Download Chapter 5

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Page 351
Analytic Trigonometry
Chapter 5
y
5.1 Using Fundamental Identities
5.2 Verifying Trigonometric
Identities
5.3 Solving Trigonometric
Equations
5.4 Sum and Difference Formulas
5.5 Multiple-Angle and
Product-to-Sum Formulas
Selected Applications
Trigonometric equations and
identities have many real-life
applications. The applications listed
below represent a small sample of
the applications in this chapter.
■ Friction, Exercise 71, page 367
■ Shadow Length, Exercise 72,
page 367
■ Projectile Motion, Exercise 103,
page 378
■ Data Analysis: Unemployment
Rate, Exercise 105, page 379
■ Standing Waves, Exercise 79,
page 385
■ Harmonic Motion, Exercise 80,
page 386
■ Railroad Track, Exercise 129,
page 397
■ Mach Number, Exercise 130,
page 398
y
y
1
1
1
x
π
−1
2π
x
x
−π
π
−1
π
2π
2π
−1
You can use multiple approaches–algebraic, numerical, and graphical–to solve
trigonometric equations. In Chapter 5, you will use all three approaches to solve
trigonometric equations. You will also use trigonometric identities to evaluate
trigonometric functions and simplify trigonometric expressions.
Dan Donovan/MLB Photos/Getty Images
Trigonometry can be used to model projectile motion, such as the flight of a baseball.
Given the angle at which the ball leaves the bat and the initial velocity, you can
determine the distance the ball will travel.
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Analytic Trigonometry
5.1 Using Fundamental Identities
What you should learn
Introduction
In Chapter 4, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how
to use the fundamental identities to do the following.
1.
2.
3.
4.
Evaluate trigonometric functions.
Simplify trigonometric expressions.
Develop additional trigonometric identities.
Solve trigonometric equations.
The fundamental trigonometric identities can
be used to simplify trigonometric expressions.
For instance, Exercise 111 on page 359 shows
you how trigonometric identities can be
used to simplify an expression for the rate
of change of a function, a concept used
in calculus.
Reciprocal Identities
1
csc u
1
csc u sin u
1
sec u
1
sec u cos u
cos u 䊏
Recognize and write the fundamental
trigonometric identities.
Use the fundamental trigonometric identities to evaluate trigonometric functions,
simplify trigonometric expressions, and
rewrite trigonometric expressions.
Why you should learn it
Fundamental Trigonometric Identities
sin u 䊏
1
cot u
1
cot u tan u
tan u Quotient Identities
tan u sin u
cos u
cot u cos u
sin u
Pythagorean Identities
sin2 u cos 2 u 1
1 tan2 u sec 2 u
1 cot 2 u csc 2 u
Cofunction Identities
冢 2 u冣 cos u
tan冢 u冣 cot u
2
sec冢 u冣 csc u
2
sin
冢 2 u冣 sin u
cot冢 u冣 tan u
2
csc冢 u冣 sec u
2
cos
STUDY TIP
Even/Odd Identities
sin共u兲 sin u
cos共u兲 cos u
tan共u兲 tan u
csc共u兲 csc u
sec共u兲 sec u
cot共u兲 cot u
Pythagorean identities are sometimes used in radical form such
as
sin u ± 冪1 cos2 u
Using the Fundamental Identities
One common use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions.
or
tan u ± 冪sec2 u 1
where the sign depends on the
choice of u.
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Section 5.1
353
Using Fundamental Identities
Example 1 Using Identities to Evaluate a Function
3
Use the values sec u 2 and tan u > 0 to find the values of all six trigonometric functions.
Solution
Using a reciprocal identity, you have
cos u 1
1
2
.
sec u 3兾2
3
Using a Pythagorean identity, you have
sin2 u 1 cos 2 u
冢 3冣
1 2
Pythagorean identity
2
2
Substitute 3 for cos u.
4 5
1 .
9 9
Simplify.
TECHNOLOGY TIP
You can use a graphing utility
to check the result of Example 2.
To do this, enter y1 and y2 as
shown below.
Because sec u < 0 and tan u > 0, it follows that u lies in Quadrant III. Moreover,
because sin u is negative when u is in Quadrant III, you can choose the negative
root and obtain sin u 冪5兾3. Now, knowing the values of the sine and cosine,
you can find the values of all six trigonometric functions.
sin u 冪5
csc u 3
2
cos u 3
tan u 1
3
3冪5
冪
sin u
5
5
1
3
sec u cos u
2
sin u
冪5兾3 冪5
cos u
2兾3
2
cot u 1
2
2冪5
冪
tan u
5
5
Now try Exercise 7.
Example 2 Simplifying a Trigonometric Expression
Simplify sin x cos 2 x sin x.
Solution
Select the line style for y1 and
the path style for y2 (see figure
above). The path style, denoted
by , traces the leading edge of
the graph and draws a path. Now,
graph both equations in the same
viewing window, as shown below.
The two graphs appear to coincide, so the expressions appear
to be equivalent. Remember that
in order to be certain that two
expressions are equivalent, you
need to show their equivalence
algebraically, as in Example 2.
First factor out a common monomial factor and then use a fundamental identity.
sin x cos 2 x sin x sin x共cos2 x 1兲
Factor out monomial factor.
sin x共1 cos 2 x兲
Distributive Property
sin x共sin2 x兲
Pythagorean identity
sin3 x
Multiply.
Now try Exercise 29.
2
−
−2
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Example 3 Verifying a Trigonometric Identity
Determine whether the equation appears to be an identity.
?
cos 3x 4 cos3 x 3 cos x
Numerical Solution
Graphical Solution
Use the table feature of a graphing utility set in radian
mode to create a table that shows the values of y1 cos 3x
and y2 4 cos3 x 3 cos x for different values of x, as
shown in Figure 5.1. The values of y1 and y2 appear to
be identical, so cos 3x 4 cos3 x 3 cos x appears to be
an identity.
Use a graphing utility set in radian mode to graph
y1 cos 3x and y2 4 cos3 x 3 cos x in the same
viewing window, as shown in Figure 5.2. (Select the line
style for y1 and the path style for y2.) Because the graphs
appear to coincide, cos 3x 4 cos3 x 3 cos x appears to
be an identity.
2
y1 = cos 3x
−
Figure 5.1
−2
y2 = 4 cos3 x − 3 cos x
Note that if the values of y1 and y2 were not identical, then
the equation would not be an identity.
Figure 5.2
Note that if the graphs of y1 and y2 did not coincide, then
the equation would not be an identity.
Now try Exercise 39.
Example 4 Verifying a Trigonometric Identity
Verify the identity
cos sin csc .
1 cos sin Algebraic Solution
Graphical Solution
sin cos 共sin 兲共sin 兲 (cos 兲共1 cos 兲
1 cos sin 共1 cos 兲共sin 兲
sin2 cos2 cos 共1 cos 兲共sin 兲
Multiply.
1 cos 共1 cos 兲共sin 兲
Pythagorean identity
1
sin Divide out common factor.
csc 4
−2
y1 =
sin θ
cos θ
+
1 + cos θ sin θ
2
Use reciprocal identity.
Notice how the identity is verified. You start with the left side of the equation
(the more complicated side) and use the fundamental trigonometric identities
to simplify it until you obtain the right side.
Now try Exercise 45.
Use a graphing utility set in radian and
dot modes to graph y1 and y2 in the same
viewing window, as shown in Figure 5.3.
Because the graphs appear to coincide,
this equation appears to be an identity.
y2 = csc θ
−4
Figure 5.3
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Section 5.1
Using Fundamental Identities
When factoring trigonometric expressions, it is helpful to find a polynomial
form that fits the expression, as shown in Example 5.
Example 5 Factoring Trigonometric Expressions
Factor (a) sec 2 1 and (b) 4 tan2 tan 3.
Solution
a. Here the expression is a difference of two squares, which factors as
sec2 1 共sec 1兲共sec 1).
b. This expression has the polynomial form ax 2 bx c and it factors as
4 tan2 tan 3 共4 tan 3兲共tan 1兲.
Now try Exercise 51.
On occasion, factoring or simplifying can best be done by first rewriting the
expression in terms of just one trigonometric function or in terms of sine or
cosine alone. These strategies are illustrated in Examples 6 and 7.
Example 6 Factoring a Trigonometric Expression
Factor csc 2 x cot x 3.
Solution
Use the identity csc 2 x 1 cot 2 x to rewrite the expression in terms of the
cotangent.
csc 2 x cot x 3 共1 cot 2 x兲 cot x 3
Pythagorean identity
cot 2 x cot x 2
Combine like terms.
共cot x 2兲共cot x 1兲
Factor.
Now try Exercise 57.
Example 7 Simplifying a Trigonometric Expression
Simplify sin t cot t cos t.
Solution
Begin by rewriting cot t in terms of sine and cosine.
sin t cot t cos t sin t 冢 sin t 冣cos t
cos t
Quotient identity
sin2 t cos 2 t
sin t
Add fractions.
1
csc t
sin t
Pythagorean identity and
reciprocal identity
Now try Exercise 67.
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Analytic Trigonometry
The last two examples in this section involve techniques for rewriting expressions into forms that are used in calculus.
Example 8 Rewriting a Trigonometric Expression
Rewrite
1
so that it is not in fractional form.
1 sin x
Solution
From the Pythagorean identity cos 2 x 1 sin2 x 共1 sin x兲共1 sin x兲, you
can see that multiplying both the numerator and the denominator by 共1 sin x兲
will produce a monomial denominator.
1
1
1 sin x 1 sin x
1 sin x
1 sin x
Multiply numerator and
denominator by 共1 sin x兲.
1 sin x
1 sin2 x
Multiply.
1 sin x
cos 2 x
Pythagorean identity
1
sin x
2
cos x cos 2 x
Write as separate fractions.
1
sin x
2
cos x cos x
1
cos x
sec2 x tan x sec x
Write as separate fractions.
Reciprocal and quotient identities
Now try Exercise 69.
Example 9 Trigonometric Substitution
Use the substitution x 2 tan , 0 < < 兾2, to write 冪4 x 2 as a trigonometric function of .
Solution
Begin by letting x 2 tan . Then you can obtain
冪4 x 2 冪4 共2 tan 兲2
冪4共1 tan2 兲
冪4
sec 2
2 sec .
Substitute 2 tan for x.
Distributive Property
Pythagorean identity
sec > 0 for 0 < <
4+
x2
x
2
Now try Exercise 81.
θ = arctan
2
Figure 5.4 shows the right triangle illustration of the substitution in
Example 9. For 0 < < 兾2, you have
opp x, adj 2, and hyp 冪4 x2.
Try using these expressions to obtain the result shown in Example 9.
Figure 5.4
x
2
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Section 5.1
5.1 Exercises
Using Fundamental Identities
357
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
Fill in the blank to complete the trigonometric identity.
1.
1
_______
cos u
2.
1
_______
cot u
3.
cos u
_______
sin u
4.
1
_______
sin u
5. 1 _______ sec2 u
7. cos
6. 1 cot2 u _______
冢2 u冣 _______
8. csc
9. tan共u兲 _______
冢2 u冣 _______
10. cos共u兲 _______
In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
1
1. sin x ,
2
cos x 2. csc 2, tan 冪3
2
冪3
冪3
3
, cos x 7
5. tan x ,
24
17
7. sec ,
15
(b) ⴚ1
(c) cot x
(e) ⴚtan x
(f) sin x
2
19.
sin共x兲
cos共x兲
冪3
2
x
17.
cot2
冪2
25
sec x 24
6. cot 5, sin (a) sec x
(d) 1
15. sec x cos x
3
3. sec 冪2, sin 4. tan x In Exercises 15–20, match the trigonometric expression
with one of the following.
csc 2
16. tan x csc x
20.
sin关共兾2兲 x兴
cos关共兾2兲 x兴
In Exercises 21–26, match the trigonometric expression
with one of the following.
(a) csc x
(b) tan x
冪26
(d) sin x tan x
26
21. sin x sec x
8
sin 17
18. 共1 cos 2 x兲共csc x兲
x
23. sec x 4
tan4
(e)
sec2
x
(c) sin2 x
(f) sec2 x 1 tan2 x
22. cos2 x共sec2 x 1兲
x
sec x 1
sin2 x
24. cot x sec x
2
cos2关共兾2兲 x兴
cos x
4
3
x , cos x 8. cos
2
5
5
25.
2冪5
2
9. sin共x兲 , tan x 3
5
In Exercises 27– 38, use the fundamental identities to
simplify the expression. Use the table feature of a graphing
utility to check your result numerically.
冢
冣
10. csc共x兲 5,
cos x 冪24
5
11. tan 2, sin < 0
12. sec 3, tan < 0
13. csc is undefined,
cos < 0
14. tan is undefined,
sin > 0
26.
27. cot x sin x
28. cos tan 29. sin 共csc sin 兲
30. sec 2 x共1 sin2 x兲
31.
csc x
cot x
33. sec sin tan 32.
sec csc 34.
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Analytic Trigonometry
冢 2 x冣csc x
cos2 y
1 sin y
36. cot
38.
冢 2 x冣cos x
1
cot x 1
2
In Exercises 39– 44, verify the identity algebraically. Use the
table feature of a graphing utility to check your result
numerically.
39. sin cos cot csc 40. 共sec tan 兲共csc 1兲 cot 41.
cos sec tan 1 sin 42.
1 csc sec cot cos 43.
In Exercises 45–50, verify the identity algebraically. Use a
graphing utility to check your result graphically.
45. csc tan sec 46. sin csc sin2 cos2 sin2 cos 47. 1 1 cos 1 sec tan 2 csc 1 sec tan cot共 兲
cos 49.
csc 50.
冢2 冣
tan共 兲
63. 共csc x 1兲共csc x 1兲
64. 共5 5 sin x兲共5 5 sin x兲
65.
1
1
1 cos x 1 cos x
66.
1
1
sec x 1 sec x 1
sec2 x
tan x
cos x
1 sin x
1 sin x
cos x
In Exercises 69–72, rewrite the expression so that it is not in
fractional form.
69.
sin2 y
1 cos y
70.
5
tan x sec x
71.
3
sec x tan x
72.
tan2 x
csc x 1
Numerical and Graphical Analysis In Exercises 73 –76,
use a graphing utility to complete the table and graph the
functions in the same viewing window. Make a conjecture
about y1 and y2.
0.2
x
0.4
0.6
0.8
1.0
1.2
1.4
y1
y2
csc 73. y1 cos
In Exercises 51– 60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to
check your result graphically.
51. cot2 x cot2 x cos2 x
53.
62. 共tan x sec x兲共tan x sec x兲
68.
sin 1 cos 2 csc sin 1 cos csc
61. 共sin x cos x兲2
67. tan x sin cos cos sin sec csc 44.
sin cos 48.
In Exercises 61–68, perform the indicated operation and
use the fundamental identities to simplify.
cos2 x 4
cos x 2
52. sec2 x tan2 x sec2 x
54.
csc2 x 1
csc x 1
冢 2 x冣,
y2 sin x
74. y1 cos x sin x tan x, y2 sec x
75. y1 cos x
1 sin x
, y2 1 sin x
cos x
76. y1 sec4 x sec2 x,
y2 tan2 x tan4 x
56. 1 2 sin2 x sin4 x
In Exercises 77–80, use a graphing utility to determine
which of the six trigonometric functions is equal to the
expression.
