Sum and Difference of Angles Formulas

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  Precalculus, Enhanced Edition

  • David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  599 9.1 The Addition Formulas 9.2 The Double-Angle Formulas 9.3 The Product-to-Sum and Sum-to-Product Formulas 9.4 Trigonometric Equations 9.5 The Inverse Trigonometric Functions . . . through the improvements in algebraic sym-bolism . . . trigonometry became, in the 17th century, largely an analytic science, and as such it entered the field of higher mathematics. —David Eugene Smith in History of Mathematics (New York: Ginn and Company, 1925) This chapter is devoted to some of the more algebraic (as opposed to geometric) portions of trigonometry. In Section 9.1 we develop six basic identities known as the addition formulas . Then, in the next two sections, we consider a number of identities that follow directly from these addition formulas. In Section 9.4 we return to a topic that was introduced briefly in the previous chapter: solving trigonometric equations. In solving many of these equations, we’ll use the identities developed in the previous sections. We also make use of the inverse trigonometric functions that were intro-duced in the previous chapter (in Section 8.2). In the last section of this chapter, Section 9.5, we take a more careful look at the inverse trigonometric functions and their properties. CHAPTER 9 Analytical Trigonometry It has long been recognized that the addition formulas are the heart of trigonometry. Indeed, Professor Rademacher and others have shown that the entire body of trigonometry can be derived from the assumption that there exist functions S and C such that 1. S ( x y ) S ( x ) C ( y ) C ( x ) S ( y ) 2. C ( x y ) C ( x ) C ( y ) S ( x ) S ( y ) 3. —Professor Frederick H. Young, “The Addition Formulas,” The Mathematics Teacher, vol. L (1957), pp. 45–48. lim x S 0 S ( x ) x 1 9.1 THE ADDITION FORMULAS For any real numbers r , s , and t it is always true that r ( s t ) rs rt . This is the so-called distributive law for real numbers. If f is a function, however, it is not true in general that f ( s t ) f ( s ) f ( t ).

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Section 2 ◆ Sum or Difference of Two Angles 469 16–2 Sum or Difference of Two Angles Having established the fundamental trignometric identities, let us now move on to others that are very useful in technical work, starting with the trig functions of the sum or difference of two angles. We now wish to derive a formula for the sine of the sum of two angles, for example, ■ Exploration: Try this. Use your calculator to evaluate Does the sum of equal the sine of ■ The sine of the sum of two angles is not the sum of the sine of each angle. We will show that We start by drawing two positive acute angles, and (Fig. 16–6), small enough so that their sum is also acute. From any point P on the terminal side of we draw perpendicular AP to the x axis and draw perpendicular BP to line OB. Since the angle between two lines equals the angle between the perpendiculars to those two lines (can you demonstrate that this is true?), we note that angle APB is equal to Then But in triangle OBC, and in triangle PBD, Substituting, we obtain But in triangle OPB, and Thus, Convince yourself that this is true by using your calculator to compute sin (45º + 30º) using this identity, and comparing it with sin (75º). sin (a  b)  sin a cos b  cos a sin b PB OP  sin b OB OP  cos b sin (a  b)  OB sin a OP  PB cos a OP PD  PB cos a BC  OB sin a  BC OP  PD OP sin (a  b)  AP OP  AD  PD OP  BC  PD OP a. b (a  b) b a sin (a  b)  sin a cos b  cos a sin b sin (a  b)  sin a  sin b Common Error 50°? sin 20° and sin 30° sin 50° sin 30° sin 20° sin (a  b). O y x B P C A D  +     FIGURE 16–6 470 Chapter 16 ◆ Trigonometric Identities and Equations Cosine of the Sum of Two Angles Again using Fig.

