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3 Double-Angle, Half-Angle, and Product-Sum Formulas

3 Double-Angle, Half-Angle, and Product-Sum Formulas

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546



CHAPTER 8



| Analytic Trigonometry



▼ Double-Angle Formulas

The formulas in the following box are immediate consequences of the addition formulas,

which we proved in the preceding section.



DOUBLE-ANGLE FORMULAS

sin 2x ϭ 2 sin x cos x

cos 2x ϭ cos2 x Ϫ sin2 x

ϭ 1 Ϫ 2 sin2 x

ϭ 2 cos2 x Ϫ 1



Formula for sine:

Formulas for cosine:



tan 2x ϭ



Formula for tangent:



2 tan x

1 Ϫ tan2 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 ϭ cos1x ϩ x2

ϭ cos x cos x Ϫ sin x sin x

ϭ cos2 x Ϫ sin2 x

The second and third formulas for cos 2x are obtained from the formula we just proved

and the Pythagorean identity. Substituting cos2 x ϭ 1 Ϫ sin2 x gives



cos 2x ϭ cos2 x Ϫ sin2 x



ϭ 11 Ϫ sin2 x2 Ϫ sin2 x



ϭ 1 Ϫ 2 sin2x



The third formula is obtained in the same way, by substituting sin2 x ϭ 1 Ϫ cos2 x.



EXAMPLE 1







Using the Double-Angle Formulas



If cos x ϭ Ϫ 23 and x is in Quadrant II, find cos 2x and sin 2x.

SOLUTION



Using one of the Double-Angle Formulas for Cosine, we get

cos 2x ϭ 2 cos2 x Ϫ 1

8

2 2

1

ϭ 2 aϪ b Ϫ 1 ϭ Ϫ 1 ϭ Ϫ

3

9

9



To use the formula sin 2x ϭ 2 sin x cos x, we need to find sin x first. We have

sin x ϭ 21 Ϫ cos2 x ϭ 21 Ϫ AϪ23 B 2 ϭ



15

3



where we have used the positive square root because sin x is positive in Quadrant II.

Thus

sin 2x ϭ 2 sin x cos x

ϭ 2a

NOW TRY EXERCISE 3



15

2

415

b aϪ b ϭ Ϫ

3

3

9





| Double-Angle, Half-Angle, and Product-Sum Formulas 547



SECTION 8.3



A Triple-Angle Formula



EXAMPLE 2



Write cos 3x in terms of cos x.

SOLUTION



cos 3x ϭ cos12x ϩ x2

ϭ cos 2x cos x Ϫ sin 2x sin x



Addition formula



ϭ 12 cos2 x Ϫ 12 cos x Ϫ 12 sin x cos x2 sin x

ϭ 2 cos x Ϫ cos x Ϫ 2 sin x cos x

3



2



Double-Angle Formulas

Expand



ϭ 2 cos3 x Ϫ cos x Ϫ 2 cos x 11 Ϫ cos2 x2



Pythagorean identity



ϭ 2 cos x Ϫ cos x Ϫ 2 cos x ϩ 2 cos x



Expand



ϭ 4 cos3 x Ϫ 3 cos x



Simplify



3



3







NOW TRY EXERCISE 101



Example 2 shows that cos 3x can be written as a polynomial of degree 3 in cos x. The

identity cos 2x ϭ 2 cos2 x Ϫ 1 shows that cos 2x is a polynomial of degree 2 in cos x. In

fact, for any natural number n, we can write cos nx as a polynomial in cos x of degree n

(see the note following Exercise 101). The analogous result for sin nx is not true in

general.



