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Chapter 5
5.4
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Analytic Trigonometry
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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Analytic Trigonometry
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 54
and 74.
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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Analytic Trigonometry
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.
tan54 tan12
1 tan54 tan12
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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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 arctanba, 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 arctanba and a > 0
84. a sin B b cos B a 2 b2 cosB C,
where C arctanab 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.
cos6 h cos6
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