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  [Hint: Use tan1 s  t 2  sin1 s  t 2 cos 1 s  t 2 and divide the numerator and denominator by cos s cos t.] 7.3 DOUBLE-ANGLE, HALF-ANGLE, AND PRODUCT-SUM FORMULAS ■ Double-Angle Formulas ■ Half-Angle Formulas ■ Evaluating Expressions Involving Inverse Trigonometric Functions ■ Product-Sum Formulas The identities we consider in this section are consequences of the addition formulas. The Double-Angle Formulas allow us to find the values of the trigonometric functions at 2x from their values at x. The Half-Angle Formulas relate the values of the trigono- metric functions at 1 2 x to their values at x. The Product-Sum Formulas relate products of sines and cosines to sums of sines and cosines. ■ Double-Angle Formulas The formulas in the box on the next page are immediate consequences of the addition formulas, which we proved in Section 7.2. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 554 CHAPTER 7 ■ Analytic Trigonometry DOUBLE-ANGLE FORMULAS Formula for sine: sin 2 x  2 sin x cos x Formulas for cosine: cos 2 x  cos 2 x  sin 2 x  1  2 sin 2 x  2 cos 2 x  1 Formula for tangent: tan 2 x  2 tan x 1  tan 2 x The proofs for the formulas for cosine are given here. You are asked to prove the remaining formulas in Exercises 35 and 36. Proof of Double-Angle Formulas for Cosine cos 2x  cos 1 x  x 2  cos x cos x  sin x sin x  cos 2 x  sin 2 x The second and third formulas for cos 2 x are obtained from the formula we just proved and the Pythagorean identity.

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  Algebra & Trig

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 7.5 Multiple-Angle and Product-to-Sum Formulas 545 GO DIGITAL Product-to-Sum and Sum-to-Product Formulas Each of the product-to-sum formulas listed below can be proved using the sum and difference formulas discussed in the preceding section. Product-to-sum formulas are used in calculus to solve problems involving the products of sines and cosines of two different angles. EXAMPLE 7 Writing Products as Sums Rewrite the product cos 5x sin 4x as a sum or difference. Solution Use the product-to-sum formula for cos u sin v with u = 5x and v = 4x. cos 5x sin 4x = 1 2 [sin(5x + 4x) - sin(5x - 4x)] = 1 2 sin 9x - 1 2 sin x Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Rewrite the product sin 5x cos 3x as a sum or difference. Occasionally, it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be accomplished with the sum-to-product formulas listed below. For a proof of the sum-to-product formulas, see Proofs in Mathematics on page 556. Product-to-Sum Formulas sin u sin v = 1 2 [cos(u - v) - cos(u + v)] cos u cos v = 1 2 [cos(u - v) + cos(u + v)] sin u cos v = 1 2 [sin(u + v) + sin(u - v)] cos u sin v = 1 2 [sin(u + v) - sin(u - v)] Sum-to-Product Formulas sin u + sin v = 2 sin ( u + v 2 ) cos ( u - v 2 ) sin u - sin v = 2 cos ( u + v 2 ) sin ( u - v 2 ) cos u + cos v = 2 cos ( u + v 2 ) cos ( u - v 2 ) cos u - cos v = -2 sin ( u + v 2 ) sin ( u - v 2 ) Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Trigonometry

  • Ron Larson(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Summarize (Section 2.4) 1. State the sum and difference formulas for sine, cosine, and tangent (page 236). For examples of using the sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations, see Examples 1–8. Solving a Trigonometric Equation Find all solutions of sin[x + (π H208624)] + sin[x - (π H208624)] = -1 in the interval [0, 2π ). Graphical Solution - 1 0 3 2 Zero X=3.9269908 Y=0 π The x-intercepts are x ≈ 3.927 and x ≈ 5.498. y = sin x + + sin x - + 1 π 4 ( π 4 ( ( ( Use the x-intercepts of y = sin[x + (π H208624)] + sin[x - (π H208624)] + 1 to conclude that the approximate solutions in the interval [0, 2π ) are x ≈ 3.927 ≈ 5π 4 and x ≈ 5.498 ≈ 7π 4 . Algebraic Solution Use sum and difference formulas to rewrite the equation. sin x cos π 4 + cos x sin π 4 + sin x cos π 4 - cos x sin π 4 = -1 2 sin x cos π 4 = -1 2(sin x) parenleft.alt4 radical.alt22 2 parenright.alt4 = -1 sin x = - 1 radical.alt22 sin x = - radical.alt22 2 So, the solutions in the interval [0, 2π ) are x = 5π 4 and x = 7π 4 . Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Find all solutions of sin[x + (π H208622)] + sin[x - (3π H208622)] = 1 in the interval [0, 2π ). Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2.4 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. 240 Chapter 2 Analytic Trigonometry Vocabulary: Fill in the blank. 1. sin(u - v) = ______ 2. cos(u + v) = ______ 3. tan(u + v) = ______ 4.