EXAMPLE 3

Prove the identity



Proving an Identity

sin 3x

ϭ 4 cos x Ϫ sec x.

sin x cos x



We start with the left-hand side:



SOLUTION



sin1x ϩ 2x2

sin 3x

ϭ

sin x cos x

sin x cos x

ϭ

ϭ



sin x cos 2x ϩ cos x sin 2x

sin x cos x



sin x 12 cos2 x Ϫ 12 ϩ cos x 12 sin x cos x2

sin x cos x



ϭ



sin x 12 cos2 x Ϫ 12

cos x 12 sin x cos x2

ϩ

sin x cos x

sin x cos x



ϭ



2 cos2 x Ϫ 1

ϩ 2 cos x

cos x



ϭ 2 cos x Ϫ



1

ϩ 2 cos x

cos x



ϭ 4 cos x Ϫ sec x

NOW TRY EXERCISE 81



Addition Formula



Double-Angle Formulas



Separate fraction



Cancel



Separate fraction

Reciprocal identity







▼ Half-Angle Formulas

The following formulas allow us to write any trigonometric expression involving

even powers of sine and cosine in terms of the first power of cosine only. This technique

is important in calculus. The Half-Angle Formulas are immediate consequences of these

formulas.



548



CHAPTER 8



| Analytic Trigonometry



FORMULAS FOR LOWERING POWERS

sin2 x ϭ



1 Ϫ cos 2x

2

tan2 x ϭ



cos2 x ϭ



1 ϩ cos 2x

2



1 Ϫ cos 2x

1 ϩ cos 2x



P R O O F The first formula is obtained by solving for sin2 x in the double-angle formula

cos 2x ϭ 1 Ϫ 2 sin2 x. Similarly, the second formula is obtained by solving for cos2 x in

the Double-Angle Formula cos 2x ϭ 2 cos2 x Ϫ 1.

The last formula follows from the first two and the reciprocal identities:



1 Ϫ cos 2x

sin x

2

1 Ϫ cos 2x

tan2 x ϭ

ϭ

ϭ

1 ϩ cos 2x

1 ϩ cos 2x

cos2 x

2

2



Lowering Powers in a Trigonometric Expression



EXAMPLE 4

2







2



Express sin x cos x in terms of the first power of cosine.

SOLUTION



We use the formulas for lowering powers repeatedly:

sin2 x cos2 x ϭ a



1 Ϫ cos 2x

1 ϩ cos 2x

ba

b

2

2



ϭ



1 Ϫ cos2 2x

1

1

ϭ Ϫ cos2 2x

4

4

4



ϭ



1

1 1 ϩ cos 4x

1

1

cos 4x

Ϫ a

b ϭ Ϫ Ϫ

4

4

2

4

8

8



ϭ



1

1

1

Ϫ cos 4x ϭ 11 Ϫ cos 4x 2

8

8

8



Another way to obtain this identity is to use the Double-Angle Formula for Sine in the

form sin x cos x ϭ 12 sin 2x. Thus

sin2 x cos2 x ϭ

ϭ



1

1 1 Ϫ cos 4x

sin2 2x ϭ a

b

4

4

2

1

11 Ϫ cos 4x 2

8





NOW TRY EXERCISE 11



HALF-ANGLE FORMULAS

sin



u

1 Ϫ cos u

ϭϮ

2

B

2

tan



cos



u

1 ϩ cos u

ϭϮ

2

B

2



u

1 Ϫ cos u

sin u

ϭ

ϭ

2

sin u

1 ϩ cos u



The choice of the ϩ or Ϫ sign depends on the quadrant in which u/2 lies.



SECTION 8.3



| Double-Angle, Half-Angle, and Product-Sum Formulas 549



P R O O F We substitute x ϭ u/2 in the formulas for lowering powers and take the square

root of each side. This gives the first two Half-Angle Formulas. In the case of the HalfAngle Formula for Tangent we get



tan



u

1 Ϫ cos u

ϭϮ

2

B 1 ϩ cos u

a



1 Ϫ cos u

1 Ϫ cos u

ba

b

B 1 ϩ cos u

1 Ϫ cos u



Multiply numerator and

denominator by 1 Ϫ cos u



11 Ϫ cos u 2 2



B 1 Ϫ cos2 u



Simplify



ϭϮ

ϭϮ



ϭϮ



0 1 Ϫ cos u 0

0 sin u 0



2A 2 ϭ 0 A 0

and 1 Ϫ cos2 u ϭ sin2 u



Now, 1 Ϫ cos u is nonnegative for all values of u. It is also true that sin u and tan1u/22 always have the same sign. (Verify this.) It follows that

u

1 Ϫ cos u

tan ϭ

2

sin u

The other Half-Angle Formula for Tangent is derived from this by multiplying the numerator and denominator by 1 ϩ cos u.