57. sin4 x cos4 x
77. cos x cot x sin x
55. tan4 x 2 tan2 x 1
58. sec4 x tan4 x
59. csc3 x csc2 x csc x 1
60. sec x sec x sec x 1
3
2
79. sec x cos x
1 sin x
78. sin x共cot x tan x兲
80.
cos 1 1 sin 2 cos 1 sin 冢
冣
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Section 5.1
In Exercises 81– 92, use the trigonometric substitution to
write the algebraic expression as a trigonometric function
of ␪, where 0 < ␪ < ␲/2.
x 5 sin 81. 冪25 x 2,
82. 冪64 x 3 sec 83. 冪x 2 9,
x 10 tan 84. 冪x 2 100,
85. 冪9 x2,
x 3 sin 86. 冪4 x2,
x 2 cos 87. 冪4x2 9,
2x 3 tan 88. 冪9x2 4,
3x 2 tan True or False? In Exercises 113 and 114, determine
whether the statement is true or false. Justify your answer.
4x 3 sec 90. 冪9x2 25,
3x 5 sec x 冪2 sin 92. 冪5 x2,
x 冪5 cos 91. 冪2 111. Rate of Change The rate of change of the function
f 共x兲 csc x sin x is given by the expression
csc x cot x cos x. Show that this expression can also be
written as cos x cot 2 x.
Synthesis
89. 冪16x2 9,
x2,
114. cos sec 1
113. sin csc 1
In Exercises 115–118, fill in the blanks. (Note: x → c1
indicates that x approaches c from the right, and x → cⴚ
indicates that x approaches c from the left.)
In Exercises 93–96, use a graphing utility to solve the
equation for ␪, where 0 } ␪ < 2␲.
93. sin 冪1 cos2 115. As x →
, sin x → 䊏 and csc x → 䊏.
2
116. As x → 0 , cos x → 䊏 and sec x → 䊏.
, tan x → 䊏 and cot x → 䊏.
2
94. cos 冪1 sin2 117. As x →
95. sec 冪1 tan2 118. As x → , sin x → 䊏 and csc x → 䊏.
96. tan 冪sec2
1
In Exercises 97–100, rewrite the expression as a single
logarithm and simplify the result.
ⱍ ⱍ ⱍ ⱍ
lnⱍcsc ⱍ lnⱍtan ⱍ
ln共1 sin x兲 lnⱍsec xⱍ
lnⱍcot tⱍ ln共1 tan2 t兲
97. ln cos ln sin 98.
99.
359
112. Rate of Change The rate of change of the function
f 共x兲 sec x cos x is given by the expression
sec x tan x sin x. Show that this expression can also be
written as sin x tan2 x.
x 2 cos 16x 2,
Using Fundamental Identities
119. Write each of the other trigonometric functions of in
terms of sin .
120. Write each of the other trigonometric functions of in
terms of cos .
121. Use the definitions of sine and cosine to derive the
Pythagorean identity sin2 cos2 1.
In Exercises 101–106, show that the identity is not true for
all values of ␪. (There are many correct answers.)
122. Writing Use the Pythagorean identity sin2 cos2 1 to derive the other Pythagorean identities
1 tan2 sec2 and 1 cot2 csc2 . Discuss
how to remember these identities and other fundamental
identities.
101. cos 冪1 sin2 102. tan 冪sec2 1
103. sin 冪1 cos2 104. sec 冪1 tan2 Skills Review
105. csc 冪1 cot2 106. cot 冪csc2 1
100.
In Exercises 107–110, use the table feature of a graphing
utility to demonstrate the identity for each value of ␪.
107. csc2 cot2 1, (a) 132 (b) 2
7
108. tan2 1 sec2 , (a) 346 (b) 3.1
109. cos
冢
sin , (a) 80 (b) 0.8
2
冣
110. sin共 兲 sin , (a) 250 (b) 1
2
In Exercises 123–126, sketch the graph of the function.
(Include two full periods.)
123. f 共x兲 1
sin x
2
124. f 共x兲 2 tan
x
2
125. f 共x兲 1
cot x 2
4
126. f 共x兲 3
cos共x 兲 3
2
冢
冣
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5.2 Verifying Trigonometric Identities
What you should learn
Introduction
In this section, you will study techniques for verifying trigonometric identities. In
the next section, you will study techniques for solving trigonometric equations.
The key to both verifying identities and solving equations is your ability to
use the fundamental identities and the rules of algebra to rewrite trigonometric
expressions.
Remember that a conditional equation is an equation that is true for only
some of the values in its domain. For example, the conditional equation
sin x 0
Conditional equation
䊏
Verify trigonometric identities.
Why you should learn it
You can use trigonometric identities to
rewrite trigonometric expressions. For
instance, Exercise 72 on page 367 shows you
how trigonometric identities can be used to
simplify an equation that models the length
of a shadow cast by a gnomon (a device used
to tell time).
is true only for x n, where n is an integer. When you find these values, you
are solving the equation.
On the other hand, an equation that is true for all real values in the domain
of the variable is an identity. For example, the familiar equation
sin2 x 1 cos 2 x
Identity
is true for all real numbers x. So, it is an identity.
Verifying Trigonometric Identities
BSCHMID/Getty Images
Verifying that a trigonometric equation is an identity is quite different from
solving an equation. There is no well-defined set of rules to follow in verifying
trigonometric identities, and the process is best learned by practice.
Guidelines for Verifying Trigonometric Identities
1. Work with one side of the equation at a time. It is often better to
work with the more complicated side first.
2. Look for opportunities to factor an expression, add fractions, square a
binomial, or create a monomial denominator.
3. Look for opportunities to use the fundamental identities. Note which
functions are in the final expression you want. Sines and cosines pair
up well, as do secants and tangents, and cosecants and cotangents.
4. If the preceding guidelines do not help, try converting all terms to
sines and cosines.
5. Always try something. Even making an attempt that leads to a dead
end provides insight.
Verifying trigonometric identities is a useful process if you need to convert a
trigonometric expression into a form that is more useful algebraically. When you
verify an identity, you cannot assume that the two sides of the equation are equal
because you are trying to verify that they are equal. As a result, when verifying
identities, you cannot use operations such as adding the same quantity to each
side of the equation or cross multiplication.
Prerequisite Skills
To review the differences among
an identity, an expression, and an
equation, see Appendix B.3.
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Section 5.2
Verifying Trigonometric Identities
361
Example 1 Verifying a Trigonometric Identity
Verify the identity
sec2 1
sin2 .
sec2 Solution
Because the left side is more complicated, start with it.
sec2 1 共tan2 1兲 1
sec2 sec2 tan2 sec2 Pythagorean identity
STUDY TIP
Simplify.
tan2 共cos 2 兲
Reciprocal identity
sin2 共cos2兲
cos2 Quotient identity
sin2 Simplify.
Remember that an identity is
true only for all real values in
the domain of the variable. For
instance, in Example 1 the identity is not true when 兾2
because sec2 is not defined
when 兾2.
Now try Exercise 5.
There can be more than one way to verify an identity. Here is another way to
verify the identity in Example 1.
sec2 1 sec2 1
2
2
sec sec sec2 Rewrite as the difference of fractions.
1 cos 2 Reciprocal identity
sin2 Pythagorean identity
Example 2 Combining Fractions Before Using Identities
Verify the identity
1
1
2 sec2 .
1 sin 1 sin Algebraic Solution
Numerical Solution
1
1
1 sin 1 sin 1 sin 1 sin 共1 sin 兲共1 sin 兲
Add fractions.
2
1 sin2 Simplify.
2
cos2 Pythagorean identity
2 sec2 Now try Exercise 31.
Use the table feature of a graphing utility set in
radian mode to create a table that shows the
values of y1 1兾共1 sin x兲 1兾共1 sin x兲
and y2 2兾cos2 x for different values of x, as
shown in Figure 5.5. From the table, you
can see that the values appear to be identical,
so 1兾共1 sin x兲 1兾共1sin x兲 2 sec 2 x appears to be an identity.
Reciprocal identity
Figure 5.5
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Analytic Trigonometry
Example 3 Verifying a Trigonometric Identity
Verify the identity 共tan2 x 1兲共cos 2 x 1兲 tan2 x.
Algebraic Solution
Graphical Solution
By applying identities before multiplying, you obtain the following.
共
tan2 x
1兲共
cos 2 x
1兲 共
兲共
sec2 x
兲
sin2 x
sin2 x
cos 2 x
冢cos x冣
sin x
Pythagorean identities
Reciprocal identity
2
tan2 x
Rule of exponents
Quotient identity
Use a graphing utility set in radian mode
to graph the left side of the identity
y1 共tan2 x 1兲共cos2 x 1兲 and the right
side of the identity y2 tan2 x
in the same viewing window, as shown in
Figure 5.6. (Select the line style for y1
and the path style for y2.) Because the
graphs appear to coincide, 共tan2 x 1兲 共cos2 x 1兲 tan2 x appears to be an
identity.
2
y1 = (tan2 x + 1)(cos2 x − 1)
−2
2
−3
Now try Exercise 39.
y2 = −tan2 x
Figure 5.6
Example 4 Converting to Sines and Cosines
Verify the identity tan x cot x sec x csc x.
Solution
In this case there appear to be no fractions to add, no products to find, and no
opportunities to use the Pythagorean identities. So, try converting the left side to
sines and cosines.
TECHNOLOGY TIP
sin x
cos x
cos x
sin x
Quotient identities
sin2 x cos 2 x
cos x sin x
Add fractions.
1
cos x sin x
Pythagorean identity
tan x cot x 1
cos x
1
sin x
sec x csc x
Now try Exercise 41.
Although a graphing utility can
be useful in helping to verify an
identity, you must use algebraic
techniques to produce a valid
proof. For example, graph the
two functions
y1 sin 50x
Product of fractions
Reciprocal identities
y2 sin 2x
in a trigonometric viewing
window. Although their graphs
seem identical, sin 50x sin 2x.
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Section 5.2
Verifying Trigonometric Identities
363
Recall from algebra that rationalizing the denominator using conjugates is,
on occasion, a powerful simplification technique. A related form of this technique
works for simplifying trigonometric expressions as well. For instance, to simplify
1兾共1 cos x兲, multiply the numerator and the denominator by 1 cos x.
1
1
1 cos x
1 cos x 1 cos x 1 cos x
冢
1 cos x
1 cos2 x
1 cos x
sin2 x
冣
csc2 x共1 cos x兲
As shown above, csc2 x共1 cos x兲 is considered a simplified form of 1兾共1 cos x兲
because the expression does not contain any fractions.
Example 5 Verifying a Trigonometric Identity
Verify the identity
sec x tan x TECHNOLOGY SUPPORT
For instructions on how to
use the radian and dot modes,
see Appendix A; for specific
keystrokes, go to this textbook’s
Online Study Center.
cos x
.
1 sin x
Algebraic Solution
Graphical Solution
Begin with the right side because you can create a monomial
denominator by multiplying the numerator and denominator by
共1 sin x兲.
Use a graphing utility set in the radian and dot
modes to graph y1 sec x tan x and
y2 cos x兾共1 sin x兲 in the same viewing window,
as shown in Figure 5.7. Because the graphs appear
to coincide, sec x tan x cos x兾共1 sin x兲
appears to be an identity.
cos x
1 sin x
cos x
1 sin x 1 sin x 1 sin x
冢
冣
Multiply numerator and
denominator by
共1 sin x兲.
cos x cos x sin x
1 sin2 x
Multiply.
cos x cos x sin x
cos 2 x
Pythagorean identity
cos x sin x
cos x
2
cos x
cos2 x
Write as separate fractions.
5
−
7
2
9
2
−5
1
sin x
cos x cos x
Simplify.
sec x tan x
Identities
y1 = sec x + tan x
Figure 5.7
Now try Exercise 47.
In Examples 1 through 5, you have been verifying trigonometric identities by
working with one side of the equation and converting it to the form given on the
other side. On occasion it is practical to work with each side separately to obtain
one common form equivalent to both sides. This is illustrated in Example 6.
y2 =
cos x
1 − sin x
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Analytic Trigonometry
Example 6 Working with Each Side Separately
Verify the identity
1 sin cot 2 .
1 csc sin Algebraic Solution
Numerical Solution
Working with the left side, you have
Use the table feature of a graphing utility set in radian
mode to create a table that shows the values of
y1 cot2 x兾共1 csc x兲 and y2 共1 sin x兲/sin x
for different values of x, as shown in Figure 5.8. From
the table you can see that the values appear to be
identical, so cot2 x兾共1 csc x兲 共1 sin x兲兾sin x
appears to be an identity.
cot 2 csc2 1
1 csc 1 csc Pythagorean identity
共csc 1兲共csc 1兲
1 csc csc 1.
Factor.
Simplify.
Now, simplifying the right side, you have
1 sin 1
sin sin sin sin Write as separate fractions.
csc 1.
Reciprocal identity
The identity is verified because both sides are equal to
csc 1.
Now try Exercise 49.
In Example 7, powers of trigonometric functions are rewritten as more
complicated sums of products of trigonometric functions. This is a common
procedure used in calculus.
Example 7 Examples from Calculus
Verify each identity.
b. sin3 x cos4 x 共cos4 x cos 6 x兲sin x
a. tan4 x tan2 x sec2 x tan2 x
Solution
a. tan4 x 共tan2 x兲共tan2 x兲
Write as separate factors.
tan2 x共sec2 x 1兲
tan2
x sec2
x
Pythagorean identity
tan2
x
Multiply.
b. sin3 x cos4 x sin2 x cos4 x sin x
共1 cos2
x兲
cos4
x sin x
共cos4 x cos6 x兲sin x
Write as separate factors.
Pythagorean identity
Multiply.
Now try Exercise 63.
TECHNOLOGY TIP Remember that you can use a graphing utility to assist
in verifying an identity by creating a table or by graphing.
Figure 5.8
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Section 5.2
5.2 Exercises
Verifying Trigonometric Identities
365
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
In Exercises 1 and 2, fill in the blanks.
1. An equation that is true for only some values in its domain is called a _______ equation.
2. An equation that is true for all real values in its domain is called an _______ .
In Exercises 3–10, fill in the blank to complete the trigonometric identity.
3.
1
_______
tan u
4.
1
_______
csc u
5.
sin u
_______
cos u
6.
1
_______
sec u
7. sin2 u _______ 1
8. tan
9. sin共u兲 _______
冢2 u冣 _______
10. sec共u兲 _______
13. y1 csc x sin x,
y2 cos x cot x
1. sin t csc t 1
14. y1 sec x cos x,
y2 sin x tan x
2. sec y cos y 1
15. y1 sin x cos x cot x,
In Exercises 1–10, verify the identity.
3.
4.
16. y1 cos x sin x tan x, y2 sec x
csc2 x
csc x sec x
cot x
sin2 t
17. y1 1
1
,
tan x cot x
18. y1 1
1
, y2 csc x sin x
sin x csc x
cos2 t
tan2 t
5. cos 2 sin2 1 2 sin2 6. cos 2 sin2 2 cos 2 1
Error Analysis
7. tan2 6 sec2 5
y2 tan x cot x
In Exercises 19 and 20, describe the error.