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  Precalculus with Limits

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5.5 Multiple-Angle and Product-to-Sum Formulas 385 GO DIGITAL Product-to-Sum and Sum-to-Product Formulas Each of the product-to-sum formulas listed below can be proved using the sum and difference formulas discussed in the preceding section. Product-to-sum formulas are used in calculus to solve problems involving the products of sines and cosines of two different angles. EXAMPLE 7 Writing Products as Sums Rewrite the product cos 5x sin 4x as a sum or difference. Solution Use the product-to-sum formula for cos u sin v with u = 5x and v = 4x. cos 5x sin 4x = 1 2 [sin(5x + 4x) - sin(5x - 4x)] = 1 2 sin 9x - 1 2 sin x Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Rewrite the product sin 5x cos 3x as a sum or difference. Occasionally, it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be accomplished with the sum-to-product formulas listed below. For a proof of the sum-to-product formulas, see Proofs in Mathematics on page 396. Product-to-Sum Formulas sin u sin v = 1 2 [cos(u - v) - cos(u + v)] cos u cos v = 1 2 [cos(u - v) + cos(u + v)] sin u cos v = 1 2 [sin(u + v) + sin(u - v)] cos u sin v = 1 2 [sin(u + v) - sin(u - v)] Sum-to-Product Formulas sin u + sin v = 2 sin ( u + v 2 ) cos ( u - v 2 ) sin u - sin v = 2 cos ( u + v 2 ) sin ( u - v 2 ) cos u + cos v = 2 cos ( u + v 2 ) cos ( u - v 2 ) cos u - cos v = -2 sin ( u + v 2 ) sin ( u - v 2 ) Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Trigonometry

  • Charles P. McKeague(Author)
  • 2020(Publication Date)
  • XYZ Textbooks

    (Publisher)

  260 Chapter 5 Identities and Formulas Th e formula for tan( A − B ) can be derived in a similar manner: tan( A − B ) = tan A − tan B _____________ 1 + tan A tan B To summarize: tan ( A + B ) = tan A + tan B _____________ 1 − tan A tan B tan ( A − B ) = tan A − tan B _____________ 1 + tan A tan B Example 7 If sin A = _ 3 5 with A in QI and cos B = − __ 5 13 with B in QIII, fin d tan ( A + B ) by using the formula tan ( A + B ) = tan A + tan B _____________ 1 − tan A tan B Solution Th e angles A and B as given here are the same ones used previously in Example 6. Looking over Example 6 again, w e fin d that tan A = 3 __ 4 tan B = 12 ___ 5 Th erefore, tan( A + B ) = tan A + tan B __________ 1 − tan A tan B ) = 3 __ 4 + 12 ___ 5 _ 1 − 3 __ 4 ∙ 12 ___ 5 = 15 __ 20 + 48 ___ 20 _ 1 − 9 ___ 5 = 63 ___ 20 _ − 4 ___ 5 = − 63 ___ 20 ∙  − 5 __ 4  = − 63 ___ 16 Which is the same result we obtained previously. 5.2 Sum and Difference Formulas 261 Example 8 Graph y = 4 sin 6 x cos 4 x − 4 cos 6 x sin 4 x from x = 0 to x = 2 π . Solution To write the equation in the form y = A sin Bx , we factor 4 from each term on the right and then apply the formula for sin ( A − B ) to the remaining expression to write it as a single trigonometric function. y = 4 sin 6 x cos 4 x − 4 cos 6 x sin 4 x = 4 sin (6 x − 4 x ) = 4 sin 2 x Th e graph is shown in Figure 2. y x 4 – 4 y = 4 sin 2 x Amplitude = 4 Period = π FIGURE 2 π 4 4 3 π π 4 5 π 2 3 π 4 7 π 2 π π 2 262 Problem Set 5.2 Vocabulary Use the vocabulary words below to fill in the blanks in the sentences. not sin A sin B cos A cos B 1. The statement sin ( A + B ) = sin A + sin B is an identity because it is not true for all values of A and B . 2. sin ( A + B ) = sin A cos B + sin B 3. sin ( A − B ) = cos B − sin B 4. cos ( A + B ) = cos A cos B − sin B 5. cos ( A − B ) = cos B + sin A Find exact values for each of the following: 1. sin 15° 2. sin 75° 3. cos 15° 4. cos 75° 5. tan 15° 6.