EXAMPLE 5



Using a Half-Angle Formula



Find the exact value of sin 22.5Њ.

S O L U T I O N Since 22.5Њ is half of 45Њ, we use the Half-Angle Formula for Sine with

u ϭ 45Њ. We choose the ϩ sign because 22.5Њ is in the first quadrant:



sin



45°

1 Ϫ cos 45°

ϭ

2

B

2



Half-Angle Formula



ϭ



1 Ϫ 12/2

B

2



cos 45° ϭ 12/2



ϭ



2 Ϫ 12

B

4



Common denominator



ϭ 12 32 Ϫ 22



Simplify







NOW TRY EXERCISE 17



EXAMPLE 6



Using a Half-Angle Formula



Find tan1u/22 if sin u ϭ 25 and u is in Quadrant II.

S O L U T I O N To use the Half-Angle Formula for Tangent, we first need to find cos u.

Since cosine is negative in Quadrant II, we have



cos u ϭ Ϫ 21 Ϫ sin2 u

ϭ Ϫ21 Ϫ A 25 B 2 ϭ Ϫ

Thus



tan



u

1 Ϫ cos u

ϭ

2

sin u

ϭ



NOW TRY EXERCISE 37



121

5



1 ϩ 121/5

2

5



ϭ



5 ϩ 121

2





550



CHAPTER 8



| Analytic Trigonometry



▼ Evaluating Expressions Involving Inverse Trigonometric Functions

Expressions involving trigonometric functions and their inverses arise in calculus. In the

next examples we illustrate how to evaluate such expressions.



EXAMPLE 7



Simplifying an Expression Involving an Inverse

Trigonometric Function



Write sin12 cosϪ1 x 2 as an algebraic expression in x only, where Ϫ1 Յ x Յ 1.

1

1-≈

œ∑∑∑∑∑



S O L U T I O N Let u ϭ cosϪ1 x, and sketch a triangle as in Figure 1. We need to find

sin 2u, but from the triangle we can find trigonometric functions of u only, not 2u. So

we use the Double-Angle Formula for Sine.



sin12 cosϪ1 x 2 sin 2u



ă

x



FIGURE 1



cos1 x u



2 sin u cos u



Double-Angle Formula



ϭ 2x 21 Ϫ x



From the triangle



2







NOW TRY EXERCISES 43 AND 47



EXAMPLE 8



Evaluating an Expression Involving Inverse

Trigonometric Functions



Evaluate sin 2u, where cos u ϭ Ϫ25 with u in Quadrant II.

S O L U T I O N We first sketch the angle u in standard position with terminal side in

Quadrant II as in Figure 2. Since cos u ϭ x/r ϭ Ϫ25, we can label a side and the hypotenuse of the triangle in Figure 2. To find the remaining side, we use the Pythagorean

Theorem:



P (x, y)



5

ă

_2



x 2 ϩ y2 ϭ r 2



1Ϫ22 2 ϩ y 2 ϭ 52



FIGURE 2



Pythagorean Theorem

x ϭ Ϫ2,



rϭ5



y ϭ Ϯ 221



Solve for y2



y ϭ ϩ 221



Because y Ͼ 0



We can now use the Double-Angle Formula for Sine:



sin 2u ϭ 2 sin u cos u

ϭ 2a

ϭϪ



2

221

b aϪ b

5

5



4221

25



Double-Angle Formula

From the triangle



Simplify



NOW TRY EXERCISE 51







▼ Product-Sum Formulas

It is possible to write the product sin u cos √ as a sum of trigonometric functions. To see