19. 共1 tan x兲关1 cot共x兲兴
8. 2 csc 2 z 1 cot2 z
共1 tan x兲共1 cot x兲
9. 共1 sin x兲共1 sin x兲 cos2 x
1 cot x tan x tan x cot x
10. tan y共csc y 1兲 1
2
2
Numerical, Graphical, and Algebraic Analysis In Exercises
11–18, use a graphing utility to complete the table and
graph the functions in the same viewing window. Use both
the table and the graph as evidence that y1 ⴝ y2 . Then verify
the identity algebraically.
1 cot x tan x 1
2 cot x tan x
20.
1 sec共 兲
1 sec sin共 兲 tan共 兲 sin tan x
0.2
0.4
0.6
0.8
1.0
y1
y2
1
, y2 csc x sin x
11. y1 sec x tan x
12. y1 csc x 1
,
1 sin x
y2 csc x
y2 csc x
1.2
1.4
1 sec 冤
共sin 兲 1 冢cos1 冣冥
1 sec sin 共1 sec 兲
1
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Analytic Trigonometry
In Exercises 21–30, verify the identity.
21.
sin1兾2
22.
sec6
23. cot
x cos x sin
5兾2
x cos x cos x冪sin x
40. csc x共csc x sin x兲 3
x共sec x tan x兲 sec4 x共sec x tan x兲 sec5 x tan3 x
冢 2 x冣csc x sec x
sin x cos x
cot x csc2 x
sin x
41.
cot x tan x
csc x
sin x
42.
1 csc cot cos sec 24.
sec关共兾2兲 x兴
sec x
tan关共兾2兲 x兴
43. csc4 x 2 csc2 x 1 cot4 x
25.
csc共x兲
cot x
sec共x兲
45. sec4 tan4 1 2 tan2 26. 共1 sin y兲关1 sin共y兲兴 cos2 y
44. sin x共1 2 cos2 x cos4 x兲 sin5 x
46. csc4 cot4 2 csc2 1
47.
sin 1 cos 1 cos sin 48.
1 csc共兲
sec cos共兲 cot共兲
csc 1
cot csc 1
cot 49.
29.
tan x tan y
sin x cos y cos x sin y
cos x cos y sin x sin y 1 tan x tan y
tan3 1
tan2 tan 1
tan 1
50.
30.
tan x tan y
cot x cot y
1 tan x tan y cot x cot y 1
sin3 cos3 1 sin cos sin cos 27.
28.
cos共兲
sec tan 1 sin共兲
In Exercises 31–38, verify the identity algebraically. Use the
table feature of a graphing utility to check your result
numerically.
31.
32.
sin x sin y
cos x cos y
0
sin x sin y
cos x cos y
tan x cot y
tan y cot x
tan x cot y
33.
1 sin 冪11 sin
sin ⱍcos ⱍ
34.
1 cos 冪11 cos
cos ⱍsin ⱍ
35. sin2
冢 2 x冣 sin
2
x1
Conjecture In Exercises 51–54, use a graphing utility to
graph the trigonometric function. Use the graph to make a
conjecture about a simplification of the expression. Verify
the resulting identity algebraically.
51. y 1
1
cot x 1 tan x 1
52. y cos x
sin x cos x
1 tan x sin x cos x
53. y 1
cos2 x
sin x
sin x
54. y sin t cot2 t
csc t
In Exercises 55–58, use the properties of logarithms and
trigonometric identities to verify the identity.
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
55. ln cot ln cos ln sin ⱍ
ⱍ
ⱍ
36. sec2 y cot 2
y 1
2
56. ln sec ln cos 37. sin x csc
x tan x
2
58. ln csc cot ln csc cot 冢
冢
38. sec2
冢
冣
冣
x 1 cot2 x
2
冣
ⱍ
57. ln共1 cos 兲 ln共1 cos 兲 2 ln sin ⱍ
ⱍ
ⱍ
ⱍ
In Exercises 59–62, use the cofunction identities to evaluate
the expression without using a calculator.
59. sin2 35 sin2 55
In Exercises 39–50, verify the identity algebraically. Use a
graphing utility to check your result graphically.
39. 2 sec2 x 2 sec2 x sin2 x sin2 x cos 2 x 1
ⱍ
60. cos 2 14 cos 2 76
61. cos2 20 cos2 52 cos2 38 cos2 70
62. sin2 18 sin2 40 sin2 50 sin2 72
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Section 5.2
Verifying Trigonometric Identities
367
In Exercises 63–66, powers of trigonometric functions are
rewritten to be useful in calculus. Verify the identity.
76. sin共x2兲 sin2共x兲
63. tan5 x tan3 x sec2 x tan3 x
In Exercises 77–80, (a) verify the identity and (b) determine
if the identity is true for the given value of x. Explain.
tan2
x共
tan2
x
x
65.
cos 3
x sin x 共sin x sin x兲cos x
66.
sin 4
x
2
2
cos 4
x12
tan4
x兲
64.
sec4
sec2
x
4
cos 2
x2
In Exercises 67–70, verify the identity.
67. tan共sin1 x兲 x
冪1 x2
68. cos共sin1 x兲 冪1 x2
冢
x1
x1
4
冪16 共x 1兲2
冢
冪4 共x 1兲2
x1
2
x1
69. tan sin1
70. tan cos1
77.
sin x
1 cos x
,
1 cos x
sin x
78.
sec x
tan x
, x
tan x
sec x cos x
cos 4 x
冣
79. csc x cot x 80.
冣
71. Friction The forces acting on an object weighing W units
on an inclined plane positioned at an angle of with the
horizontal (see figure) are modeled by
sin x
,
1 cos x
2
In Exercises 81–84, use the trigonometric substitution to
write the algebraic expression as a trigonometric function
of ␪, where 0 < ␪ < ␲/2. Assume a > 0.
81. 冪a2 u2,
u a sin 82. 冪a2 u2,
u a cos W cos W sin 83.
u a tan where is the coefficient of friction. Solve the equation for
and simplify the result.
84. 冪u2 a2,
u a sec θ
x
cot x 1 1 tan x
, x
cot x 1 1 tan x
4
u2,
W
x0
冪a2
Think About It In Exercises 85 and 86, explain why the
equation is not an identity and find one value of the variable
for which the equation is not true.
85. 冪tan2 x tan x
86. sin 冪1 cos2 冤
87. Verify that for all integers n, cos
冤
88. Verify that for all integers n, sin
共2n 1兲
0.
2
冥
共12n 1兲
1
.
6
2
冥
Skills Review
72. Shadow Length The length s of the shadow cast by a
vertical gnomon (a device used to tell time) of height h
when the angle of the sun above the horizon is can be
modeled by the equation
s
h sin共90 兲
.
sin Show that the equation is equivalent to s h cot .
Synthesis
True or False? In Exercises 73–76, determine whether the
statement is true or false. Justify your answer.
In Exercises 89–92, find a polynomial function with real
coefficients that has the given zeros. (There are many
correct answers.)
89. 1, 8i, 8i
90. i, i, 4i, 4i
91. 4, 6 i, 6 i
92. 0, 0, 2, 1 i
In Exercises 93–96, sketch the graph of the function by
hand.
93. f 共x兲 2x 3
94. f 共x兲 2x3
95. f 共x兲 2x 1
96. f 共x兲 2x1 3
73. There can be more than one way to verify a trigonometric
identity.
In Exercises 97–100, state the quadrant in which ␪ lies.
74. Of the six trigonometric functions, two are even.
97. csc > 0 and tan < 0
98. cot > 0 and cos < 0
99. sec > 0 and sin < 0
100. cot > 0 and sec < 0
75. The equation sin2 cos2 1 tan2 is an identity,
because sin2共0兲 cos2共0兲 1 and 1 tan2共0兲 1.
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Analytic Trigonometry
5.3 Solving Trigonometric Equations
What you should learn
Introduction
To solve a trigonometric equation, use standard algebraic techniques such as
collecting like terms and factoring. Your preliminary goal is to isolate the trigonometric function involved in the equation.
䊏
䊏
䊏
Example 1 Solving a Trigonometric Equation
2 sin x 1 0
䊏
Original equation
2 sin x 1
Why you should learn it
Add 1 to each side.
sin x 12
Divide each side by 2.
1
To solve for x, note in Figure 5.9 that the equation sin x 2 has solutions
x 兾6 and x 5兾6 in the interval 关0, 2兲. Moreover, because sin x has a
period of 2, there are infinitely many other solutions, which can be written as
x
2n
6
x
and
5
2n
6
Use standard algebraic techniques to
solve trigonometric equations.
Solve trigonometric equations of
quadratic type.
Solve trigonometric equations involving
multiple angles.
Use inverse trigonometric functions to
solve trigonometric equations.
You can use trigonometric equations to solve
a variety of real-life problems.For instance,
Exercise 100 on page 378 shows you how
solving a trigonometric equation can help
answer questions about the position of the
sun in Cheyenne,Wyoming.
General solution
where n is an integer, as shown in Figure 5.9.
y
π
x = − 2π
6
y=
x=
1
2
1
π
6
−π
x=
π
+ 2π
6
x
π
x = 5π − 2 π
6
SuperStock
x = 5π
6
−1
x = 5π + 2π
6
y = sin x
Figure 5.9
Now try Exercise 25.
1
Figure 5.10 verifies that the equation sin x 2 has infinitely many solutions.
Any angles that are coterminal with 兾6 or 5兾6 are also solutions of the equation.
sin 5π + 2nπ = 1
2
6
(
Figure 5.10
(
5π
6
π
6
sin π + 2nπ = 1
2
6
(
(
Prerequisite Skills
If you have trouble finding coterminal
angles, review Section 4.1.
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Section 5.3
369
Solving Trigonometric Equations
Example 2 Collecting Like Terms
Find all solutions of sin x 冪2 sin x in the interval 关0, 2兲.
Algebraic Solution
Numerical Solution
Rewrite the equation so that sin x is isolated on one side of
the equation.
Use the table feature of a graphing utility set in radian
mode to create a table that shows the values of
y1 sin x 冪2 and y2 sin x for different values
of x. Your table should go from x 0 to x 2 using
increments of 兾8, as shown in Figure 5.11. From the
table, you can see that the values of y1 and y2 appear to be
identical when x ⬇ 3.927 ⬇ 5兾4 and x ⬇ 5.4978 ⬇
7兾4. These values are the approximate solutions of
sin x 冪2 sin x.
sin x 冪2 sin x
sin x sin x 冪2
2 sin x 冪2
sin x 冪2
2
Write original equation.
Add sin x to and subtract
冪2 from each side.
Combine like terms.
Divide each side by 2.
The solutions in the interval 关0, 2兲 are
x
5
4
x
and
7
.
4
Now try Exercise 35.
Figure 5.11
Example 3 Extracting Square Roots
Solve 3
tan2
TECHNOLOGY SUPPORT
For instructions on how to use
the table feature, see Appendix A;
for specific keystrokes, go to this
textbook’s Online Study Center.
x 1 0.
Solution
Rewrite the equation so that tan x is isolated on one side of the equation.
3 tan2 x 1
tan2 x Add 1 to each side.
1
3
tan x ±
Divide each side by 3.
1
Extract square roots.
冪3
6
Because tan x has a period of , first find all solutions in the interval 关0, 兲. These
are x 兾6 and x 5兾6. Finally, add multiples of to each of these solutions
to get the general form
x n
6
and
5
x
n
6
−
General solution
where n is an integer. The graph of y 3 tan2 x 1, shown in Figure 5.12,
confirms this result.
Now try Exercise 37.
Recall that the solutions of an equation correspond to the x-intercepts of the
graph of the equation. For instance, the graph in Figure 5.12 has x-intercepts at
兾6, 5兾6, 7兾6, and so on.
y = 3 tan2 x − 1
5
2
2
−2
Figure 5.12
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Analytic Trigonometry
The equations in Examples 1, 2, and 3 involved only one trigonometric
function. When two or more functions occur in the same equation, collect all
terms on one side and try to separate the functions by factoring or by using appropriate identities. This may produce factors that yield no solutions, as illustrated in
Example 4.
Example 4 Factoring
Solve cot x cos2 x 2 cot x.
Solution
Begin by rewriting the equation so that all terms are collected on one side of the
equation.
cot x cos 2 x 2 cot x
Write original equation.
cot x cos 2 x 2 cot x 0
Subtract 2 cot x from each side.
cot x共cos2 x 2兲 0
cot x 0
and
−
3
x20
cos2 x 2
−3
cos x ± 冪2
The equation cot x 0 has the solution x 兾2 关in the interval 共0, 兲兴. No solution is obtained for cos x ± 冪2 because ± 冪2 are outside the range of the
cosine function. Because cot x has a period of , the general form of the solution
is obtained by adding multiples of to x 兾2, to get
x
y = cot x cos2 x − 2 cot x
Factor.
By setting each of these factors equal to zero, you obtain the following.
cos2
3
n
2
General solution
where n is an integer. The graph of y cot x cos 2 x 2 cot x (in dot mode),
shown in Figure 5.13, confirms this result.
Now try Exercise 39.
Equations of Quadratic Type
Many trigonometric equations are of quadratic type ax2 bx c 0. Here are
a few examples.
Quadratic in sin x
2 sin2 x sin x 1 0
Quadratic in sec x
sec2 x 3 sec x 2 0
2共sin x兲2 sin x 1 0
共sec x兲2 3 共sec x兲 2 0
To solve equations of this type, factor the quadratic or, if factoring is not
possible, use the Quadratic Formula.
Figure 5.13
Exploration
Using the equation in Example
4, explain what would happen
if you divided each side of the
equation by cot x. Why is this
an incorrect method to use
when solving an equation?
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Section 5.3
371
Solving Trigonometric Equations
Example 5 Factoring an Equation of Quadratic Type
Find all solutions of 2 sin2 x sin x 1 0 in the interval 关0, 2兲.
Algebraic Solution
Graphical Solution
Treating the equation as a quadratic in sin x and
factoring produces the following.
Use a graphing utility set in radian mode to graph y 2 sin2 x sin x 1 for 0 ≤ x < 2, as shown in Figure 5.14.
Use the zero or root feature or the zoom and trace features to
approximate the x-intercepts to be
2 sin2 x sin x 1 0
共2 sin x 1兲共sin x 1兲 0
Write original
equation.
Factor.
Setting each factor equal to zero, you obtain the
following solutions in the interval 关0, 2兲.
2 sin x 1 0
sin x x
and sin x 1 0
1
2
x ⬇ 1.571 ⬇
7
11
, x ⬇ 3.665 ⬇
, and x ⬇ 5.760 ⬇
.
2
6
6
These values are the approximate solutions of 2 sin2 x sin x 1 0 in the interval [0, 2兲.
sin x 1
7 11
,
6 6
x
3
2
y = 2 sin2 x − sin x − 1
2
0
−2
Now try Exercise 49.
Figure 5.14
When working with an equation of quadratic type, be sure that the equation
involves a single trigonometric function, as shown in the next example.
Example 6 Rewriting with a Single Trigonometric Function
Solve 2 sin2 x 3 cos x 3 0.
Solution
Begin by rewriting the equation so that it has only cosine functions.
