For a proof of the sum and difference formulas, see Proofs in Mathematics on ...
Examples 1 and 2 show how sum and difference formulas can be used to.
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Sum and Difference Formulas
What you should learn • Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations.
Why you should learn it You can use identities to rewrite trigonometric expressions. For instance, in Exercise 75 on page 405, you can use an identity to rewrite a trigonometric expression in a form that helps you analyze a harmonic motion equation.
Using Sum and Difference Formulas In this and the following section, you will study the uses of several trigonometric identities and formulas.
Sum and Difference Formulas sinu v sin u cos v cos u sin v sinu v sin u cos v cos u sin v cosu v cos u cos v sin u sin v cosu v cos u cos v sin u sin v
tanu v
tan u tan v 1 tan u tan v
tanu v
tan u tan v 1 tan u tan v
For a proof of the sum and difference formulas, see Proofs in Mathematics on page 424.
Exploration Use a graphing utility to graph y1 cosx 2 and y2 cos x cos 2 in the same viewing window. What can you conclude about the graphs? Is it true that cosx 2 cos x cos 2? Use a graphing utility to graph y1 sinx 4 and y2 sin x sin 4 in the same viewing window. What can you conclude about the graphs? Is it true that sinx 4 sin x sin 4?
Richard Megna/Fundamental Photographs
Examples 1 and 2 show how sum and difference formulas can be used to find exact values of trigonometric functions involving sums or differences of special angles.
Example 1
Evaluating a Trigonometric Function
Find the exact value of cos 75.
Solution To find the exact value of cos 75, use the fact that 75 30 45. Consequently, the formula for cosu v yields cos 75 cos30 45 cos 30 cos 45 sin 30 sin 45
3 2
1 2
2 2 2 2
6 2
4
.
Try checking this result on your calculator. You will find that cos 75 0.259. Now try Exercise 1.
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Section 5.4
The Granger Collection, New York
Example 2
Sum and Difference Formulas
401
Evaluating a Trigonometric Expression
Find the exact value of sin
. 12
Solution Using the fact that
12 3 4 together with the formula for sinu v, you obtain
Historical Note Hipparchus, considered the most eminent of Greek astronomers, was born about 160 B.C. in Nicaea. He was credited with the invention of trigonometry. He also derived the sum and difference formulas for sinA ± B and cosA ± B.
sin
sin 12 3 4 sin cos cos sin 3 4 3 4 3 2 1 2 2 2 2 2 6 2 . 4
Now try Exercise 3.
Example 3
Evaluating a Trigonometric Expression
Find the exact value of sin 42 cos 12 cos 42 sin 12.
Solution Recognizing that this expression fits the formula for sinu v, you can write sin 42 cos 12 cos 42 sin 12 sin42 12 sin 30 1 2. Now try Exercise 31. 2
1
Example 4 u
An Application of a Sum Formula
Write cosarctan 1 arccos x as an algebraic expression. 1
Solution This expression fits the formula for cosu v. Angles u arctan 1 and v arccos x are shown in Figure 5.7. So
1
v x FIGURE
5.7
1 − x2
cosu v cosarctan 1 cosarccos x sinarctan 1 sinarccos x 1 1 x 2 1 x 2 2 x 1 x 2 . 2 Now try Exercise 51.
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Example 5 shows how to use a difference formula to prove the cofunction identity cos
2 x sin x. Proving a Cofunction Identity
Example 5
Prove the cofunction identity cos
2 x sin x.
Solution Using the formula for cosu v, you have cos
2 x cos 2 cos x sin 2 sin x 0cos x 1sin x sin x. Now try Exercise 55.
Sum and difference formulas can be used to rewrite expressions such as
sin
n 2
cos
and
n , 2
where n is an integer
as expressions involving only sin or cos . The resulting formulas are called reduction formulas.
Example 6 Activities 1. Use the sum and difference formulas to find the exact value of cos 15. 6 2 Answer: 4 2. Rewrite the expression using the sum and difference formulas. tan 40 tan 10 1 tan 40 tan 10 Answer: tan40 10 tan 50 3. Verify the identity sin cos . 2 Answer: sin sin cos cos sin 2 2 2 1 cos 0 sin cos
Deriving Reduction Formulas
Simplify each expression.
a. cos
3 2
b. tan 3
Solution a. Using the formula for cosu v, you have
cos
3 3 3 sin sin cos cos 2 2 2
cos 0 sin 1 sin . b. Using the formula for tanu v, you have tan 3
tan tan 3 1 tan tan 3 tan 0 1 tan 0
tan . Now try Exercise 65.
