Sum and Difference Formulas Exploration

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400 Chapter 5 Analytic Trigonometry
5.4
What you should learn
• Use sum and difference
formulas to evaluate
trigonometric functions,
verify identities, and solve
trigonometric equations.
Why you should learn it
Sum and Difference Formulas
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.
Richard Megna/Fundamental Photographs
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
For a proof of the sum and difference formulas, see Proofs in Mathematics on
page 424.
Exploration
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.
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 2 1
2 2 2
2
6 2
.
4
tanu v
tanu v
tan u tan v
1 tan u tan v
tan u tan v
1 tan u tan v
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?
Example 1
Try checking this result on your calculator. You will find that cos 75 0.259.
Now try Exercise 1.

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u
v
FIGURE 5.7
The Granger Collection, New York
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.
2
1
1
1
1 − x2
x
Example 2
Find the exact value of
Solution
Using the fact that
12
Evaluating a Trigonometric Expression
together with the formula for sinu v, you obtain
sin
12
Now try Exercise 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
Now try Exercise 31.
An Application of a Sum Formula
Write cosarctan 1 arccos x as an algebraic expression.
Solution
3 4
Example 3
Example 4
sin
3
sin
3
4
cos
4
3
2 2 1
2 2 2
6 2
.
4
sin
12 .
cos sin
3 4
2
This expression fits the formula for cosu v.
v arccos x are shown in Figure 5.7. So
Angles u arctan 1 and
cosu v cosarctan 1 cosarccos x sinarctan 1 sinarccos x
1 1
x
2 2 1 x 2
x 1 2 x
.
2
Section 5.4 Sum and Difference Formulas 401
Now try Exercise 51.
1
sin 30
2.

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402 Chapter 5 Analytic Trigonometry
Activities
1. Use the sum and difference formulas
to find the exact value of
Answer:
2. Rewrite the expression using the sum
and difference formulas.
Answer:
3. Verify the identity
Answer:
sin
1 cos 0 sin
cos
sin
sin cos cos sin
2 2 2
cos 15.
6 2
4
tan 40 tan 10
1 tan 40 tan 10
tan40 10 tan 50
cos .
2
Example 5 shows how to use a difference formula to prove the cofunction
identity
cos
x sin x.
2
Example 5
Proving a Cofunction Identity
Prove the cofunction identity cos
Solution
Using the formula for cosu v, you have
x sin x.
2
cos
2
Now try Exercise 55.
Sum and difference formulas can be used to rewrite expressions such as
and cos where n is an integer
2
as expressions involving only sin or cos . The resulting formulas are called
reduction formulas.
,
n
sin
2
Example 6
Deriving Reduction Formulas
Simplify each expression.
a. cos b.
2
Solution
a. Using the formula for cosu v, you have
3
cos
x cos
2
3
2
b. Using the formula for tanu v, you have
tan 3
0cos x 1sin x sin x.
3
3
cos cos sin sin
2 2
cos 0 sin 1
sin .
tan tan 3
1 tan tan 3
tan 0
1 tan 0
tan .
cos x sin sin x
2
n
tan 3
Now try Exercise 65.

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3
2
1
−1
−2
−3
y
FIGURE 5.8
π
2
π 2π
π
( (
4
(
π
4
= sin x + + sin x − (y + 1
x
Example 7
Solving a Trigonometric Equation
Find all solutions of sinx sinx in the interval 0, 2.
Solution
Using sum and difference formulas, rewrite the equation as
sin x cos
4
So, the only solutions in the interval 0, 2 are
5
x
4
and
You can confirm this graphically by sketching the graph of
y sinx
1 for 0 ≤ x < 2,
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
Verify that
sinx h sin x
h
where h 0.
Solution
cos x sin
4
4
4
x 7
4 .
sin x
An Application from Calculus
cos x
4
sin h
h
Using the formula for sinu v, you have
Section 5.4 Sum and Difference Formulas 403
4 1
sin x cos cos x sin
4 4 1
2 sin x cos
4 1
2sin x 2
2 1
sin x 2
2 .
sin x 1
2
1 cos h
sin x h
sin h
1 cos h
cos x h sin x h
Now try Exercise 91.
.
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

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404 Chapter 5 Analytic Trigonometry
In Exercises 1– 6, find the exact value of each expression.
1. (a) (b)
2. (a) (b)
3. (a) (b)
4. (a) (b)
5. (a) (b) sin
6. (a) sin315 60 (b) sin 315 sin 60
7
sin
sin
6 3
7
sin
6 3
3
sin
5
sin
4 6
3
cos
5
4 6
cos
cos
4 3
cos120 45
cos 120 cos 45
sin135 30
sin 135 cos 30
4 3
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. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
12
13
12
7
285
105
165
15
12
12
105 60 45
165 135 30
195 225 30
255 300 45
3
12 4 6
12 3 4
9 5
12 4 6
12 6 4
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.
26.
5.4
11
17
13
tan 325 tan 86
1 tan 325 tan 86
tan 140 tan 60
1 tan 140 tan 60
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.
7
5
27.
28. cos
29.
tan 2x tan x
1 tan 2x tan x
30. cos 3x cos 2y sin 3x sin 2y
cos sin sin
7 5 7 5
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.
34.
35.
36.
sin 3 cos 1.2 cos 3 sin 1.2
sin
12
cos
16
cos
4
cos 3
16
cos sin
12 4
tan 25 tan 110
1 tan 25 tan 110
3
sin sin
16 16
tan54 tan12
1 tan54 tan12
In Exercises 37–44, find the exact value of the trigonometric
function given that and cos v (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
3
sin u 5.
5
13
In Exercises 45–50, find the exact value of the trigonometric
function given that and cos v (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
4
5 .
7
sin u 25