2 sin2 x 3 cos x 3 0
Write original equation.
2共1 cos 2 x兲 3 cos x 3 0
2
cos 2
Pythagorean identity
Combine like terms and
multiply each side by 1.
x 3 cos x 1 0
共2 cos x 1兲共cos x 1兲 0
Factor.
By setting each factor equal to zero, you can find the solutions in the interval
关0, 2兲 to be x 0, x 兾3, and x 5兾3. Because cos x has a period of 2,
the general solution is
x 2n,
x
2n,
3
x
5
2n
3
−
y = 2 sin2 x + 3 cos x − 3
3
2
General solution
where n is an integer. The graph of y 2 sin2 x 3 cos x 3, shown in Figure
5.15, confirms this result.
Now try Exercise 51.
1
−6
Figure 5.15
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Analytic Trigonometry
Sometimes you must square each side of an equation to obtain a quadratic.
Because this procedure can introduce extraneous solutions, you should check any
solutions in the original equation to see whether they are valid or extraneous.
Example 7 Squaring and Converting to Quadratic Type
Find all solutions of cos x 1 sin x in the interval 关0, 2兲.
Solution
It is not clear how to rewrite this equation in terms of a single trigonometric function. Notice what happens when you square each side of the equation.
cos x 1 sin x
x 2 cos x 1 1 Use a graphing utility to
confirm the solutions found in
Example 7 in two different
ways. Do both methods produce
the same x-values? Which
method do you prefer? Why?
Write original equation.
cos 2 x 2 cos x 1 sin2 x
cos 2
Exploration
cos 2
Square each side.
x
Pythagorean identity
2 cos 2 x 2 cos x 0
Combine like terms.
2 cos x共cos x 1兲 0
Factor.
1. Graph both sides of the
equation and find the
x-coordinates of the points at
which the graphs intersect.
Setting each factor equal to zero produces the following.
2 cos x 0
and
cos x 1 0
cos x 0
x
cos x 1
3
,
2 2
Left side: y cos x 1
Right side: y sin x
x
2. Graph the equation
y cos x 1 sin x and
find the x-intercepts of the
graph.
Because you squared the original equation, check for extraneous solutions.
Check
cos
cos
?
1 sin
2
2
Substitute 兾2 for x.
011
Solution checks.
3
3
?
1 sin
2
2
0 1 1
?
cos 1 sin 1 1 0
✓
Substitute 3兾2 for x.
Solution does not check.
y = cos x + 1 − sin x
Substitute for x.
Solution checks.
✓
Of the three possible solutions, x 3兾2 is extraneous. So, in the interval
关0, 2兲, the only solutions are x 兾2 and x . The graph of
y cos x 1 sin x, shown in Figure 5.16, confirms this result because the
graph has two x-intercepts 共at x 兾2 and x 兲 in the interval 关0, 2兲.
Now try Exercise 53.
3
0
−1
Figure 5.16
2
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Section 5.3
373
Solving Trigonometric Equations
Functions Involving Multiple Angles
The next two examples involve trigonometric functions of multiple angles of the
forms sin ku and cos ku. To solve equations of these forms, first solve the equation for ku, then divide your result by k.
Example 8 Functions of Multiple Angles
Solve 2 cos 3t 1 0.
Solution
2 cos 3t 1 0
Write original equation.
2 cos 3t 1
cos 3t Add 1 to each side.
1
2
Divide each side by 2.
In the interval 关0, 2兲, you know that 3t 兾3 and 3t 5兾3 are the only solutions. So in general, you have 3t 兾3 2n and 3t 5兾3 2n. Dividing
this result by 3, you obtain the general solution
t
2n
9
3
and
t
5 2n
9
3
2
y = 2 cos 3t − 1
2
0
General solution
−4
where n is an integer. This solution is confirmed graphically in Figure 5.17.
Figure 5.17
Now try Exercise 65.
Example 9 Functions of Multiple Angles
Solve 3 tan
x
3 0.
2
Solution
3 tan
x
30
2
Write original equation.
3 tan
x
3
2
Subtract 3 from each side.
tan
x
1
2
Divide each side by 3.
In the interval 关0, 兲, you know that x兾2 3兾4 is the only solution. So in general, you have x兾2 3兾4 n. Multiplying this result by 2, you obtain the
general solution
x
3
2n
2
20
−2
−20
Figure 5.18
x
+3
2
2
General solution
where n is an integer. This solution is confirmed graphically in Figure 5.18.
Now try Exercise 71.
y = 3 tan
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Analytic Trigonometry
Using Inverse Functions
Example 10 Using Inverse Functions
Find all solutions of sec2 x 2 tan x 4.
Solution
sec2 x 2 tan x 4
Write original equation.
1 tan2 x 2 tan x 4 0
Pythagorean identity
tan2 x 2 tan x 3 0
Combine like terms.
共tan x 3兲共tan x 1兲 0
Factor.
Setting each factor equal to zero, you obtain two solutions in the interval
共 兾2, 兾2兲. [Recall that the range of the inverse tangent function is
共 兾2, 兾2兲.]
tan x 3
tan x 1
and
x arctan共1兲 x arctan 3
y = sec2 x − 2 tan x − 4
4
6
Finally, because tan x has a period of , add multiples of to obtain
x arctan 3 n
x n
General solution
and
4
−
2
2
−4
where n is an integer. This solution is confirmed graphically in Figure 5.19.
Figure 5.19
Now try Exercise 47.
With some trigonometric equations, there is no reasonable way to find the
solutions algebraically. In such cases, you can still use a graphing utility to
approximate the solutions.
Example 11 Approximating Solutions
Approximate the solutions of x 2 sin x in the interval 关 , 兴.
Solution
Use a graphing utility to graph y x 2 sin x in the interval 关 , 兴. Using the
zero or root feature or the zoom and trace features, you can see that the solutions
are x ⬇ 1.8955, x 0, and x ⬇ 1.8955. (See Figure 5.20.)
3
−
−
−3
Figure 5.20
3
3
−3
y ⴝ x ⴚ 2 sin x
Now try Exercise 85.
−
−3
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Section 5.3
375
Solving Trigonometric Equations
Example 12 Surface Area of a Honeycomb
θ
The surface area of a honeycomb is given by the equation
3 冪3 cos S 6hs s 2
,
2
sin 冢
冣
0 < ≤ 90
where h 2.4 inches, s 0.75 inch, and is the angle indicated in Figure 5.21.
h = 2.4 in.
a. What value of gives a surface area of 12 square inches?
b. What value of gives the minimum surface area?
Solution
s = 0.75 in.
a. Let h 2.4, s 0.75, and S 12.
Figure 5.21
3 冪3 cos S 6hs s2
2
sin 冢
冣
冪3 cos 3
12 6共2.4兲共0.75兲 共0.75兲2
2
sin 冢
12 10.8 0.84375
0 0.84375
冢
冢
冪3 cos sin 冪3 cos sin 冣
冣
TECHNOLOGY SUPPORT
For instructions on how to use
the degree mode and the minimum feature, see Appendix A;
for specific keystrokes, go to this
textbook’s Online Study Center.
冣 1.2
Using a graphing utility set to degree mode, you can graph the function
y 0.84375
冢
冪3 cos x
sin x
冣 1.2.
Using the zero or root feature or the zoom and trace features, you can determine that ⬇ 49.9 and ⬇ 59.9. (See Figure 5.22.)
0.05
y = 10.8 + 0.84375
90
0
y ⴝ 0.84375
冪3 ⴚ cos x
冸
sin x
冹 ⴚ 1.2
b. From part (a), let h 2.4 and s 0.75 to obtain
S 10.8 0.84375
冢
冪3 cos sin 冣.
Graph this function using a graphing utility set to degree mode. Use the
minimum feature or the zoom and trace features to approximate the minimum
point on the graph, which occurs at ⬇ 54.7, as shown in Figure 5.23.
Now try Exercise 93.
150
11
−0.02
Figure 5.22
(
90
0
−0.02
3 − cos x
sin x
14
0.05
0
(
Figure 5.23
STUDY TIP
By using calculus, it can be
shown that the exact minimum
value is
arccos
冢冪3冣 ⬇ 54.7356.
1
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Analytic Trigonometry
5.3 Exercises
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
Fill in the blanks.
1. The equation 2 cos x 1 0 has the solutions x 5
2n and x 2n,
3
3
which are called _______ solutions.
2. The equation tan2 x 5 tan x 6 0 is an equation of _______ type.
3. A solution of an equation that does not satisfy the original equation is called an _______ solution.
In Exercises 1– 6, verify that each x-value is a solution of the
equation.
1. 2 cos x 1 0
(a) x 3
2. sec x 2 0
(a) x 3
5
(b) x 3
(b) x 5
3
5
(b) x 12
(b) x 2
3
(b) x 1
2
7
6
5
3
8. cos x 1
9. cos x 2
冪3
冪3
15. cot x 1
2
24. csc x 冪2
25. 2 cos x 1 0
26. 冪2 sin x 1 0
27. 冪3 sec x 2 0
28. cot x 1 0
10. sin x 14. sin x 16. sin x csc2
x40
sin2 x
3
30. 3 cot2 x 1 0
32. cos x共cos x 1兲 0
cos2 x
35. tan x 冪3 0
36. 2 sin x 1 0
37. csc2 x 2 0
38. tan2 x 1 0
tan3
x tan x
40. 2 sin2 x 2 cos x
41. sec2 x sec x 2
42. sec x csc x 2 csc x
43. 2 sin x csc x 0
44. sec x tan x 1
45. cos x sin x tan x 2
46. sin2 x cos x 1 0
47.
sec2
x tan x 3
48. 2 cos2 x cos x 1 0
2
冪2
2
In Exercises 13–24, find all solutions of the equation in the
interval [0, 2␲冈.
13. cos x 23. tan x 1
39. 3
12. tan x 冪3
11. tan x 1
22. sec x 2
In Exercises 35–48, find all solutions of the equation in the
interval [0, 2␲冈 algebraically. Use the table feature of a
graphing utility to check your answers numerically.
In Exercises 7–12, find all solutions of the equation in the
interval [0ⴗ, 360ⴗ冈.
7. sin x 2
34. 共3 tan2 x 1兲共tan2 x 3兲 0
6. sec4 x 3 sec2 x 4 0
(a) x 冪2
20. sec x 冪2
33.
7
(b) x 8
2
18. cos x 21. cot x 冪3
31. 4 cos2 x 1 0
5. 2 sin2 x sin x 1 0
(a) x 3
19. csc x 2
29. 3
4. 4 cos2 2x 2 0
(a) x 8
冪3
In Exercises 25–34, solve the equation.
3. 3 tan2 2x 1 0
(a) x 12
17. tan x 1
2
冪3
2
In Exercises 49–56, use a graphing utility to approximate
the solutions of the equation in the interval [0, 2␲冈 by setting
the equation equal to 0, graphing the new equation, and
using the zero or root feature to approximate the x-intercepts of the graph.
49. 2 sin2 x 3 sin x 1 0
50. 2 sec2 x tan2 x 3 0
51. 4 sin2 x 2 cos x 1
52. csc2 x 3 csc x 4
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Section 5.3
53. csc x cot x 1
Solving Trigonometric Equations
In Exercises 77–84, use a graphing utility to approximate
the solutions of the equation in the interval [0, 2␲冈.
54. 4 sin x cos x 2
55.
cos x cot x
3
1 sin x
77. 2 cos x sin x 0
56.
cos x
1 sin x
4
cos x
1 sin x
79. x tan x 1 0
78. 2 sin x cos x 0
80. 2x sin x 2 0
In Exercises 57–60, (a) use a graphing utility to graph each
function in the interval [0, 2␲冈, (b) write an equation whose
solutions are the points of intersection of the graphs, and (c)
use the intersect feature of the graphing utility to find the
points of intersection (to four decimal places).
58. y cos x, y x x2
59. y sin2 x,
y ex 4x
60. y cos2 x,
y ex x 1
62. sin
63. sin 4x 1
冪3
88. 2 sec2 x tan x 6 0,
69. tan 3x共tan x 1兲 0
70. cos 2x共2 cos x 1兲 0
冪2
x
2
2
72. tan
x
1
3
In Exercises 73–76, approximate the x-intercepts of the
graph. Use a graphing utility to check your solutions.
x
1
2
74. y sin x cos x
3
2
−1
−2
3
4
−1
75. y tan2
冤 2 , 2 冥
冤 2 , 2 冥
冤 2 , 2 冥
66. sec 4x 2
2
68. tan2 3x 3
73. y sin
84. 3 tan2 x 4 tan x 4 0
87. 4 cos2 x 2 sin x 1 0,
x
0
2
67. 2 sin2 2x 1
71. cos
83. 12 sin2 x 13 sin x 3 0
86. cos2 x 2 cos x 1 0, 关0, 兴
64. cos 2x 1
65. sin 2x 82. csc2 x 0.5 cot x 5
85. 3 tan2 x 5 tan x 4 0,
In Exercises 61–72, solve the multiple-angle equation.
x
0
4
81. sec2 x 0.5 tan x 1
In Exercises 85–88, use a graphing utility to approximate
the solutions (to three decimal places) of the equation in the
given interval.
57. y sin 2x, y x2 2x
61. cos
377
In Exercises 89–94, (a) use a graphing utility to graph the
function and approximate the maximum and minimum
points (to four decimal places) of the graph in the interval
[0, 2␲ 兴, and (b) solve the trigonometric equation and verify
that the x-coordinates of the maximum and minimum
points of f are among its solutions (the trigonometric equation is found using calculus).
Function
Trigonometric Equation
89. f 共x兲 sin 2x
2 cos 2x 0
90. f 共x兲 cos 2x
2 sin 2x 0
91. f 共x兲 sin2 x cos x
2 sin x cos x sin x 0
92. f 共x兲 cos2 x sin x
2 sin x cos x cos x 0
93. f 共x兲 sin x cos x
cos x sin x 0
94. f 共x兲 2 sin x cos 2x
2 cos x 4 sin x cos x 0
−2
x
3
6
冢 冣
76. y sec4
2
−3
冢 冣
4
3
−4
Fixed Point In Exercises 95 and 96, find the smallest positive fixed point of the function f. [A fixed point of a function
f is a real number c such that f 冇c冈 ⴝ c.]
x
4
8
95. f 共x兲 tan
−3
3
−4
x
4
96. f 共x兲 cos x
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Analytic Trigonometry
97. Graphical Reasoning
100. Position of the Sun Cheyenne, Wyoming has a latitude
of 41 N. At this latitude, the position of the sun at sunrise
can be modeled by
Consider the function
1
f 共x兲 cos
x
D 31 sin
and its graph shown in the figure.
y
2
1
−π
π
x
2
t 1.4冣
冢365
where t is the time (in days) and t 1 represents January
1. In this model, D represents the number of degrees north
or south of due east at which the sun rises. Use a graphing
utility to determine the days on which the sun is more than
20 north of due east at sunrise.
101. Harmonic Motion A weight is oscillating on the end of
a spring (see figure). The position of the weight relative to
the point of equilibrium is given by
−2
(a) What is the domain of the function?
(b) Identify any symmetry or asymptotes of the graph.
(c) Describe the behavior of the function as x → 0.