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Section 5.4
Example 7
Sum and Difference Formulas
403
Solving a Trigonometric Equation
Find all solutions of sin x
sin x 1 in the interval 0, 2. 4 4
Solution Using sum and difference formulas, rewrite the equation as sin x cos
cos x sin sin x cos cos x sin 1 4 4 4 4 2 sin x cos 1 4 2 2sin x 1 2
y
1 2 2 sin x . 2 sin x
3 2 1 π 2
−1
π
2π
−3
(
FIGURE
5.8
So, the only solutions in the interval 0, 2 are 5 7 and x . 4 4 You can confirm this graphically by sketching the graph of x
−2
y = sin x +
x
π π + sin x − +1 4 4
(
(
(
y sin x
sin x 1 for 0 ≤ x < 2, 4 4
as shown in Figure 5.8. From the graph you can see that the x-intercepts are 5 4 and 7 4. Now try Exercise 69. The next example was taken from calculus. It is used to derive the derivative of the sine function.
Example 8
An Application from Calculus
Verify that sinx h sin x sin h 1 cos h cos x sin x h h h where h 0.
Solution Using the formula for sinu v, you have sinx h sin x sin x cos h cos x sin h sin x h h cos x sin h sin x1 cos h h sin h 1 cos h cos x sin x . h h Now try Exercise 91.
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Exercises
VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity. 1. sinu v ________
2. cosu v ________
3. tanu v ________
4. sinu v ________
5. cosu v ________
6. tanu v ________
PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, find the exact value of each expression. 1. (a) cos120 45
(b) cos 120 cos 45
2. (a) sin135 30
(b) sin 135 cos 30
3. (a) cos 4 3
(b) cos
cos 4 3
(b) sin
3 5 sin 4 6
(b) sin
7 sin 6 3
4. (a) sin 5. (a) sin
3
4
5 6
7 6 3
6. (a) sin315 60
(b) sin 315 sin 60
In Exercises 7–22, find the exact values of the sine, cosine, and tangent of the angle by using a sum or difference formula. 7. 105 60 45
8. 165 135 30
9. 195 225 30
10. 255 300 45
11.
11 3 12 4 6
12.
7 12 3 4
13.
17 9 5 12 4 6
14.
12 6 4
15. 285
16. 105
17. 165
18. 15
13 19. 12
7 20. 12
21.
13 12
22.
5 12
In Exercises 23–30, write the expression as the sine, cosine, or tangent of an angle. 23. cos 25 cos 15 sin 25 sin 15 24. sin 140 cos 50 cos 140 sin 50 25.
tan 325 tan 86 1 tan 325 tan 86
26.
tan 140 tan 60 1 tan 140 tan 60
27. sin 3 cos 1.2 cos 3 sin 1.2 28. cos 29.
cos sin sin 7 5 7 5
tan 2x tan x 1 tan 2x tan x
30. cos 3x cos 2y sin 3x sin 2y In Exercises 31–36, find the exact value of the expression. 31. sin 330 cos 30 cos 330 sin 30 32. cos 15 cos 60 sin 15 sin 60 33. sin
cos cos sin 12 4 12 4
34. cos
3 3 cos sin sin 16 16 16 16
35.
tan 25 tan 110 1 tan 25 tan 110
36.
tan5 4 tan 12 1 tan5 4 tan 12
In Exercises 37–44, find the exact value of the trigonometric 5 3 function given that sin u 13 and cos v 5. (Both u and v are in Quadrant II.) 37. sinu v
38. cosu v
39. cosu v
40. sinv u
41. tanu v
42. cscu v
43. secv u
44. cotu v
In Exercises 45–50, find the exact value of the trigonometric 7 4 function given that sin u 25 and cos v 5. (Both u and v are in Quadrant III.) 45. cosu v
46. sinu v
47. tanu v
48. cotv u
49. secu v
50. cosu v
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Section 5.4 In Exercises 51–54, write the trigonometric expression as an algebraic expression. 51. sinarcsin x arccos x
52. sinarctan 2x arccos x
53. cosarccos x arcsin x 54. cosarccos x arctan x In Exercises 55– 64, verify the identity.
x cos x 56. sin 2
55. sin3 x sin x 57. sin
6 x 2 cos x 3 sin x
58. cos
1
2 5 cos x sin x x 4 2
59. cos sin 60. tan
Model It 75. Harmonic Motion A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by y
1 1 sin 2t cos 2t 3 4
where y is the distance from equilibrium (in feet) and t is the time (in seconds). (a) Use the identity a sin B b cos B a 2 b2 sinB C
2 0
where C arctanb a, a > 0, to write the model in the form
1 tan
4 1 tan
405
Sum and Difference Formulas
y a2 b2 sinBt C.