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  Trigonometry

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Multiply the tan(α + β) = 4 ( √ _ 2 _ 4 − √ _ 15 ) ______________ 4 ( 1 + √ _ 30 _ 4 ) = √ _ 2 − 4 √ _ 15 ___________ 4 + √ _ 30 numerator and the denominator by 4. The expression tan(α + β) = √ _ 2 − 4 √ _ 15 ___________ 4 + √ _ 30 can be simplified further if we rationalize the denominator. Section 5.2 Summary In this section, we derived the sum and difference identities for the cosine function using the distance formula. We then used these identities to derive the cofunction identities. The cofunction identities and sum and difference identities for the cosine function were used to derive the sum and difference identities for the sine function. The sine and cosine sum and difference identities were combined to determine the tangent sum and difference identities. The sum and difference identities enabled us to evaluate trigonometric expressions exactly for arguments that are integer multiples of 15° ( i.e., π _ 12 ) . cos (A + B) = cos A cos B − sin A sin B cos (A − B) = cos A cos B + sin A sin B sin (A + B) = sin A cos B + cos A sin B sin (A − B) = sin A cos B − cos A sin B tan(A + B) = tan A + tan B ___________ 1 − tan A tan B tan(A − B) = tan A − tan B ___________ 1 + tan A tan B Section 5.2 Exercises Skills In Exercises 1–20, find exact values for each trigonometric expression. 1. sin ( π _ 12 ) 2. cos ( π _ 12 ) 3. cos ( − 5π _ 12 ) 4. sin ( − 5π _ 12 ) 5. sin ( 7π _ 12 ) 6. cos ( 7π _ 12 ) 7. tan ( − π _ 12 ) 8. tan ( 13π _ 12 ) 9. sin 105° 10. cos 195° 11. tan (−105° ) 12. tan 165° 13. cot ( π _ 12 ) 14. cot ( − 5π _ 12 ) 15. sec ( − 11π _ 12 ) 16. sec ( − 13π _ 12 ) 17. csc ( 17π _ 12 ) 18. csc ( 23π _ 12 ) 19. sec (−195° ) 20. csc 285° It is important to note in Example 5 that right triangles have been superimposed in the Cartesian plane. The coordinate pair (x, y) can have positive or negative values, but the radius r is always positive.

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  The expression tan(α + β) = √ _ 2 − 4 √ _ 15 ___________ 4 + √ _ 30 can be simplified further if we rationalize the denominator. Section 6.2 Summary In this section, we derived the sum and difference identities for the cosine function using the distance formula. We then used these identities to derive the cofunction identities. The cofunction identities and sum and difference identities for the cosine function were used to derive the sum and difference identities for the sine function. The sine and cosine sum and difference identities were combined to determine the tangent sum and difference identities. The sum and difference identities enabled us to evaluate trigonometric expressions exactly for arguments that are integer multiples of 15° ( i.e., π _ 12 ) . cos (A + B) = cos A cos B − sin A sin B cos (A − B) = cos A cos B + sin A sin B sin (A + B) = sin A cos B + cos A sin B sin (A − B) = sin A cos B − cos A sin B tan(A + B) = tan A + tan B ___________ 1 − tan A tan B tan(A − B) = tan A − tan B ___________ 1 + tan A tan B Section 6.2 Exercises Skills In Exercises 1–16, find exact values for each trigonometric expression. 1. sin ( π _ 12 ) 2. cos ( π _ 12 ) 3. cos ( − 5π _ 12 ) 4. sin ( − 5π _ 12 ) 5. tan ( − π _ 12 ) 6. tan ( 13π _ 12 ) 7. sin 105° 8. cos 195° 9. tan (−105° ) 10. tan 165° 11. cot ( π _ 12 ) 12. cot ( − 5π _ 12 ) 13. sec ( − 11π _ 12 ) 14. sec ( − 13π _ 12 ) 15. csc(−255°) 16. csc(−15°) In Exercises 17–28, write each expression as a single trigonometric function. 17. sin (2x) sin (3x) + cos (2x) cos (3x) 18. sin x sin(2x) − cos x cos(2x) 19. sin x cos (2x) − cos x sin (2x) 20. sin(2x) cos(3x) + cos(2x) sin(3x) 21. cos(π − x) sin x + sin(π − x) cos x 22. sin ( π __ 3 x ) cos ( − π __ 2 x ) − cos ( π __ 3 x ) sin ( − π __ 2 x ) 23. (sin A − sin B) 2 + (cos A − cos B) 2 − 2 24. (sin A + sin B) 2 + (cos A + cos B) 2 − 2 25. 2 − (sin A + cos B) 2 − (cos A + sin B) 2 26. 2 − (sin A − cos B) 2 − (cos A + sin B) 2 27.