(d) How many solutions does the equation cos共1兾x兲 0
have in the interval 关1, 1兴? Find the solutions.
1
y 12
共cos 8t 3 sin 8t兲
where y is the displacement (in meters) and t is the time
(in seconds). Find the times at which the weight is at the
point of equilibrium 共y 0兲 for 0 ≤ t ≤ 1.
(e) Does the equation cos共1兾x兲 0 have a greatest solution? If so, approximate the solution. If not, explain.
98. Graphical Reasoning
f 共x兲 Consider the function
Equilibrium
sin x
x
y
and its graph shown in the figure.
y
3
2
−π
−1
−2
−3
π
x
(a) What is the domain of the function?
(b) Identify any symmetry or asymptotes of the
graph.
(c) Describe the behavior of the function as x → 0.
(d) How many solutions does the equation 共sin x)/x 0
have in the interval 关8, 8兴? Find the solutions.
99. Sales The monthly sales S (in thousands of units) of lawn
mowers are approximated by
S 74.50 43.75 cos
102. Damped Harmonic Motion The displacement from
equilibrium of a weight oscillating on the end of a spring
is given by y 1.56e0.22t cos 4.9t, where y is the
displacement (in feet) and t is the time (in seconds). Use
a graphing utility to graph the displacement function for
0 ≤ t ≤ 10. Find the time beyond which the displacement does not exceed 1 foot from equilibrium.
103. Projectile Motion A batted baseball leaves the bat at
an angle of with the horizontal and an initial velocity of
v0 100 feet per second. The ball is caught by an
outfielder 300 feet from home plate (see figure). Find if
the range r of a projectile is given by
r
1 2
v sin 2.
32 0
t
6
where t is the time (in months), with t 1 corresponding
to January. Determine the months during which sales
exceed 100,000 units.
θ
r = 300 ft
Not drawn to scale
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Section 5.3
g共x兲 0.45x 2 5.52x 13.70.
A 2x cos x, 0 ≤ x ≤ .
2
Use a graphing utility to graph f and g in the same
viewing window. Describe the result.
y
(c) Use the Quadratic Formula to find the zeros of g.
Compare the zero in the interval 关0, 6兴 with the result
of part (a).
y = cos x
π
2
x
π
2
x
Synthesis
True or False? In Exercises 107–109, determine whether
the statement is true or false. Justify your answer.
−1
(a) Use a graphing utility to graph the area function, and
approximate the area of the largest inscribed rectangle.
(b) Determine the values of x for which A ≥ 1.
105. Data Analysis: Unemployment Rate The table shows
the unemployment rates r in the United States for selected
years from 1990 through 2004. The time t is measured in
years, with t 0 corresponding to 1990. (Source: U.S.
Bureau of Labor Statistics)
Time, t
Rate, r
Time, t
Rate, r
0
2
4
6
5.6
7.5
6.1
5.4
8
10
12
14
4.5
4.0
5.8
5.5
(a) Use a graphing utility to create a scatter plot of the
data.
(b) Which of the following models best represents the
data? Explain your reasoning.
107. All trigonometric equations have either an infinite number
of solutions or no solution.
108. The solutions of any trigonometric equation can always
be found from its solutions in the interval 关0, 2兲.
109. If you correctly solve a trigonometric equation down to
the statement sin x 3.4, then you can finish solving the
equation by using an inverse trigonometric function.
110. Writing Describe the difference between verifying an
identity and solving an equation.
Skills Review
In Exercises 111–114, convert the angle measure from
degrees to radians. Round your answer to three decimal
places.
111. 124
112. 486
113. 0.41
114. 210.55
In Exercises 115 and 116, solve for x.
115.
116.
(1) r 1.24 sin共0.47t 0.40兲 5.45
14
(2) r 1.24 sin共0.47t 0.01兲 5.45
x
(4) r 896 sin共0.57t 2.05兲 6.48
(c) What term in the model gives the average unemployment rate? What is the rate?
(d) Economists study the lengths of business cycles, such
as unemployment rates. Based on this short span of
time, use the model to determine the length of this
cycle.
(e) Use the model to estimate the next time the unemployment rate will be 5% or less.
Consider the function
f 共x兲 3 sin共0.6x 2兲.
(a) Approximate the zero of the function in the interval
关0, 6兴.
10
30°
(3) r sin共0.10t 5.61兲 4.80
106. Quadratic Approximation
379
(b) A quadratic approximation agreeing with f at x 5 is
104. Area The area of a rectangle inscribed in one arc of the
graph of y cos x (see figure) is given by
−
Solving Trigonometric Equations
x
70°
117. Distance From the 100-foot roof of a condominium on
the coast, a tourist sights a cruise ship. The angle of
depression is 2.5. How far is the ship from the shoreline?
118.
Make a Decision To work an extended application
analyzing the normal daily high temperatures in Phoenix
and in Seattle, visit this textbook’s Online Study Center.
(Data Source: NOAA)
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Analytic Trigonometry
5.4 Sum and Difference Formulas
What you should learn
Using Sum and Difference Formulas
In this section and the following section, you will study the uses of several trigonometric identities and formulas.
Sum and Difference Formulas
(See the proofs on page 404.)
sin共u v兲 sin u cos v cos u sin v
tan共u v兲 tan u tan v
1 tan u tan v
tan共u v兲 tan u tan v
1 tan u tan v
sin共u v兲 sin u cos v cos u sin v
cos共u v兲 cos u cos v sin u sin v
cos共u v兲 cos u cos v sin u sin v
䊏
Use sum and difference formulas to
evaluate trigonometric functions, verify
trigonometric identities, and solve
trigonometric equations.
Why you should learn it
You can use sum and difference formulas
to rewrite trigonometric expressions. For
instance, Exercise 79 on page 385 shows how
to use sum and difference formulas to rewrite
a trigonometric expression in a form that
helps you find the equation of a standing
wave.
Exploration
Use a graphing utility to graph y1 cos共x 2兲 and y2 cos x cos 2 in
the same viewing window. What can you conclude about the graphs? Is it
true that cos共x 2兲 cos x cos 2?
Use a graphing utility to graph y1 sin共x 4兲 and y2 sin x sin 4 in
the same viewing window. What can you conclude about the graphs? Is it
true that sin共x 4兲 sin x sin 4?
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.
Richard Megna/Fundamental Photographs
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 cos共u v兲 yields
cos 75 cos共30 45兲
cos 30 cos 45 sin 30 sin 45
冪3 冪2
1 冪2
2 冢 2 冣 2冢 2 冣
冪6 冪2
4
.
Try checking this result on your calculator. You will find that cos 75 ⬇ 0.259.
Now try Exercise 1.
Prerequisite Skills
To review sines, cosines, and tangents
of special angles, see Section 4.3.
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Section 5.4
381
Sum and Difference Formulas
Example 2 Evaluating a Trigonometric Function
Find the exact value of sin
.
12
Solution
Using the fact that 兾12 兾3 兾4 together with the formula for sin共u v兲,
you obtain
sin
sin
sin cos cos sin
12
3
4
3
4
3
4
冢
y
冣
冪3 冪2
1 冪2
2 冢 2 冣 2冢 2 冣
冪6 冪2
4
.
5
u
4
x
Now try Exercise 3.
52 −
42 =
3
Example 3 Evaluating a Trigonometric Expression
Find the exact value of sin共u v兲 given
4
sin u , where 0 < u <
5
2
12
and cos v , where < v < .
13
2
Figure 5.24
y
Solution
Because sin u 4兾5 and u is in Quadrant I, cos u 3兾5, as shown in Figure
5.24. Because cos v 12兾13 and v is in Quadrant II, sin v 5兾13, as shown
in Figure 5.25. You can find sin共u v兲 as follows.
13 2 − 12 2 = 5
13
v
12
sin共u v兲 sin u cos v cos u sin v
x
12
15
33
冢45冣冢 13
冣 冢35冣冢135 冣 48
65 65
65
Now try Exercise 35.
Figure 5.25
Example 4 An Application of a Sum Formula
Write cos共arctan 1 arccos x兲 as an algebraic expression.
Solution
This expression fits the formula for cos共u v兲. Angles u arctan 1 and
v arccos x are shown in Figure 5.26.
2
1
u
1
cos共u v兲 cos共arctan 1兲 cos共arccos x兲 sin共arctan 1兲 sin共arccos x兲
1
冪2
x
1
冪2
冪1 x 2 Now try Exercise 43.
x 冪1 x2
.
冪2
1
v
x
Figure 5.26
1 − x2
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Analytic Trigonometry
Example 5 Proving a Cofunction Identity
Prove the cofunction identity cos
冢 2 x冣 sin x.
Solution
Using the formula for cos共u v兲, you have
cos
冢 2 x冣 cos 2 cos x sin 2 sin x
共0兲共cos x兲 共1兲共sin x兲
sin x.
Now try Exercise 63.
Sum and difference formulas can be used to derive reduction formulas
involving expressions such as
冢
sin n
2
冣
and
冢
cos n
, where n is an integer.
2
冣
Example 6 Deriving Reduction Formulas
Simplify each expression.
冢
a. cos 3
2
冣
b. tan共 3兲
Solution
a. Using the formula for cos共u v兲, you have
冢
cos 3
3
3
cos cos
sin sin
2
2
2
冣
共cos 兲共0兲 共sin 兲共1兲
sin .
b. Using the formula for tan共u v兲, you have
tan共 3兲 tan tan 3
1 tan tan 3
tan 0
1 共tan 兲共0兲
tan .
Note that the period of tan is , so the period of tan共 3兲 is the same as
the period of tan .
Now try Exercise 67.
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Section 5.4
383
Sum and Difference Formulas
Example 7 Solving a Trigonometric Equation
冢
Find all solutions of sin x sin x 1 in the interval 关0, 2兲.
4
4
冣
冢
冣
Algebraic Solution
Graphical Solution
Using sum and difference formulas, rewrite the equation as
Use a graphing utility set in radian mode to
sin x 1,
graph y sin x 4
4
as shown in Figure 5.27. Use the zero or root
feature or the zoom and trace features to
approximate the x-intercepts in the interval
关0, 2兲 to be
sin x cos
冢
cos x sin sin x cos cos x sin 1
4
4
4
4
2 sin x cos
2共sin x兲
1
4
冪2
冢 2 冣 1
sin x sin x x ⬇ 3.927 ⬇
冣
冢
冣
5
7
.
and x ⬇ 5.498 ⬇
4
4
1
冪2
冪2
2
( π4 ( + sin (x − π4 ( + 1
y = sin x +
.
3
So, the only solutions in the interval 关0, 2兲 are
x
5
4
and
x
7
.
4
2
0
−1
Now try Exercise 71.
Figure 5.27
The next example was taken from calculus. It is used to derive the formula
for the derivative of the cosine function.
Example 8 An Application from Calculus
cos共x h兲 cos x
cos h 1
sin h
共cos x兲
共sin x兲
, h 0.
Verify
h
h
h
冢
冣
冢
冣
Solution
Using the formula for cos共u v兲, you have
cos共x h兲 cos x cos x cos h sin x sin h cos x
h
h
cos x共cos h 1兲 sin x sin h
h
共cos x兲
冢
cos h 1
sin h
共sin x兲
.
h
h
Now try Exercise 93.
冣
冢
冣
TECHNOLOGY SUPPORT
For instructions on how to use
the zero or root feature and
the zoom and trace features,
see Appendix A; for specific
keystrokes, go to this textbook’s
Online Study Center.
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Analytic Trigonometry
5.4 Exercises
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
Fill in the blank to complete the trigonometric formula.
1. sin共u v兲 _______
2. cos共u v兲 _______
3. tan共u v兲 _______
4. sin共u v兲 _______
5. cos共u v兲 _______
6. tan共u v兲 _______
In Exercises 1–6, find the exact value of each expression.
1. (a) cos共240 0兲
(b) cos 240 cos 0
2. (a) sin共405 120兲
(b) sin 405 sin 120
3. (a) cos
4
3
(b) cos
cos
4
3
(b) sin
5
2
sin
3
6
冢
4. (a) sin
2
冢3
冣
5
6
冣
5. (a) sin共315 60兲
(b) sin 315 sin 60
7 6. (a) sin
6
3
(b) sin
冢
冣
7
sin
6
3
In Exercises 7– 22, find the exact values of the sine, cosine,
and tangent of the angle.
7. 105 60 45
8. 165 135 30
9. 195 225 30
10. 255 300 45
11.
11 3 12
4
6
13. 12
6
4
12.
17 7 12
6
4
14. 19 2 9
12
3
4
15. 75
16. 15
17. 225
18. 165
13
19.
12
5
20.
12
7
21. 12
13
22. 12
28. cos 0.96 cos 0.42 sin 0.96 sin 0.42
29. cos
cos sin sin
9
7
9
7
30. sin
4
4
cos cos
sin
9
8
9
8
Numerical, Graphical, and Algebraic Analysis In Exercises
31–34, use a graphing utility to complete the table and graph
the two functions in the same viewing window. Use both the
table and the graph as evidence that y1 y2 . Then verify the
identity algebraically.
0.2
x
0.4
0.6
0.8
1.0
1.2
1.4
y1
y2
31. y1 sin
冢 6 x冣,
32. y1 cos
5
冢4
y2 1
共cos x 冪3 sin x兲
2
冣
x , y2 33. y1 cos共x 兲 cos共x 兲,
34. y1 sin共x 兲 sin共x 兲,
冪2
2
共cos x sin x兲
y2 cos 2 x
y2 sin2 x
In Exercises 35–38, find the exact value of the trigonometric
5
function given that sin u 13
and cos v 35. (Both u and
v are in Quadrant II.)
35. sin共u v兲
36. cos共v u兲
In Exercises 23 – 30, write the expression as the sine, cosine,
or tangent of an angle.
37. tan共u v兲
38. sin共u v兲
23. cos 60 cos 20 sin 60 sin 20
In Exercises 39–42, find the exact value of the trigonometric
8
function given that sin u 17
and cos v 45. (Both u
and v are in Quadrant III.)
24. sin 110 cos 80 cos 110 sin 80
25.
tan 325 tan 86
1 tan 325 tan 86
26.
tan 154 tan 49
1 tan 154 tan 49
27. sin 3.5 cos 1.2 cos 3.5 sin 1.2
39. cos共u v兲
40. tan共u v兲
41. sin共v u兲
42. cos共u v兲
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Section 5.4
In Exercises 43–46, write the trigonometric expression as an
algebraic expression.
43. sin共arcsin x arccos x兲
44. cos共arccos x arcsin x兲
45. sin共arctan 2x arccos x兲
46. cos共arcsin x arctan 2x兲
In Exercises 47– 54, find the value of the expression without
using a calculator.
47. sin共sin1 1 cos1 1兲
In Exercises 71–74, find the solution(s) of the equation in the
interval [0, 2冈. Use a graphing utility to verify your results.
冢
sin x 1
3
3
冢
cos x 1
6
6
71. sin x 72. cos x 冢
冣
1
52. cos冤 cos 冢 冣 sin 1冥
2
1
53. tan冢sin 0 sin
2冣
冪2
54. tan冢cos
sin 0冣
2
冢
冣
3 tan共 x兲 0
2
冣
冢
cos x 1
4
4
冢
3
cos x 0
2
2
75. cos x 1
76. sin x 1
1
冣
In Exercises 75–78, use a graphing utility to approximate
the solutions of the equation in the interval [0, 2冈.