this, consider the addition and subtraction formulas for the sine function:



sin1u ϩ √2 ϭ sin u cos √ ϩ cos u sin √

sin1u Ϫ √2 ϭ sin u cos √ Ϫ cos u sin √

Adding the left- and right-hand sides of these formulas gives



sin1u ϩ √2 ϩ sin1u Ϫ √2 ϭ 2 sin u cos √



SECTION 8.3



| Double-Angle, Half-Angle, and Product-Sum Formulas 551



Dividing by 2 gives the formula



sin u cos √ ϭ 12 3sin1u ϩ √2 ϩ sin1u Ϫ √2 4

The other three Product-to-Sum Formulas follow from the addition formulas in a similar way.



PRODUCT-TO-SUM FORMULAS

sin u cos √ ϭ 12 3sin1u ϩ √2 ϩ sin1u Ϫ √2 4



cos u sin √ ϭ 12 3sin1u ϩ √2 Ϫ sin1u Ϫ √2 4



cos u cos √ ϭ 12 3cos1u ϩ √2 ϩ cos1u Ϫ √2 4



sin u sin √ ϭ 12 3cos1u Ϫ √2 Ϫ cos1u ϩ √2 4



EXAMPLE 9



Expressing a Trigonometric Product as a Sum



Express sin 3x sin 5x as a sum of trigonometric functions.

S O L U T I O N Using the fourth Product-to-Sum Formula with u ϭ 3x and √ ϭ 5x and the

fact that cosine is an even function, we get



sin 3x sin 5x ϭ 12 3cos13x Ϫ 5x2 Ϫ cos13x ϩ 5x2 4

ϭ 12 cos1Ϫ2x2 Ϫ 12 cos 8x



ϭ 12 cos 2x Ϫ 12 cos 8x





NOW TRY EXERCISE 55



The Product-to-Sum Formulas can also be used as Sum-to-Product Formulas. This is

possible because the right-hand side of each Product-to-Sum Formula is a sum and the left

side is a product. For example, if we let



xϩy

and

2

in the first Product-to-Sum Formula, we get





√ϭ



xϪy

2



xϩy

xϪy

cos

ϭ 12 1sin x ϩ sin y2

2

2

xϩy

xϪy

sin x ϩ sin y ϭ 2 sin

cos

2

2



sin

so



The remaining three of the following Sum-to-Product Formulas are obtained in a similar manner.



SUM-TO-PRODUCT FORMULAS

sin x ϩ sin y ϭ 2 sin



xϩy

xϪy

cos

2

2



sin x Ϫ sin y ϭ 2 cos



xϩy

xϪy

sin

2

2



cos x ϩ cos y ϭ 2 cos



xϩy

xϪy

cos

2

2



cos x Ϫ cos y ϭ Ϫ2 sin



xϩy

xϪy

sin

2

2



552



CHAPTER 8



| Analytic Trigonometry



Expressing a Trigonometric Sum as a Product



EXAMPLE 10



Write sin 7x ϩ sin 3x as a product.

SOLUTION



The first Sum-to-Product Formula gives

sin 7x ϩ sin 3x ϭ 2 sin



7x ϩ 3x

7x Ϫ 3x

cos

2

2



ϭ 2 sin 5x cos 2x





NOW TRY EXERCISE 61



Proving an Identity



EXAMPLE 11

Verify the identity



sin 3x Ϫ sin x

ϭ tan x.

cos 3x ϩ cos x



S O L U T I O N We apply the second Sum-to-Product Formula to the numerator and the

third formula to the denominator:



sin 3x Ϫ sin x

LHS ϭ

ϭ

cos 3x ϩ cos x



3x ϩ x

3x Ϫ x

sin

2

2

3x ϩ x

3x Ϫ x

2 cos

cos

2

2

2 cos



Sum-to-Product Formulas



ϭ



2 cos 2x sin x

2 cos 2x cos x



Simplify



ϭ



sin x

ϭ tan x ϭ RHS

cos x



Cancel







NOW TRY EXERCISE 89



8.3 EXERCISES

7. sin x ϭ Ϫ 35, x in Quadrant III



CONCEPTS

1. If we know the values of sin x and cos x, we can find the value

of sin 2x by using the



Formula for Sine. State the



formula: sin 2x ϭ



.