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  Introduction to Trigonometry, An

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  3. Law of tangents Fig. 1 - A triangle. In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposite sides. In Figure 1, a , b , and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that The law of tangents, although not as commonly known as the law of sines or the law of cosines, is just as useful, and can be used in any case where two sides and an angle, or two angles and a side are known. ________________________ WORLD TECHNOLOGIES ________________________ The law of tangents for spherical triangles was described in the 13th century by Persian mathematician, Nasir al-Din al-Tusi (1201-74), who also presented the law of sines for plane triangles in his five volume work Treatise on the Quadrilateral . Proof To prove the law of tangents we can start with the law of sines: Let so that It follows that Using the trigonometric identity, the factor formula for sines specifically we get As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity ________________________ WORLD TECHNOLOGIES ________________________ 4. Euler's formula Euler's formula , named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x , where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes called cis ( x ). The formula is still valid if x is a complex number, and so some authors refer to the more general com-plex version as Euler's formula.

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  Precalculus

  A Prelude to Calculus

  • Sheldon Axler(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  28 Evaluate cos ( cos -1 2 3 + tan -1 3 ) . 29 Find a formula for cos ( θ + π 2 ) . 30 Find a formula for sin ( θ + π 2 ) . 31 Find a formula for cos ( θ + π 4 ) . 32 Find a formula for sin ( θ - π 4 ) . 33 Find a formula for tan ( θ + π 4 ) . 34 Find a formula for tan ( θ - π 4 ) . 35 Find a formula for tan ( θ + π 2 ) . 36 Find a formula for tan ( θ - π 2 ) . Problems 37 Show that sin 75 ◦ = √ 6+ √ 2 4 . 38 Show (without using a calculator) that sin 10 ◦ cos 20 ◦ + cos 10 ◦ sin 20 ◦ = 1 2 . 39 Show (without using a calculator) that sin π 7 cos 4π 21 + cos π 7 sin 4π 21 = √ 3 2 . 40 Show that √ 6 + √ 2 4 = p 2 + √ 3 2 . Do this without using a calculator and without using the knowledge that both expressions above are equal to cos 15 ◦ (see Example 2 in this section and Example 3 in Section 5.5). 41 Show that sin 15 ◦ = √ 6 - √ 2 4 . [Hint: 15 = 45 - 30] 42 Show that cos(3θ ) = 4 cos 3 θ - 3 cos θ for all θ. [Hint: cos(3θ ) = cos(2θ + θ ).] 43 Show that cos 20 ◦ is a zero of the polynomial 8x 3 - 6x - 1. [Hint: Set θ = 20 ◦ in the identity from the previous prob- lem.] 44 Show that sin(3θ ) = 3 sin θ - 4 sin 3 θ for all θ. 45 Show that sin π 18 is a zero of the polynomial 8x 3 - 6x + 1. [Hint: Use the identity from the previous problem.] 46 Show that cos(5θ ) = 16 cos 5 θ - 20 cos 3 θ + 5 cos θ for all θ. 47 Use the previous problem to find a polynomial of degree 5 with integer coefficients that has cos 12 ◦ as a zero. 48 Find a nice formula for sin(5θ ) in terms of sin θ. 49 Show that sin u sin v = cos(u - v) - cos(u + v) 2 for all u, v. 50 Show that cos u sin v = sin(u + v) - sin(u - v) 2 for all u, v. 51 Show that cos x + cos y = 2 cos x+y 2 cos x-y 2 for all x, y. [Hint: Take u = x+y 2 and v = x-y 2 in the formula given by Example 5.] 52 Show that cos x - cos y = 2 sin x+y 2 sin y-x 2 for all x, y. 420 Chapter 5 Trigonometric Algebra and Geometry 53 Show that sin x - sin y = 2 cos x+y 2 sin x-y 2 for all x, y.

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