1
1
51. sin sin1 cos1
2
2
1
冣
74. 2 sin x 50. cos关cos1共1兲 cos1 1兴
冢
73. tan共x 兲 2 sin共x 兲 0
49. sin关sin1 1 cos1共1兲兴
1
冣
冢
48. cos关sin1共1兲 cos1 0兴
冣
冢
冣
冣
冢
冣
冢
77. tan共x 兲 cos x 1
冢
0
2
冣
78. tan共 x兲 2 cos x In Exercises 55– 58, evaluate the trigonometric function
without using a calculator.
55. sin
sin1共1兲
2
56. sin关cos 共1兲 兴
57. cos共 sin1 1兲
58. cos关 cos1共1兲兴
冤
冥
1
In Exercises 59– 62, use right triangles to evaluate the
expression.
冢
60. cos冢sin
61. sin冢tan
62. tan冢sin
59. sin cos1
3
5
sin1
5
13
冣
1
12
8
cos1
13
17
1
3
3
sin1
4
5
1
y1 A cos 2
冣
冢
t
x
2
A cos 2
冢T 冣
t
2 x
2 t
cos
.
T
y1 + y2
y2
t=0
冣
y1
64. sin共3 x兲 sin x
1 tan 4
1 tan 冣
67. sin共x y兲 sin共x y兲 2 sin x cos y
68. cos共x y兲 cos共x y兲 2 cos x cos y
69. cos共x y兲 cos共x y兲 cos2 x sin2 y
70. sin共x y兲 sin共x y兲 sin2 x sin2 y
y1 + y2
y2
t = 18 T
65. tan共x 兲 tan共 x兲 2 tan x
66. tan
冢T 冣 and y
y1 y2 2A cos
y1
冢2 x冣 cos x
冣
show that
In Exercises 63– 70, verify the identity.
63. sin
3
0
2
79. 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
冣
4
5
cos1
5
13
385
Sum and Difference Formulas
y1
t = 28 T
y1 + y2
y2
x
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Analytic Trigonometry
80. 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
93. Verify the following identity used in calculus.
sin共x h兲 sin x cos x sin h sin x共1 cos h兲
h
h
h
94. Exploration Let x 兾3 in the identity in Exercise 93
and define the functions f and g as follows.
1
1
sin 2t cos 2t
3
4
sin共兾3 h兲 sin共兾3兲
h
where y is the distance from equilibrium (in feet) and t is
the time (in seconds).
f 共h兲 (a) Use a graphing utility to graph the model.
g共h兲 cos
(b) Use the identity
a sin B b cos 冪a2 b2 sin共B C兲
where C arctan共b兾a兲, a > 0, to write the model in
the form y 冪a2 b2 sin共Bt C兲. Use a graphing
utility to verify your result.
(c) Find the amplitude of the oscillations of the weight.
sin h
1 cos h
sin
3 h
3
h
冢
冣
冢
冣
(a) What are the domains of the functions f and g?
(b) Use a graphing utility to complete the table.
0.01
h
0.02
0.05
0.1
0.2
0.5
f 共h兲
(d) Find the frequency of the oscillations of the weight.
g共h兲
Synthesis
(c) Use a graphing utility to graph the functions f and g.
True or False? In Exercises 81 and 82, determine whether
the statement is true or false. Justify your answer.
81. cos共u ± v兲 cos u ± cos v
95. Conjecture Three squares of side s are placed side by
side (see figure). Make a conjecture about the relationship
between the sum u v and w. Prove your conjecture by
using the identity for the tangent of the sum of two angles.
11
82. sin x cos x
2
冢
(d) Use the table and graph to make a conjecture about
the values of the functions f and g as h → 0.
冣
In Exercises 83– 86, verify the identity.
83. cos共n 兲 共1兲n cos , n is an integer.
84. sin共n 兲 共1兲n sin ,
n is an integer.
s
85. a sin B b cos B 冪a 2 b2 sin共B C兲, where
C arctan共b兾a兲 and a > 0.
86. a sin B b cos B
C arctan共a兾b兲 and b > 0.
冪a 2
b2
cos共B C兲, where
(a) 冪a2 1 b2 sin冇B 1 C 冈
(b) 冪a2 1 b2 cos冇B C 冈
87. sin cos 88. 3 sin 2 4 cos 2
89. 12 sin 3 5 cos 3
90. sin 2 cos 2
In Exercises 91 and 92, use the formulas given in Exercises 85
and 86 to write the trigonometric expression in the form
a sin B 1 b cos B.
冢
2
冣
v
s
w
s
s
96. (a) Write a sum formula for sin共u v w兲.
In Exercises 87– 90, use the formulas given in Exercises 85
and 86 to write the expression in the following forms. Use a
graphing utility to verify your results.
91. 2 sin u
冢
92. 5 cos 4
冣
(b) Write a sum formula for tan共u v w兲.
Skills Review
In Exercises 97–100, find the x- and y-intercepts of the
graph of the equation. Use a graphing utility to verify your
results.
97. y 12共x 10兲 14
ⱍ
ⱍ
99. y 2x 9 5
98. y x2 3x 40
100. y 2x冪x 7
In Exercises 101–104, evaluate the expression without using
a calculator.
冢 23冣
101. arccos
103. sin1 1
冪
102. arctan共 冪3兲
104. tan1 0
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Section 5.5
Multiple-Angle and Product-to-Sum Formulas
387
5.5 Multiple-Angle and Product-to-Sum Formulas
What you should learn
Multiple-Angle Formulas
In this section, you will study four additional categories of trigonometric
identities.
1. The first category involves functions of multiple angles such as sin ku and
cos ku.
2. The second category involves squares of trigonometric functions such as
sin2 u.
3. The third category involves functions of half-angles such as sin共u兾2兲.
4. The fourth category involves products of trigonometric functions such as
sin u cos v.
You should learn the double-angle formulas below because they are used
most often.
Double-Angle Formulas
(See the proofs on page 405.)
䊏
䊏
䊏
䊏
Use multiple-angle formulas to rewrite
and evaluate trigonometric functions.
Use power-reducing formulas to rewrite
and evaluate trigonometric functions.
Use half-angle formulas to rewrite and
evaluate trigonometric functions.
Use product-to-sum and sum-to-product
formulas to rewrite and evaluate
trigonometric functions.
Why you should learn it
You can use a variety of trigonometric formulas to rewrite trigonometric functions in more
convenient forms.For instance, Exercise 130
on page 398 shows you how to use a halfangle formula to determine the apex angle
of a sound wave cone caused by the speed
of an airplane.
cos 2u cos2 u sin2 u
sin 2u 2 sin u cos u
2 cos2 u 1
2 tan u
tan 2u 1 tan2 u
1 2 sin2 u
Example 1 Solving a Multiple-Angle Equation
Solve 2 cos x sin 2x 0.
NASA-Liaison/Getty Images
Solution
Begin by rewriting the equation so that it involves functions of x (rather than 2x).
Then factor and solve as usual.
2 cos x sin 2x 0
Write original equation.
2 cos x 2 sin x cos x 0
Double-angle formula
2 cos x共1 sin x兲 0
cos x 0
x
Factor.
1 sin x 0
3
,
2 2
x
3
2
Set factors equal to zero.
Solutions in 关0, 2兲
So, the general solution is
x
2n
2
and
x
3
2n
2
General solution
where n is an integer. Try verifying this solution graphically.
Now try Exercise 3.
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Analytic Trigonometry
Example 2 Using Double-Angle Formulas
to Analyze Graphs
Analyze the graph of y 4 cos2 x 2 in the interval 关0, 2兴.
Solution
Using a double-angle formula, you can rewrite the original function as
y 4 cos2 x 2
2共2 cos2 x 1兲
2 cos 2x.
Using the techniques discussed in Section 4.5, you can recognize that the graph
of this function has an amplitude of 2 and a period of . The key points in the
interval 关0, 兴 are as follows.
Maximum
共0, 2兲
Intercept
,0
4
冢 冣
Minimum
, 2
2
冢
Intercept
3
,0
4
冣
冢
冣
Maximum
共, 2兲
Two cycles of the graph are shown in Figure 5.28.
y = 4 cos2 x − 2
3
2
0
−3
Figure 5.28
Now try Exercise 7.
Example 3 Evaluating Functions Involving Double Angles
Use the following to find sin 2, cos 2, and tan 2.
cos 5
,
13
3
< < 2
2
Solution
In Figure 5.29, you can see that sin y兾r 12兾13. Consequently, using
each of the double-angle formulas, you can write the double angles as follows.
冢 冣冢 冣
sin 2 2 sin cos 2 cos 2 2 cos2 1 2
12
13
5
120
13
169
冢169冣 1 169
25
119
2 tan 2共12兾5兲
120
tan 2 1 tan2 1 共12兾5兲2 119
Now try Exercise 13.
and
θ
−4
x
−2
2
4
−2
−4
−6
13
−8
−10
The double-angle formulas are not restricted to the angles 2 and . Other
double combinations, such as 4 and 2 or 6 and 3, are also valid. Here are two
examples.
sin 4 2 sin 2 cos 2
y
cos 6 cos2 3 sin2 3
By using double-angle formulas together with the sum formulas derived in the
preceding section, you can form other multiple-angle formulas.
−12
Figure 5.29
(5, −12)
6
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Section 5.5
Multiple-Angle and Product-to-Sum Formulas
389
Example 4 Deriving a Triple-Angle Formula
sin 3x sin共2x x兲
Rewrite as a sum.
sin 2x cos x cos 2x sin x
Sum formula
2 sin x cos x cos x 共1 2 sin2 x兲 sin x
Double-angle formula
2 sin x
Multiply.
cos2
x sin x 2
sin3
x
2 sin x共1 sin2 x兲 sin x 2 sin3 x
Pythagorean identity
2 sin x 2
Multiply.
sin3 x
sin x 2
sin3
x
3 sin x 4 sin3 x
Simplify.
Now try Exercise 19.
Power-Reducing Formulas
The double-angle formulas can be used to obtain the following power-reducing
formulas.
Power-Reducing Formulas
sin2 u 1 cos 2u
2
(See the proofs on page 405.)
cos2 u 1 cos 2u
2
tan2 u 1 cos 2u
1 cos 2u
Example 5 Reducing a Power
Rewrite sin4 x as a sum of first powers of the cosines of multiple angles.
Solution
sin4 x 共sin2 x兲2
冢
1 cos 2x
2
Property of exponents
冣
2
Power-reducing formula
1
共1 2 cos 2x cos2 2x兲
4
Expand binomial.
1
1 cos 4x
1 2 cos 2x 4
2
1 1
1 1
cos 2x cos 4x
4 2
8 8
Distributive Property
3 1
1
cos 2x cos 4x
8 2
8
Simplify.
冢
1
共3 4 cos 2x cos 4x兲
8
Now try Exercise 23.
冣
Power-reducing formula
Factor.
STUDY TIP
Power-reducing formulas
are often used in calculus.
Example 5 shows a typical
power reduction that is used in
calculus. Note the repeated use
of power-reducing formulas.
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Analytic Trigonometry
Half-Angle Formulas
You can derive some useful alternative forms of the power-reducing formulas by
replacing u with u兾2. The results are called half-angle formulas.
Half-Angle Formulas
冪1 2cos u
sin
u
±
2
tan
u 1 cos u
sin u
2
sin u
1 cos u
The signs of sin
cos
u
±
2
冪1 2cos u
u
u
u
and cos depend on the quadrant in which lies.
2
2
2
Example 6 Using a Half-Angle Formula
Find the exact value of sin 105.
STUDY TIP
y
105°
210°
x
To find the exact value of a
trigonometric function with an
angle in DM S form using a
half-angle formula, first convert
the angle measure to decimal
degree form. Then multiply the
angle measure by 2.
Figure 5.30
Solution
Begin by noting that 105 is half of 210. Then, using the half-angle formula for
sin共u兾2兲 and the fact that 105 lies in Quadrant II (see Figure 5.30), you have
冪
1 共cos 30兲
冪
2
1 共 3兾2兲 冪2 冪
2
2
sin 105 1 cos 210
2
冪
冪3
.
The positive square root is chosen because sin is positive in Quadrant II.
Now try Exercise 39.
TECHNOLOGY TIP
Use your calculator to verify the
result obtained in Example 6.
That is, evaluate sin 105 and
共冪2 冪3 兲兾2. You will notice
that both expressions yield the
same result.
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Section 5.5
Multiple-Angle and Product-to-Sum Formulas
391
Example 7 Solving a Trigonometric Equation
Find all solutions of 2 sin2 x 2 cos2
x
in the interval 关0, 2兲.
2
Algebraic Solution
Graphical Solution
2 sin2 x 2 cos 2
冢冪
2 sin2 x 2 ±
2 sin2 x 2
x
2
1 cos x
2
冢
Write original equation.
2
冣
1 cos x
2
冣
2 sin2 x 1 cos x
2 共1 cos 2
cos 2
x兲 1 cos x
x cos x 0
cos x共cos x 1兲 0
Half-angle formula
Simplify.
Simplify.
Pythagorean identity
Use a graphing utility set in radian mode to graph
y 2 sin2 x 2 cos2 共x兾2兲, as shown in Figure
5.31. Use the zero or root feature or the zoom and trace
features to approximate the x-intercepts in the interval
关0, 2兲 to be
x 0, x ⬇ 1.5708 ⬇
3
, and x ⬇ 4.7124 ⬇ .
2
2
These values are the approximate solutions of
x
2 sin2 x 2 cos2 in the interval 关0, 2兲.
2
Simplify.
y = 2 − sin2 x − 2 cos2
Factor.
3
By setting the factors cos x and 共cos x 1兲 equal to zero, you
find that the solutions in the interval 关0, 2兲 are
x
3
, x
, and x 0.
2
2
−
2
2
−1
Now try Exercise 57.
Figure 5.31
Product-to-Sum Formulas
Each of the following product-to-sum formulas is easily verified using the sum
and difference formulas discussed in the preceding section.
Product-to-Sum Formulas
1
sin u sin v 关cos共u v兲 cos共u v兲兴
2
1
cos u cos v 关cos共u v兲 cos共u v兲兴
2
1
sin u cos v 关sin共u v兲 sin共u v兲兴
2
1
cos u sin v 关sin共u v兲 sin共u v兲兴
2
Product-to-sum formulas are used in calculus to evaluate integrals involving
the products of sines and cosines of two different angles.
x
2
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Analytic Trigonometry
Example 8 Writing Products as Sums
TECHNOLOGY TIP
You can use a graphing utility to
verify the solution in Example 8.
Graph y1 cos 5x sin 4x and
y2 12 sin 9x 12 sin x in the
same viewing window. Notice
that the graphs coincide. So,
you can conclude that the two
expressions are equivalent.
Rewrite the product as a sum or difference.
cos 5x sin 4x
Solution
1
cos 5x sin 4x 关sin共5x 4x兲 sin共5x 4x兲兴
2
1
1
sin 9x sin x
2
2
Now try Exercise 63.
Occasionally, it is useful to reverse the procedure and write a sum of
trigonometric functions as a product. This can be accomplished with the following
sum-to-product formulas.