2. If we know the value of cos x and the quadrant in which x/2



8. sec x ϭ 2, x in Quadrant IV

9. tan x ϭ Ϫ 13, cos x Ͼ 0

10. cot x ϭ 23, sin x Ͼ 0



lies, we can find the value of sin1x/22 by using the

Formula for Sine. State the formula:



11–16 ■ Use the formulas for lowering powers to rewrite

the expression in terms of the first power of cosine, as in

Example 4.



sin1x/22 ϭ



11. sin4 x



12. cos4 x



13. cos2 x sin4 x



14. cos4 x sin2 x



SKILLS



15. cos4 x sin4 x



16. cos6 x



3–10 ■ Find sin 2x, cos 2x, and tan 2x from the given

information.



17–28 ■ Use an appropriate Half-Angle Formula to find the

exact value of the expression.



.



3. sin x ϭ 135 , x in Quadrant I



17. sin 15Њ



18. tan 15Њ



4. tan x ϭ Ϫ 43, x in Quadrant II



19. tan 22.5Њ



20. sin 75Њ



5. cos x ϭ 45, csc x Ͻ 0



21. cos 165Њ



22. cos 112.5Њ



6. csc x ϭ 4, tan x Ͻ 0



SECTION 8.3

3p

8

5p

26. tan

12

11p

28. sin

12



p

8

p

25. cos

12

9p

27. sin

8

23. tan



61–66



24. cos



29–34 ■ Simplify the expression by using a Double-Angle

Formula or a Half-Angle Formula.

29. (a) 2 sin 18Њ cos 18Њ

2 tan 7°

1 Ϫ tan2 7°



30. (a)



(b)



31. (a) cos 34Њ Ϫ sin 34Њ

2



32. (a) cos2



2



u

u

Ϫ sin2

2

2



(b) cos 5u Ϫ sin 5u

2



(b) 2 sin



1 Ϫ cos 30°

B

2



(b)



Write the sum as a product.



61. sin 5x ϩ sin 3x



62. sin x Ϫ sin 4x



63. cos 4x Ϫ cos 6x



64. cos 9x ϩ cos 2x



65. sin 2x Ϫ sin 7x



66. sin 3x ϩ sin 4x



67–72







Find the value of the product or sum.



67. 2 sin 52.5Њ sin 97.5Њ



68. 3 cos 37.5Њ cos 7.5Њ



69. cos 37.5Њ sin 7.5Њ



70. sin 75Њ ϩ sin 15Њ



71. cos 255Њ Ϫ cos 195Њ



2 tan 7u

1 Ϫ tan2 7u

2



u

u

cos

2

2



1 Ϫ cos 4u

(b)

sin 4u



sin 8°

33. (a)

1 ϩ cos 8°

34. (a)



(b) 2 sin 3u cos 3u







| Double-Angle, Half-Angle, and Product-Sum Formulas 553



1 Ϫ cos 8u

B

2



35. Use the Addition Formula for Sine to prove the Double-Angle

Formula for Sine.

36. Use the Addition Formula for Tangent to prove the DoubleAngle Formula for Tangent.



72. cos

73–90



5p

p

ϩ cos

12

12







Prove the identity.



73. cos 5x Ϫ sin2 5x ϭ cos 10x

2



74. sin 8x ϭ 2 sin 4x cos 4x



75. 1sin x ϩ cos x 2 2 ϭ 1 ϩ sin 2x

76.



2 tan x

ϭ sin 2x

1 ϩ tan2 x



77.



sin 4x

ϭ 4 cos x cos 2x

sin x



1 ϩ sin 2x

ϭ 1 ϩ 12 sec x csc x

sin 2x

21tan x Ϫ cot x 2

79.