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58.
59.
60.
61.
62. sinx y sinx y) sin
63. sinx y sinx y 2 sin x cos y
64. cosx y cosx y 2 cos x cos y
2 x sin2 cosx y cosx y cos
y
2 x sin2 tan
y
0
2
1 tan
4 1 tan
In Exercises 65 –68, simplify the expression algebraically
and use a graphing utility to confirm your answer
graphically.
65. 66.
67. sin 68.
3
cos
2 3
x 2
In Exercises 69 –72, find all solutions of the equation in the
interval [0, 2.
69.
sinx 3 sinx 1 3
70.
1
sinx 6 sinx 6 2
71.
cosx 4 cosx 1 4
72. tanx 2 sinx 0
In Exercises 73 and 74, use a graphing utility to approximate
the solutions in the interval [0, 2.
73.
74.
cos 5
4
x 2cos
x sin x
2
cos sin
cosx 4 cosx 1 4
tanx cosx 0 2
cos x
tan
Section 5.4 Sum and Difference Formulas 405
In Exercises 51–54, write the trigonometric expression as
an algebraic expression.
51. 52.
53.
54.
In Exercises 55– 64, verify the identity.
55. 56.
57. sin
sin
1
x cos x 3 sin x
6 2
sinarcsin x arccos x sinarctan 2x arccos x 75. Harmonic Motion A weight is attached to a spring
cosarccos x arcsin x
suspended vertically from a ceiling. When a driving
cosarccos x arctan x
force is applied to the system, the weight moves
vertically from its equilibrium position, and this motion
is modeled by
x cos x
2
y
where y is the distance from equilibrium (in feet) and t
is the time (in seconds).
(a) Use the identity
1
Model It
1
sin 2t cos 2t
3 4
sin3 x sin x
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 A cos 2 t x
T
show that
t = 0
1
t = T
8
2
t = T
8
a sin B b cos B a 2 b 2 sinB C
where C arctanba, a > 0, to write the model
in the form
y a 2 b 2 sinBt C.
(b) Find the amplitude of the oscillations of the weight.
(c) Find the frequency of the oscillations of the weight.
y 1
y 1
y 1
and
y1 y2 2A cos 2t 2x cos
T
.
y 1 1+ y 2
y 1 1+ y 2
y 1 1+ y 2
y2 A cos 2 t x
T
y 2
y 2
y 2

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406 Chapter 5 Analytic Trigonometry
Synthesis
True or False? In Exercises 77–80, determine whether the
statement is true or false. Justify your answer.
77. sinu ± v sin u ± sin v
78. cosu ± v cos u ± cos v
79.
cosx sin x 80.
2
In Exercises 81–84, verify the identity.
81. is an integer
82. is an integer
83.
where and
84. a sin B b cos B a
where C arctanab and b > 0
2 b2 a sin B b cos B a
C arctanba a > 0
cosB C,
2 b2 n
n
sinB C,
cosn 1 n cos ,
sinn 1 n sin ,
In Exercises 85–88, use the formulas given in Exercises 83
and 84 to write the trigonometric expression in the
following forms.
(a) (b) a2 b2 a cosB C
2 b2 sinB 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 90. 5 cos
4
91. Verify the following identity used in calculus.
cosx h cos x
h
cos xcos h 1
h
92. Exploration Let x 6 in the identity in Exercise 91
and define the functions f and g as follows.
gh cos
cos6 h cos6
f h
h
cos h 1 sin h
h sin
h
6
6
(a) What are the domains of the functions f and g?
(b) Use a graphing utility to complete the table.
h
2
f h
gh
sin x sin h
h
sinx cos x
2
3
0.01 0.02 0.05 0.1 0.2 0.5
(c) Use a graphing utility to graph the functions f and g.
(d) Use the table and the graphs to make a conjecture
about the values of the functions f and g as h → 0.
In Exercises 93 and 94, use the figure, which shows two
lines whose equations are
y1 m1x b1 and y2 m2 x b2 .
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.
93. y x and y 3x
94. y x and
95. Conjecture Consider the function given by
f sin 2
Use a graphing utility to graph the function and use the
graph to create an identity. Prove your conjecture.
96. Proof
(a) Write a proof of the formula for sinu v.
(b) Write a proof of the formula for sinu v.
Skills Review
In Exercises 97–100, find the inverse function of Verify
that and f 1 f f f x x.
1 f.
x x
97. f x 5x 3
98. f x
f x x 2 8
y 1 = m 1 x + b 1
y 1
3 x
−2 2 4
4 sin2
99. 100.
4 .
7 x
8
f x x 16
In Exercises 101–104, apply the inverse properties of
and e to simplify the expression.
x
ln x
101. 102.
103. e ln 104. 12x e xx2
ln6x3
log8 83x2 log3 34x3 6
4
y
θ
y 2 = m 2 x + b 2
x