Sum-to-Product Formulas
sin u sin v 2 sin
冢
sin u sin v 2 cos
uv
uv
cos
2
2
冣 冢
冣
uv
uv
sin
2
2
冣
冢
cos u cos v 2 cos
(See the proof on page 406.)
冣 冢
冢
uv
uv
cos
2
2
cos u cos v 2 sin
冣 冢
冢
冣
uv
uv
sin
2
2
冣 冢
冣
Example 9 Using a Sum-to-Product Formula
Find the exact value of cos 195 cos 105.
Solution
Using the appropriate sum-to-product formula, you obtain
cos 195 cos 105 2 cos
冢
195 105
195 105
cos
2
2
冣 冢
2 cos 150 cos 45
冢
2 冪3
冪6
2
冪2
2 冣冢 2 冣
.
Now try Exercise 81.
冣
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Section 5.5
Multiple-Angle and Product-to-Sum Formulas
393
Example 10 Solving a Trigonometric Equation
Find all solutions of sin 5x sin 3x 0 in the interval 关0, 2兲.
Algebraic Solution
Graphical Solution
sin 5x sin 3x 0
2 sin
冢
5x 3x
5x 3x
cos
0
2
2
冣 冢
冣
2 sin 4x cos x 0
Write original
equation.
Sum-to-product
formula
Simplify.
By setting the factor sin 4x equal to zero, you can find
that the solutions in the interval 关0, 2兲 are
3
5 3 7
x 0, , , , , , , .
4 2 4
4 2 4
Moreover, the equation cos x 0 yields no additional
solutions. Note that the general solution is
x
Use a graphing utility set in radian mode to graph
y sin 5x sin 3x, as shown in Figure 5.32. Use the zero or
root feature or the zoom and trace features to approximate the
x-intercepts in the interval 关0, 2兲 to be
x ⬇ 0, x ⬇ 0.7854 ⬇
, x ⬇ 1.5708 ⬇ ,
4
2
3
5
, x ⬇ 3.1416 ⬇ , x ⬇ 3.9270 ⬇
,
4
4
3
7
x ⬇ 4.7124 ⬇ , x ⬇ 5.4978 ⬇
.
2
4
x ⬇ 2.3562 ⬇
These values are the approximate solutions of sin 5x sin 3x 0 in the interval 关0, 2兲.
n
4
3
y = sin 5x + sin 3x
where n is an integer.
−
4
2
−3
Now try Exercise 85.
Figure 5.32
Example 11 Verifying a Trigonometric Identity
Verify the identity
sin t sin 3t
tan 2t.
cos t cos 3t
Algebraic Solution
Numerical Solution
Using appropriate sum-to-product formulas,
you have
Use the table feature of a graphing utility set in radian mode to create a
table that shows the values of y1 共sin x sin 3x兲兾共cos x cos 3x兲
and y2 tan 2x for different values of x, as shown in Figure 5.33. In
the table, you can see that the values appear to be identical, so
共sin x sin 3x兲兾共cos x cos 3x兲 tan 2x appears to be an identity.
sin t sin 3t
2 sin 2t cos共t兲
cos t cos 3t 2 cos 2t cos共t兲
sin 2t
cos 2t
tan 2t.
Now try Exercise 105.
Figure 5.33
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Analytic Trigonometry
5.5 Exercises
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
Fill in the blank to complete the trigonometric formula.
1. sin 2u _______
2. cos2 u _______
3. _______ 1 2 sin2 u
4. _______ 5. tan 2u _______
6. cos u cos v _______
7. _______ 1 cos 2u
2
sin u
1 cos u
8. _______ ±
9. sin u cos v _______
冪1 2cos u
10. sin u sin v _______
In Exercises 1 and 2, use the figure to find the exact value of
each trigonometric function.
In Exercises 13–18, find the exact values of sin 2u, cos 2u,
and tan 2u using the double-angle formulas.
3
13. sin u 5, 0 < u < 兾2
2
14. cos u 7, 兾2 < u < 1.
1
15. tan u 2, < u < 3兾2
3
16. cot u 6, 3兾2 < u < 2
θ
5
17. sec u 2, 兾2 < u < 4
(a) sin (b) cos (c) cos 2
(d) sin 2
(e) tan 2
(f) sec 2
(g) csc 2
(h) cot 2
18. csc u 3, 兾2 < u < In Exercises 19–22, use a double-angle formula to rewrite the
expression. Use a graphing utility to graph both expressions
to verify that both forms are the same.
19. 8 sin x cos x
2.
20. 4 sin x cos x 1
θ
5
21. 6 12 sin2 x
22. 共cos x sin x兲共cos x sin x兲
12
(a) sin (b) cos (c) sin 2
(d) cos 2
(e) tan 2
(f) cot 2
(g) sec 2
(h) csc 2
In Exercises 3–12, use a graphing utility to approximate the
solutions of the equation in the interval [0, 2␲冈. If possible,
find the exact solutions algebraically.
3. sin 2x sin x 0
4. sin 2x cos x 0
5. 4 sin x cos x 1
6. sin 2x sin x cos x
7. cos 2x cos x 0
8. tan 2x cot x 0
9. sin 4x 2 sin 2x
11. cos 2x sin x 0
10. 共sin 2x cos 2x兲2 1
12. tan 2x 2 cos x 0
In Exercises 23–36, rewrite the expression in terms of the
first power of the cosine. Use a graphing utility to graph
both expressions to verify that both forms are the same.
23. cos4 x
24. sin4 x
25. sin2 x cos2 x
27.
sin2
x
cos4
x
29. sin2 2x
31. cos2
x
2
33. sin2 2x cos2 2x
35. sin4
x
2
26. cos 6 x
28. sin4 x cos2 x
30. cos2 2x
x
2
x
x
34. sin2 cos2
2
2
32. sin2
36. cos4
x
2
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Section 5.5
Multiple-Angle and Product-to-Sum Formulas
In Exercises 37 and 3 8, use the figure to find the exact value
of each trigonometric function.
8
49. tan u 5,
37.
51. csc u 53, < u < 3兾2
3兾2 < u < 2
50. cot u 7, < u < 3兾2
52. sec u 72, 兾2 < u < 8
θ
In Exercises 53–56, use the half-angle formulas to simplify
the expression.
15
(a) cos
2
(b) sin
2
53.
(c) tan
2
6x
冪1 cos
2
(d) sec
2
54.
(e) csc
2
(f) cot
2
4x
冪1 cos
2
55. 8x
冪11 cos
cos 8x
56. 冪1 cos2共x 1兲
(g) 2 sin
cos
2
2
(h) 2 cos
tan
2
2
38.
θ
7
24
(a) sin
(c) tan
(e) sec
2
(b) cos
2
(d) cot
2
(f) csc
(g) 2 sin
cos
2
2
2
2
2
In Exercises 57–60, find the solutions of the equation in the
interval [0, 2␲冈. Use a graphing utility to verify your
answers.
57. sin
x
cos x 0
2
58. sin
x
cos x 1 0
2
59. cos
x
sin x 0
2
60. tan
x
sin x 0
2
(h) cos 2
In Exercises 61–72, use the product-to-sum formulas to
write the product as a sum or difference.
In Exercises 39–46, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the
angle.
61. 6 sin
cos
3
3
39. 15
40. 165
62. 4 sin
41. 112 30
42. 157 30
5
cos
3
6
43.
8
44.
12
45.
3
8
46.
7
12
In Exercises 47–52, find the exact values of sin 冇u/2冈,
cos 冇u/ 2冈, and tan 冇u/ 2冈 using the half-angle formulas.
5
47. sin u 13
,
395
兾2 < u < 7
48. cos u 25
, 0 < u < 兾2
63. sin 5 cos 3
64. 5 sin 3 sin 4
65. 10 cos 75 cos 15
66. 6 sin 45 cos 15
67. 5 cos共5兲 cos 3
68. cos 2 cos 4
69. sin共x y兲 sin共x y兲
70. sin共x y兲 cos共x y兲
71. cos共 兲 sin共 兲
72. sin共 兲 sin共 兲
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Analytic Trigonometry
sec2 2 sec2 In Exercises 73–80, use the sum-to-product formulas to
write the sum or difference as a product.
94. sec 2 73. sin 5 sin 95. cos2 2 sin2 2 cos 4
74. sin 3 sin 96. cos4 x sin4 x cos 2x
75. cos 6x cos 2x
97. 共sin x cos x兲2 1 sin 2x
76. sin x sin 7x
98. sin
77. sin共 兲 sin共 兲
99. 1 cos 10y 2 cos2 5y
78. cos共
2兲 cos 冢
cos 2
2
冢
sin x 2
2
79. cos 80. sin x 冣
冢
冣
冢
2
1
cos sin
3
3
2
3
冣
100.
冣
In Exercises 81–84, use the sum-to-product formulas to find
the exact value of the expression.
81. sin 195 sin 105
cos 3
1 4 sin2 cos 冪tan 2u tan sinu u
101. sec
u
±
2
102. tan
u
csc u cot u
2
103. cos 3 cos3 3 sin2 cos 104. sin 4 4 sin cos 共1 2 sin2 兲
82. cos 165 cos 75
105.
cos 4x cos 2x
sin x
2 sin 3x
106.
cos 3x cos x
tan 2x
sin 3x sin x
107.
cos 4x cos 2x
cot 3x
sin 4x sin 2x
In Exercises 85– 88, find the solutions of the equation in the
interval [0, 2␲冈. Use a graphing utility to verify your
answers.
108.
cos t cos 3t
cot t
sin 3t sin t
85. sin 6x sin 2x 0
109. sin
5
83. cos
cos
12
12
84. sin
11
7
sin
12
12
86. cos 2x cos 6x 0
冢6 x冣 sin冢6 x冣 cos x
冢3 x冣 cos冢3 x冣 cos x
cos 2x
87.
10
sin 3x sin x
110. cos
88. sin2 3x sin2 x 0
In Exercises 111–114, rewrite the function using the powerreducing formulas. Then use a graphing utility to graph the
function.
In Exercises 89–92, use the figure and trigonometric identities to find the exact value of the trigonometric function in
two ways.
111. f 共x兲 sin2 x
112. f 共x兲 cos2 x
113. f 共x兲 cos 4 x
3
5
β 4
α
114. f 共x兲 sin 3 x
In Exercises 115–120, write the trigonometric expression as
an algebraic expression.
12
89. sin2 90. cos2 91. sin cos 92. cos sin 115. sin共2 arcsin x兲
116. cos共2 arccos x兲
117. cos共2 arcsin x兲
In Exercises 93–110, verify the identity algebraically. Use a
graphing utility to check your result graphically.
118. sin共2 arccos x兲
csc 93. csc 2 2 cos 120. sin共2 arctan x兲
119. cos共2 arctan x兲
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Section 5.5
In Exercises 121–124, (a) use a graphing utility to graph the
function and approximate the maximum and minimum
points of the graph in the interval [0, 2␲], and (b) solve
the trigonometric equation and verify that the x-coordinates of the maximum and minimum points of f are among
its solutions (calculus is required to find the trigonometric
equation).
Function
x
121. f 共x兲 4 sin cos x
2
Trigonometric Equation
x
2 cos sin x 0
2
122. f 共x兲 cos 2x 2 sin x
2 cos x 共2 sin x1兲 0
123. f 共x兲 2 cos
x
sin 2x
2
2 cos 2x sin
124. f 共x兲 2 sin
x
2
10 sin 2x 冢
5 cos 2x 冢
4
x
0
2
x
cos 0
4
2
冣
冣
In Exercises 125 and 126, the graph of a function f is shown
over the interval [0, 2␲]. (a) Find the x-intercepts of the
graph of f algebraically. Verify your solutions by using the
zero or root feature of a graphing utility. (b) The x-coordinates of the extrema or turning points of the graph of f are
solutions of the trigonometric equation (calculus is required
to find the trigonometric equation). Find the solutions of
the equation algebraically. Verify the solutions using the
maximum and minimum features of a graphing utility.
125. Function: f 共x兲 sin 2x sin x
Trigonometric equation: 2 cos 2x cos x 0
3
f
2
0
Multiple-Angle and Product-to-Sum Formulas
127. Projectile Motion The range of a projectile fired at an
angle with the horizontal and with an initial velocity of
v0 feet per second is given by
r
1 2
v sin 2
32 0
where r is measured in feet.
(a) Rewrite the expression for the range in terms of .
(b) Find the range r if the initial velocity of a projectile is
80 feet per second at an angle of 42.
(c) Find the initial velocity required to fire a projectile
300 feet at an angle of 40.
(d) For a given initial velocity, what angle of elevation
yields a maximum range? Explain.
128. Geometry The length of each of the two equal sides of
an isosceles triangle is 10 meters (see figure). The angle
between the two sides is .
10 m
θ
10 m
(a) Write the area of the triangle as a function of 兾2.
(b) Write the area of the triangle as a function of and
determine the value of such that the area is a
maximum.
129. Railroad Track When two railroad tracks merge, the
overlapping portions of the tracks are in the shape of a
circular arc (see figure). The radius of each arc r (in feet)
and the angle are related by
x
2r sin2 .
2
2
Write a formula for x in terms of cos .
−3
126. Function: f 共x兲 cos 2x sin x
Trigonometric equation: 2 sin 2x cos x 0
r
3
r
θ
2
0
f
−3
397
θ
x
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Analytic Trigonometry
130. Mach Number The mach number M of an airplane is the
ratio of its speed to the speed of sound. When an airplane
travels faster than the speed of sound, the sound waves form
a cone behind the airplane (see figure). The mach number
is related to the apex angle of the cone by
sin
(d) Rewrite the result of part (c) in terms of the sine of a
double angle. Use a graphing utility to rule out incorrectly rewritten functions.
1
.
2 M
(e) When you rewrite a trigonometric expression, the
result may not be the same as a friend’s. Does this
mean that one of you is wrong? Explain.
θ
(a) Find the angle that corresponds to a mach number of 1.
(b) Find the angle that corresponds to a mach number of
4.5.
(c) The speed of sound is about 760 miles per hour.
Determine the speed of an object having the mach
numbers in parts (a) and (b).
(d) Rewrite the equation as a trigonometric function of .
Synthesis
135. Writing Describe how you can use a double-angle
formula or a half-angle formula to derive a formula for the
area of an isosceles triangle. Use a labeled sketch to
illustrate your derivation. Then write two examples that
show how your formula can be used.
136. (a) Write a formula for cos 3.
(b) Write a formula for cos 4.
Skills Review
In Exercises 137–140, (a) plot the points, (b) find the
distance between the points, and (c) find the midpoint of the
line segment connecting the points.
137. 共5, 2兲, 共1, 4兲
True or False? In Exercises 131 and 132, determine
whether the statement is true or false. Justify your answer.
131. sin
(c) Add a trigonometric term to the function so that it
becomes a perfect square trinomial. Rewrite the
function as a perfect square trinomial minus the term
that you added. Use a graphing utility to rule out
incorrectly rewritten functions.
冪
x
2
1 cos x
,
2
≤ x ≤ 2
138. 共4, 3兲, 共6, 10兲
1
4 5
139. 共0, 2 兲, 共3, 2 兲
140.