ϭ sin 2 x

tan2 x Ϫ cot 2 x

78.



37–42







x

x

x

Find sin , cos , and tan from the given information.

2

2

2



37. sin x ϭ 35, 0Њ Ͻ x Ͻ 90Њ

38. cos x ϭ Ϫ 45, 180Њ Ͻ x Ͻ 270Њ

39. csc x ϭ 3, 90Њ Ͻ x Ͻ 180Њ

40. tan x ϭ 1, 0Њ Ͻ x Ͻ 90Њ



80. cot 2 x ϭ



1 Ϫ tan2 x

2 tan x



81. tan 3x ϭ



3 tan x Ϫ tan3 x

1 Ϫ 3 tan2 x



41. sec x ϭ 32, 270Њ Ͻ x Ͻ 360Њ



82. 41sin6 x ϩ cos6 x2 ϭ 4 Ϫ 3 sin2 2x



42. cot x ϭ 5, 180Њ Ͻ x Ͻ 270Њ



83. cos4 x Ϫ sin4 x ϭ cos 2x



43–46







Write the given expression as an algebraic expression in x.



43. sin12 tanϪ1 x 2



44. tan12 cosϪ1 x 2

46. cos12 sinϪ1 x 2



45. sinA 12 cosϪ1 xB

47–50







47. sinA2

49. secA2

51–54







Find the exact value of the given expression.

cosϪ1 257 B



48. cosA2



sinϪ1 14 B



tanϪ1 125 B



50. tanA 12 cosϪ1 23 B



52. sin1u/22 ; tan u ϭ Ϫ125 , u in Quadrant IV

53. sin 2u; sin u ϭ

54. tan 2u; cos u ϭ





1

7,

3

5,



u in Quadrant II

u in Quadrant I



Write the product as a sum.



55. sin 2x cos 3x



56. sin x sin 5x



57. cos x sin 4x



58. cos 5x cos 3x

x

x

60. 11 sin cos

2

4



59. 3 cos 4x cos 7x



x

p

1 ϩ sin x

ϩ b ϭ

2

4

1 Ϫ sin x



85.



sin x ϩ sin 5x

ϭ tan 3x

cos x ϩ cos 5x



86.



sin 3x ϩ sin 7x

ϭ cot 2x

cos 3x Ϫ cos 7x



87.



cos 5x

sin 10x

ϭ

sin 9x ϩ sin x

cos 4x



88.



sin x ϩ sin 3x ϩ sin 5x

ϭ tan 3x

cos x ϩ cos 3x ϩ cos 5x



89.



sin x ϩ sin y

xϩy

ϭ tan a

b

cos x ϩ cos y

2



Evaluate each expression under the given conditions.



51. cos 2u; sin u ϭ Ϫ35, u in Quadrant III



55–60



84. tan2 a



90. tan y ϭ



sin1x ϩ y 2 Ϫ sin1x Ϫ y 2



cos1x ϩ y2 ϩ cos1x Ϫ y2



91. Show that sin 130Њ Ϫ sin 110Њ ϭ Ϫsin 10Њ.

92. Show that cos 100Њ Ϫ cos 200Њ ϭ sin 50Њ.

93. Show that sin 45Њ ϩ sin 15Њ ϭ sin 75Њ.

94. Show that cos 87Њ ϩ cos 33Њ ϭ sin 63Њ.



554



CHAPTER 8



| Analytic Trigonometry



95. Prove the identity

sin x ϩ sin 2x ϩ sin 3x ϩ sin 4x ϩ sin 5x

ϭ tan 3x

cos x ϩ cos 2x ϩ cos 3x ϩ cos 4x ϩ cos 5x



(c) Find the dimensions of the inscribed rectangle with the

largest possible area.



96. Use the identity

sin 2x ϭ 2 sin x cos x

n times to show that



ă



sin12nx 2 2n sin x cos x cos 2 x cos 4x . . . cos 2nϪ1 x



sin 3x

cos 3x

Ϫ

97. (a) Graph f 1x2 ϭ

and make a conjecture.

sin x

cos x

(b) Prove the conjecture you made in part (a).