61. cosx y cosx y cos2 x sin2 y 62. sinx y sinx y) sin2 x sin 2 y
(b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weight.
63. sinx y sinx y 2 sin x cos y 64. cosx y cosx y 2 cos x cos y In Exercises 65 –68, simplify the expression algebraically and use a graphing utility to confirm your answer graphically. 65. cos 67. sin
3
2
3
2
66. cos x
y1 A cos 2
68. tan
show that
x
In Exercises 69 –72, find all solutions of the equation in the interval [0, 2.
sin x 1 69. sin x 3 3
70. sin x
72. tanx 2 sinx 0 In Exercises 73 and 74, use a graphing utility to approximate the solutions in the interval [0, 2.
cos x 1 73. cos x 4 4
y1
74. tanx cos x
0 2
t
x
and
y2 A cos 2
T
2 t 2 x cos . T y1 + y2
y2
t=0
T
y1 y2 2A cos
1 sin x 6 6 2
cos x 1 71. cos x 4 4
76. Standing Waves The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A, period T, and wavelength . If the models for these waves are
y1
y1 + y2
y2
t = 18 T y1 t = 28 T
y1 + y2
y2
t
x
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Synthesis
(c) Use a graphing utility to graph the functions f and g.
True or False? In Exercises 77–80, determine whether the statement is true or false. Justify your answer.
In Exercises 93 and 94, use the figure, which shows two lines whose equations are
77. sinu ± v sin u ± sin v 78. cosu ± v cos u ± cos v
79. cos x
sin x 2
(d) Use the table and the graphs to make a conjecture about the values of the functions f and g as h → 0.
80. sin x
cos x 2
In Exercises 81–84, verify the identity.
y1 m1 x b1
y2 m2 x b2.
and
Assume that both lines have positive slopes. Derive a formula for the angle between the two lines.Then use your formula to find the angle between the given pair of lines.
81. cosn 1n cos , n is an integer 82. sinn 1 sin , n
y 6
n is an integer
83. a sin B b cos B a 2 b2 sinB C,
y1 = m1x + b1 4
where C arctanb a and a > 0
84. a sin B b cos B a 2 b2 cosB C, where C arctana b and b > 0
(b) a 2 b2 cosB C
85. sin cos
86. 3 sin 2 4 cos 2
87. 12 sin 3 5 cos 3
88. sin 2 cos 2
In Exercises 89 and 90, use the formulas given in Exercises 83 and 84 to write the trigonometric expression in the form a sin B b cos B.
89. 2 sin 2
3 90. 5 cos 4
f h gh
0.05
1 3
x
95. Conjecture Consider the function given by
f sin2
sin2 . 4 4
0.1
In Exercises 97–100, find the inverse function of f. Verify that f f 1x x and f 1f x x. 97. f x 5x 3 99. f x x 2 8
(b) Use a graphing utility to complete the table. 0.02
94. y x and y
Skills Review
(a) What are the domains of the functions f and g?
0.01
93. y x and y 3 x
(b) Write a proof of the formula for sinu v.
cos 6 h cos 6 f h h cos h 1 sin h gh cos sin 6 h 6 h
h
y2 = m2 x + b2
(a) Write a proof of the formula for sinu v.
92. Exploration Let x 6 in the identity in Exercise 91 and define the functions f and g as follows.
4
96. Proof
cosx h cos x h cos xcos h 1 sin x sin h h h
2
Use a graphing utility to graph the function and use the graph to create an identity. Prove your conjecture.
91. Verify the following identity used in calculus.
x
−2
In Exercises 85–88, use the formulas given in Exercises 83 and 84 to write the trigonometric expression in the following forms. (a) a 2 b2 sinB C
θ
0.2
98. f x
7x 8
100. f x x 16
In Exercises 101–104, apply the inverse properties of ln x and e x to simplify the expression. 0.5
2
101. log3 34x3
102. log8 83x
103. eln6x3
104. 12x eln xx2
