos c = 2 and a sin c = 5. We can solve for a as follows: 1a cos c a sin c2 2 = a 2 cos 2 c + a 2 sin 2 c = 29 a 2 1cos 2 c + sin 2 c2 = 29 Pytogorean identity If we coose a to be positive, ten cos c = 2/9 and sin c = 5/9. We can identify an acute angle c wit tose specifications as eiter cos -1 /92 or sin -1 15/ 92, wic are equal. So, an exact sinusoid for ƒ is ƒ1x2 = 2 sin x + 5 cos x = a sin 1bx + c2 2 sin x + 5 cos x = a 1sin bx cos c + cos bx sin c2 = 1a cos c2 sin bx + 1a sin c2 cos bx. a 2 = 29 a = 9 = 9 sin 1x + cos -1 /922 or 9 sin 1x + sin -1 15/922 Now try Exercise 3. QUICK REVIEW 5.3 (For elp, go to Sections.2 and 5.1.) Exercise numbers wit a gray background indicate problems tat te autors ave designed to be solved witout a calculator. In Exercises 1 6, express te angle as a sum or difference of special angles (multiples of 30, 5, p/6, or p/). Answers are not unique p/ 5. 5p/ 6. 7p/ In Exercises 7 10, tell weter or not te identity ƒ1x + y2 = ƒ1x2 + ƒ1y2 olds for te function ƒ. 7. ƒ1x2 = ln x 8. ƒ1x2 = e x 9. ƒ1x2 = 32x 10. ƒ1x2 = x + 10 SECTION 5.3 EXERCISES In Exercises 1 10, use a sum or difference identity to find an exact value. 1. sin tan sin 75. cos cos p 6. sin 7p 7. tan 5p 8. tan 11p 9. cos 7p 10. sin -p In Exercises 11 22, write te expression as te sine, cosine, or tangent of an angle. 11. sin 2 cos 17 - cos 2 sin 17. cos 9 cos 18 + sin 9 sin sin p 5 cos p 2 + sin p 2 cos p 5 1. sin p 3 cos p 7 - sin p 7 cos p 3 tan 19 + tan tan 19 tan 7

6 26 CHAPTER 5 Analytic Trigonometry tan 1p/52 - tan 1p/ tan 1p/52 tan 1p/ cos p 7 cos x + sin p 7 sin x 18. cos x cos p 7 - sin x sin p sin 3x cos x - cos 3x sin x 20. cos 7y cos 3y - sin 7y sin 3y tan 2y + tan 3x tan 2y tan 3x tan 3a - tan 2b 1 + tan 3a tan 2b In Exercises 35 and 36, use sum or difference identities (and not your graper) to solve te equation exactly. 35. sin 2x cos x = cos 2x sin x 36. cos 3x cos x = sin 3x sin x In Exercises 37 2, prove te reduction formula. 37. sin a p - ub = cos u 38. tan a p - ub = cot u cot a p - ub = tan u 0. sec a p - ub = csc u 2 2 In Exercises 23 30, prove te identity. 1. csc a p - ub = sec u 2. 2 cos ax + p 2 b = -sin x 23. sin ax - p 2. tan ax - p 2 b = -cos x 2 b = -cot x 25. cos ax - p 2 b = sin x 26. cos ca p - xb - y d = sin 1x + y sin ax + p 6 b = 13 2 sin x cos x cos ax - p b = 1cos x + sin x2 2 tan au + p b = 1 + tan u 1 - tan u 30. cos au + p 2 b = -sin u In Exercises 31 3, matc eac grap wit a pair of te following equations. Use your knowledge of identities and transformations, not your graper. (a) y = cos 13-2x2 y = sin x cos 1 + cos x sin 1 (c) y = cos 1x - 32 (d) y = sin (2x - 52 (e) y = cos x cos 3 + sin x sin 3 (f) y = sin 1x + (g) y = cos 3 cos 2x + sin 3 sin 2x () y = sin 2x cos 5 - cos 2x sin [ 2π, 2 π ] by [ 1, 1] [ 2π, 2 π ] by [ 1, 1] In Exercises 3 6, express te function as a sinusoid in te form y = a sin 1bx + c2. 3. y = 3 sin x + cos x. y = 5 sin x - cos x 5. y = cos 3x + 2 sin 3x 6. y = 3 cos 2x - 2 sin 2x In Exercises 7 55, prove te identity. 7. sin 1x - y2 + sin 1x + y2 = 2 sin x cos y 8. cos 1x - y2 + cos 1x + y2 = 2 cos x cos y 9. cos 3x = cos 3 x - 3 sin 2 x cos x 50. sin 3u = 3 cos 2 u sin u - sin 3 u 51. cos 3x + cos x = 2 cos 2x cos x 52. sin x + sin 2x = 2 sin 3x cos x 53. tan 1x + y2 tan 1x - y2 = tan2 x - tan 2 y 1 - tan 2 x tan 2 y 5. tan 5u tan 3u = tan2 u - tan 2 u 1 - tan 2 u tan 2 u 55. sin 1x + y2 sin 1x - y2 Standardized Test Questions 56. True or False If A and B are supplementary angles, ten cos A + cos B = 0. Justify your answer. 