98. (a) Graph f 1x2 ϭ cos 2x ϩ 2 sin2 x and make a conjecture.

(b) Prove the conjecture you made in part (a).

99. Let f 1x 2 ϭ sin 6x ϩ sin 7x.

(a) Graph y ϭ f 1x2 .

(b) Verify that f 1x2 ϭ 2 cos 12 x sin 132 x.

(c) Graph y ϭ 2 cos 12 x and y ϭ Ϫ2 cos 12 x, together with

the graph in part (a), in the same viewing rectangle. How

are these graphs related to the graph of f ?

100. Let 3x ϭ p/3 and let y ϭ cos x. Use the result of Example 2

to show that y satisfies the equation



5 cm



A P P L I C AT I O N S

105. Sawing a Wooden Beam A rectangular beam is to be

cut from a cylindrical log of diameter 20 in.

(a) Show that the cross-sectional area of the beam is

modeled by the function

A1u2 ϭ 200 sin 2u

where u is as shown in the figure.

(b) Show that the maximum cross-sectional area of such a

beam is 200 in2. [Hint: Use the fact that sin u achieves

its maximum value at u ϭ p/2.]



8y 3 Ϫ 6y Ϫ 1 ϭ 0

NOTE This equation has roots of a certain kind that are used

to show that the angle p/3 cannot be trisected by using a

ruler and compass only.

101. (a) Show that there is a polynomial P1t2 of degree 4 such

that cos 4x ϭ P1cos x 2 (see Example 2).

(b) Show that there is a polynomial Q1t 2 of degree 5 such

that cos 5x ϭ Q1cos x 2 .

NOTE In general, there is a polynomial Pn 1t 2 of degree n

such that cos nx ϭ Pn 1cos x 2 . These polynomials are called

Tchebycheff polynomials, after the Russian mathematician

P. L. Tchebycheff (1821–1894).

102. In triangle ABC (see the figure) the line segment s bisects angle C. Show that the length of s is given by



20 in.



20 in.

ă



ă



106. Length of a Fold The lower right-hand corner of a long

piece of paper 6 in. wide is folded over to the left-hand edge

as shown. The length L of the fold depends on the angle u.

Show that





3

sin u cos2 u



2ab cos x

ab

[Hint: Use the Law of Sines.]

s



ă



C

a



x



x

s



L



b



B



A

6 in.



103. If A, B, and C are the angles in a triangle, show that

sin 2A ϩ sin 2B ϩ sin 2C ϭ 4 sin A sin B sin C

104. A rectangle is to be inscribed in a semicircle of radius 5 cm

as shown in the following figure.

(a) Show that the area of the rectangle is modeled by the

function

A1u 2 ϭ 25 sin 2u

(b) Find the largest possible area for such an inscribed

rectangle.



107. Sound Beats When two pure notes that are close in frequency are played together, their sounds interfere to produce

beats; that is, the loudness (or amplitude) of the sound alternately increases and decreases. If the two notes are given by

f1 1t 2 ϭ cos 11t



and



f2 1t2 ϭ cos 13t



the resulting sound is f 1t2 ϭ f1 1t2 ϩ f2 1t2 .

(a) Graph the function y ϭ f 1t 2 .

(b) Verify that f 1t2 ϭ 2 cos t cos 12t.



SECTION 8.4

(c) Graph y ϭ 2 cos t and y ϭ Ϫ2 cos t, together with the

graph in part (a), in the same viewing rectangle. How do

these graphs describe the variation in the loudness

of the sound?



DISCOVERY



2



3



Low

770 Hz

frequency

852 Hz

f1



4



5



6



7



8



9



941 Hz



*



0



#







WRITING



C



A



ă

1



O



1



B



[Hint: Find the area of triangle ABC in two different ways.

You will need the following facts from geometry:

An angle inscribed in a semicircle is a right angle, so

ЄACB is a right angle.

The central angle subtended by the chord of a circle is

twice the angle subtended by the chord on the circle, so

ЄBOC is 2u.]



High frequency f2

1209 1336 1477 Hz

1



DISCUSSION



109. Geometric Proof of a Double-Angle Formula

Use the figure to prove that sin 2u ϭ 2 sin u cos u.



108. Touch-Tone Telephones When a key is pressed on a

touch-tone telephone, the keypad generates two pure tones,

which combine to produce a sound that uniquely identifies

the key. The figure shows the low frequency f1 and the high

frequency f2 associated with each key. Pressing a key produces the sound wave y ϭ sin12pf1t 2 ϩ sin12pf2t2 .

(a) Find the function that models the sound produced when

the 4 key is pressed.

(b) Use a Sum-to-Product Formula to express the sound generated by the 4 key as a product of a sine and a cosine

function.

(c) Graph the sound wave generated by the 4 key, from

t ϭ 0 to t ϭ 0.006 s.



697 Hz







| Basic Trigonometric Equations 555



❍ DISCOVERY

PROJECT



Where to Sit at the Movies



In this project we use trigonometry to find the best

location to observe such things as a painting or a movie.

You can find the project at the book companion website:

www.stewartmath.com



8.4 B ASIC T RIGONOMETRIC E QUATIONS

Basic Trigonometric Equations ᭤ Solving Trigonometric Equations

by Factoring

An equation that contains trigonometric functions is called a trigonometric equation.

For example, the following are trigonometric equations:



sin2 u ϩ cos2 u ϭ 1



2 sin u Ϫ 1 ϭ 0



tan 2u Ϫ 1 ϭ 0



The first equation is an identity—that is, it is true for every value of the variable u. The

other two equations are true only for certain values of u. To solve a trigonometric equation, we find all the values of the variable that make the equation true.



▼ Basic Trigonometric Equations

Solving any trigonometric equation always reduces to solving a basic trigonometric

equation—an equation of the form T1u 2 ϭ c, where T is a trigonometric function and c

is a constant. In the next three examples we solve such basic equations.



EXAMPLE 1



Solving a Basic Trigonometric Equation



Solve the equation sin u



1

.

2



556



CHAPTER 8



ă=5

6



y

1



| Analytic Trigonometry

SOLUTION



1

2



Find the solutions in one period. Because sine has period 2p, we first find the solutions

in any interval of length 2p. To find these solutions, we look at the unit circle in Figure 1. We

see that sin u ϭ 12 in Quadrants I and II, so the solutions in the interval 3 0, 2p 2 are





ă=

6

_1



0



u

1



x



p

6



5p

6



u



Find all solutions. Because the sine function repeats its values every 2p units, we get all

solutions of the equation by adding integer multiples of 2p to these solutions:



p

ϩ 2kp

6







_1



5p

ϩ 2kp

6







where k is any integer. Figure 2 gives a graphical representation of the solutions.



FIGURE 1



y



y=ß ¨



1



_





6



1



y= 2

5π π

6



π

6



13π

6



17π

6



25π

6



¨



_1



FIGURE 2







NOW TRY EXERCISE 5



EXAMPLE 2



Solving a Basic Trigonometric Equation



Solve the equation cos u ϭ Ϫ



22

, and list eight specific solutions.

2



SOLUTION

Find the solutions in one period. Because cosine has period 2p, we first find the

solutions in any interval of length 2p. From the unit circle in Figure 3 we see that

cos u ϭ Ϫ 22/2 in Quadrants II and III, so the solutions in the interval 3 0, 2p 2 are



u



ă=



3p

4



u



5p

4



y

1



5

4



3

ă=

4

_1

_



1



x







2



2

_1



FIGURE 3



Find all solutions. Because the cosine function repeats its values every 2p units, we get

all solutions of the equation by adding integer multiples of 2p to these solutions:







3p

ϩ 2kp

4







5p

ϩ 2kp

4



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