57. True or False If cos A + cos B = 0, ten A and B are supplementary angles. Justify your answer. You sould answer tese questions witout using a calculator. 58. Multiple Coice If cos A cos B = sin A sin B, ten cos 1A + B2 = (A) 0. (B) 1. (C) cos A + cos B. = 1tan x + tan y2 1tan x - tan y2 (E) cos A cos B + sin A sin B. (D) cos B + cos A. 59. Multiple Coice Te function y = sin x cos 2x + cos x sin 2x as amplitude (A) 1. (B) 1.5. (C) 2. (D) 3. (E) 6. [ 2π, 2 π ] by [ 1, 1] [ 2π, 2 π ] by [ 1, 1]

7 SECTION 5.3 Sum and Difference Identities Multiple Coice 1 (A). (C) sin 15 = 71. tan A + tan B + tan C = tan A tan B tan C (B) 72. (D) (E). 61. Multiple Coice A function wit te property ƒ1 + ƒ2 ƒ = is 1 - ƒ1ƒ2 (A) ƒ1x2 = sin x. (B) ƒ1x2 = tan x. (C) ƒ1x2 = sec x. (D) ƒ1x2 = e x. (E) ƒ1x2 = -1. cos A cos B cos C - sin A sin B cos C - sin A cos B sin C - cos A sin B sin C = Writing to Learn Te figure sows graps of y 1 = cos 5x cos x and y 2 = -sin 5x sin x in one viewing window. Discuss te question, How many solutions are tere to te equation cos 5x cos x = -sin 5x sin x in te interval 3-2p, 2p? Give an algebraic argument tat answers te question more convincingly tan te grap does. Ten support your argument wit an appropriate grap. Explorations 62. Prove te identity tan 1u + v2 = 63. Prove te identity tan 1u - v2 = 6. Writing to Learn Explain wy te identity in Exercise 62 cannot be used to prove te reduction formula tan 1x + p/22 = -cot x. Ten prove te reduction formula. 65. Writing to Learn Explain wy te identity in Exercise 63 cannot be used to prove te reduction formula tan 1x - 3p/22 = -cot x. Ten prove te reduction formula. 66. An Identity for Calculus Prove te following identity, wic is used in calculus to prove an important differentiation formula. sin 1x sin x = sin x a cos - 1 b + cos x sin 67. An Identity for Calculus Prove te following identity, wic is used in calculus to prove anoter important differentiation formula. cos 1x cos x = cos x a cos - 1 b - sin x sin 68. Group Activity Place 2 points evenly spaced around te unit circle, starting wit te point 11, 02. Using only your knowledge of te special angles and te sum and difference identities, work wit your group to find te exact coordinates of all 2 points. Extending te Ideas tan u + tan v 1 - tan u tan v. tan u - tan v 1 + tan u tan v. In Exercises 69 72, assume tat A, B, and C are te tree angles of some ABC. 1Note, ten, tat A + B + C = p.2 Prove te following identities. 69. sin 1A + B2 = sin C 70. cos C = sin A sin B - cos A cos B [ 2π, 2 π ] by [ 1, 1] 7. Harmonic Motion Alternating electric current, an oscillating spring, or any oter armonic oscillator can be modeled by te equation were T is te time for one period and d is te pase constant. Sow tat tis motion can also be modeled by te following sum of cosine and sine, eac wit zero pase constant: were a 1 = a cos d and a 2 = -a sin d. 75. Magnetic Fields A magnetic field B can sometimes be modeled as te sum of an incident and a reflective field as B = B in + B ref, were B in = E 0 c B ref = E 0 c x = a cos a 2p T t + db, a 1 cos a 2p T bt + a 2 sin a 2p T b t, vx cos avt - and c b, cos avt + vx c b. E 0 vx Sow tat B = 2 cos vt cos c c.