共13, 23 兲, 共1, 32 兲
132. The graph of y 4 8 sin2 x has a maximum at 共, 4兲.
In Exercises 141–144, find (if possible) the complement and
supplement of each angle.
133. Conjecture Consider the function
141. (a) 55
(b) 162
冢
142. (a) 109
(b) 78
143. (a)
18
(b)
144. (a) 0.95
(b) 2.76
f 共x兲 2 sin x 2 cos 2
冣
x
1 .
2
(a) Use a graphing utility to graph the function.
(b) Make a conjecture about the function that is an identity with f.
(c) Verify your conjecture algebraically.
134. Exploration
Consider the function
f 共x兲 sin x cos4 x.
4
(a) Use the power-reducing formulas to write the function
in terms of cosine to the first power.
(b) Determine another way of rewriting the function. Use
a graphing utility to rule out incorrectly rewritten
functions.
9
20
145. Find the radian measure of the central angle of a circle
with a radius of 15 inches that intercepts an arc of length
7 inches.
146. Find the length of the arc on a circle of radius 21 centimeters intercepted by a central angle of 35.
In Exercises 147–150, sketch a graph of the function.
(Include two full periods.) Use a graphing utility to verify
your graph.
147. f 共x兲 3
cos 2x
2
148. f 共x兲 1
5
sin x
2
2
149. f 共x兲 1
tan 2x
2
150. f 共x兲 1
x
sec
4
2
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Chapter Summary
399
What Did You Learn?
Key Terms
sum and difference formulas, p. 380
reduction formulas, p. 382
double-angle formulas, p. 387
power-reducing formulas, p. 389
half-angle formulas, p. 390
Key Concepts
5.1
䊏
Use the fundamental trigonometric
identities
The fundamental trigonometric identities can be used to
evaluate trigonometric functions, simplify trigonometric
expressions, develop additional trigonometric identities,
and solve trigonometric equations.
5.2 䊏 Verify trigonometric identities
1. Work with one side of the equation at a time. It is
often better to work with the more complicated side
first.
5.5
product-to-sum formulas, p. 391
sum-to-product formulas, p. 392
䊏
Use multiple-angle formulas, power-reducing
formulas, half-angle formulas, product-tosum formulas, and sum-to-product formulas
Double-Angle Formulas:
sin 2u 2 sin u cos u
tan 2u 2 tan u
1 tan2 u
fractions, square a binomial, or create a monomial
denominator.
3. Look for opportunities to use the fundamental identi-
Half-Angle Formulas:
ties. Note which functions are in the final expression
you want. Sines and cosines pair up well, as do
secants and tangents, and cosecants and cotangents.
4. Try converting all terms to sines and cosines.
5. Always try something.
5.3 䊏 Solve trigonometric equations
1. Use algebraic techniques, such as collecting like
terms, extracting square roots, factoring, and the
Quadratic Formula to isolate the trigonometric
function involved in the equation.
2. If there is no reasonable way to find the solution(s)
of a trigonometric equation algebraically, use a
graphing utility to approximate the solution(s).
5.4
䊏
Use sum and difference formulas to evaluate
trigonometric functions, verify identities, and
solve trigonometric equations
sin共u ± v兲 sin u cos v ± cos u sin v
cos共u ± v兲 cos u cos v sin u sin v
tan共u ± v兲 tan u ± tan v
1 tan u tan v
2 cos2 u 1
1 2 sin2 u
Power-Reducing Formulas:
1 cos 2u
2
1
cos 2u
cos2 u 2
1
cos
2u
tan2 u 1 cos 2u
2. Look for opportunities to factor an expression, add
cos 2u cos2 u sin2 u
sin2 u 冪
u
1 cos u
u
±
cos ±
2
2
2
u 1 cos u
sin u
tan 2
sin u
1 cos u
sin
冪1 2cos u
Product-to-Sum Formulas:
sin u sin v 12关cos共u v兲 cos共u v兲兴
cos u cos v 12关cos共u v兲 cos共u v兲兴
sin u cos v 12关sin共u v兲 sin共u v兲兴
cos u sin v 12关sin共u v兲 sin共u v兲兴
Sum-to-Product Formulas:
冢u 2 v冣 cos冢u 2 v冣
uv
uv
sin u sin v 2 cos冢
sin
2 冣 冢 2 冣
uv
uv
cos u cos v 2 cos冢
cos冢
2 冣
2 冣
uv
uv
cos u cos v 2 sin冢
sin
2 冣 冢 2 冣
sin u sin v 2 sin
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Analytic Trigonometry
Review Exercises
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
5.1 In Exercises 1–10, name the trigonometric function that
is equivalent to the expression.
5.2 In Exercises 25–36, verify the identity.
25. cos x共tan2 x 1兲 sec x
1
1.
cos x
1
2.
sin x
26. sec2 x cot x cot x tan x
1
3.
sec x
1
4.
tan x
28. cot2 x cos2 x cot2 x cos2 x
5. 冪1 cos2 x
6. 冪1 tan2 x
7. csc
冢2 x冣
8. cot
9. sec共x兲
27. sin3 sin cos 2 sin 冢2 x冣
29. sin5 x cos2 x 共cos2 x 2 cos 4 x cos6 x兲 sin x
30. cos3 x sin2 x 共sin2 x sin4 x兲 cos x
31.
10. tan共x兲
1 sin 冪11 sin
sin ⱍcos ⱍ
32. 冪1 cos x In Exercises 11–14, use the given values to evaluate (if
possible) the remaining trigonometric functions of the angle.
4
11. sin x ,
5
cos x 2
12. tan ,
3
sec 3
5
冪13
3
冢
14. csc
冣
冢2 冣 3,
sin 17.
1
tan2 x 1
sin2 cos 2 sin2 sin cos 19. tan2 共csc2 1兲
21. tan
冢2 x冣 sec x
34.
1 sec共x兲
csc x
sin共x兲 tan共x兲
16.
18.
2
x
冢2 x冣 sec x csc x
37. 2 sin x 1 0
38. tan x 1 0
39. sin x 冪3 sin x
40. 4 cos x 1 2 cos x
41. 3冪3 tan x 3
42.
x1
sec x 1
43. 3 csc x 4
44. 4 tan2 x 1 tan2 x
45. 4 cos2 x 3 0
46. sin x共sin x 1兲 0
sin3 cos3 sin cos 47. sin x tan x 0
48. csc x 2 cot x 0
2
sec2
20. csc 2 x共1 cos2 x兲
22.
冢2 x冣 1 tan
5.3 In Exercises 37–48, solve the equation.
In Exercises 15 – 22, use the fundamental identities to
simplify the expression. Use the table feature of a graphing
utility to check your result numerically.
15.
csc共x兲
cot x
sec共x兲
36. tan
2冪2
3
冪1 cos x
33.
35. csc2
1
1
13. sin
x , sin x 2
冪2
冪2
ⱍsin xⱍ
sin共x兲 cot x
sin
x
2
冢
冣
23. Rate of Change The rate of change of the function
f 共x兲 2冪sin x is given by the expression sin1兾2 x cos x.
Show that this expression can also be written as
cot x冪sin x.
24. Rate of Change The rate of change of the function
f 共x兲 csc x cot x is given by the expression
csc2 x csc x cot x. Show that this expression can also be
written as 共1 cos x兲兾sin2 x.
1
2
sec x 1 0
In Exercises 49–52, find all solutions of the equation in the
interval [0, 2␲冈. Use a graphing utility to check your
answers.
49. 2 cos2 x cos x 1
50. 2 sin2 x 3 sin x 1
51. cos2 x sin x 1
52. sin2 x 2 cos x 2
In Exercises 53–58, find all solutions of the multiple-angle
equation in the interval [0, 2␲冈.
53. 2 sin 2x 冪2 0
54. 冪3 tan 3x 0
55. cos 4x共cos x 1兲 0
56. 3 csc2 5x 4
57. cos 4x 7 cos 2x 8
58. sin 4x sin 2x 0
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Review Exercises
In Exercises 59–62, solve the equation.
59. 2 sin 2x 1 0
60. 2 cos 4x 冪3 0
61. 2 sin2 3x 1 0
62. 4 cos2 2x 3 0
In Exercises 63 – 66, use the inverse functions where
necessary to find all solutions of the equation in the interval
[0, 2␲冈.
63. sin2 x 2 sin x 0
65. tan2 3 tan 10 0
66.
冢
sin x 冪2
2
2
冢
cos x 1
4
4
91. sin x 92. cos x 冣
冢
冣
冣
冢
5
93. sin u ,
7
x 6 tan x 4 0
5.4 In Exercises 67– 70, find the exact values of the sine,
cosine, and tangent of the angle.
67. 285
315
30
68. 345
300
45
31 11 3
69.
12
6
4
70.
13 11 3
12
6
4
In Exercises 71–74, write the expression as the sine, cosine,
or tangent of an angle.
71. sin 130
cos 50
cos 130
sin 50
冣
< u <
3
< u < 2
2
4
94. cos u ,
5
< u < 2
2
95. tan u ,
9
96. cos u 3
2
2
冪5
,
< u < 2
In Exercises 97–100, use double-angle formulas to verify the
identity algebraically. Use a graphing utility to check your
result graphically.
97. 6 sin x cos x 3 sin 2x
72. cos 45
cos 120
sin 45
sin 120
tan 25
tan 10
73.
1 tan 25
tan 10
In Exercises 91 and 92, find the solutions of the equation in
the interval [0, 2␲冈.
5.5 In Exercises 93–96, find the exact values of sin 2u,
cos 2u, and tan 2u using the double-angle formulas.
64. 3 cos2 x 5 cos x 0
sec2
401
tan 63
tan 118
74.
1 tan 63
tan 118
In Exercises 75– 80, find the exact value of the trigonometric
7
function given that sin u ⴝ 35 and cos v ⴝ ⴚ 25
. (Both u and v
are in Quadrant II.)
75. sin共u v兲
76. tan共u v兲
77. tan共u v兲
78. sin共u v兲
79. cos共u v兲
80. cos共u v兲
In Exercises 81– 84, find the value of the expression without
using a calculator.
98. 4 sin x cos x 2 2 sin 2x 2
99. 1 4 sin2 x cos2 x cos2 2x
100. sin 4x 8 cos3 x sin x 4 cos x sin x
101. Projectile Motion A baseball leaves the hand of the first
baseman at an angle of with the horizontal and with an
initial velocity of v0 80 feet per second. The ball is
caught by the second baseman 100 feet away. Find if the
1 2
range r of a projectile is given by r 32
v0 sin 2.
102. Projectile Motion Use the equation in Exercise 101 to
find when a golf ball is hit with an initial velocity of
v0 50 feet per second and lands 77 feet away.
83. cos关cos1 1 sin1共1兲兴
In Exercises 103–106, use the power-reducing formulas to
rewrite the expression in terms of the first power of the
cosine.
84. tan关cos1共1兲 cos1 1兴
103. sin6 x
81. sin关sin1 0 cos1共1兲兴
82. cos共cos1 1 sin1 0兲
105.
cos4
2x
104. cos4 x sin4 x
106. sin4 2x
In Exercises 85– 90, verify the identity.
冢
85. cos x 87. cot
sin x
2
冣
冢 2 x冣 tan x
89. cos 3x 4 cos3 x 3 cos x
90.
sin共 兲
tan tan cos cos 冢
86. sin x 3
cos x
2
冣
88. sin共 x兲 sin x
In Exercises 107–110, use the half-angle formulas to
determine the exact values of the sine, cosine, and tangent of
the angle.
107. 105
109.
7
8
108. 112
30
110.
11
12
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Analytic Trigonometry
冢
sin x 4
4
冢
cos x 6
6
In Exercises 111–114, find the exact values of sin冇u/ 2冈,
cos冇u/ 2冈, and tan 冇u/ 2冈 using the half-angle formulas.
125. sin x 2
126. cos x 3
111. sin u ,
5
112. tan u 0 < u <
21
,
20
< u <
3
2
2
113. cos u ,
7
< u < 2
114. sec u 6,
< u < 2
In Exercises 115 and 116, use the half-angle formulas to
simplify the expression.
115. 116.
冪
1 cos 8x
2
冣
冢
冣
冣
冢
冣
Harmonic Motion In Exercises 127–130, 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. This motion
is described by the model
y ⴝ 1.5 sin 8t ⴚ 0.5 cos 8t
where y is the distance from equilibrium in feet and t is the
time in seconds.
127. Write the model in the form
y 冪a 2 b2 sin共Bt C兲.
128. Use a graphing utility to graph the model.
sin 10x
1 cos 10x
129. Find the amplitude of the oscillations of the weight.
130. Find the frequency of the oscillations of the weight.
Geometry In Exercises 117 and 118, a trough for feeding
cattle is 4 meters long and its cross sections are isosceles
triangles with two equal sides of 12 meter (see figure). The angle
between the equal sides is ␪.
Synthesis
True or False? In Exercises 131–134, determine whether
the statement is true or false. Justify your answer.
131. If
< < , then cos < 0.
2
2
132. sin共x y兲 sin x sin y
4m
133. 4 sin共x兲 cos共x兲 2 sin 2x
1
2m
134. 4 sin 45
cos 15
1 冪3
θ
1
2
135. List the reciprocal identities, quotient identities, and
Pythagorean identities from memory.
m
117. Write the trough’s volume as a function of 兾2.
136. Is cos 冪1 sin2 an identity? Explain.
118. Write the volume of the trough as a function of and
determine the value of such that the volume is
maximum.
In Exercises 137 and 138, use the graphs of y1 and y2 to
determine how to change y2 to a new function y3 such that
y1 ⴝ y3 .
In Exercises 119–122, use the product-to-sum formulas to
write the product as a sum or difference.
137. y1 sec2
119. 6 sin
cos
4
4
121. sin 5 sin 4
冢 2 x冣
cos 3x
cos x
y2 共2 sin x兲2
y2 cot2 x
120. 4 sin 15
sin 45
5
5
y2
122. cos 6 sin 8
y1
In Exercises 123–126, use the sum-to-product formulas to
write the sum or difference as a product.
123. cos 5 cos 4
138. y1 124. sin 3 sin 2
−
−1
y2
−
y1
−4
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Chapter Test
5 Chapter Test
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Take this test as you would take a test in class. After you are finished, check your
work against the answers given in the back of the book.
1. If tan 32 and cos < 0, use the fundamental identities to evaluate the other five
trigonometric functions of .
2. Use the fundamental identities to simplify csc2 共1 cos2 兲.
3. Factor and simplify
4. Add and simplify
sec4 x tan4 x
.
sec2 x tan2 x
sin cos .
sin cos 5. Determine the values of , 0 ≤ < 2, for which tan 冪sec2 1 is true.
6. Use a graphing utility to graph the functions y1 sin x cos x cot x and y2 csc x.
Make a conjecture about y1 and y2. Verify your result algebraically.
In Exercises 7–12, verify the identity.
7. sin sec tan 8. sec2 x tan2 x sec2 x sec4 x
9.
csc sec cot tan sin cos 冢
10. cos x sin x
2
冣
11. sin共n 兲 共1兲n sin , n is an integer.
12. 共sin x cos x兲2 1 sin 2x
y
13. Find the exact value of tan 105
.
14. Rewrite sin4 x tan2 x in terms of the first power of the cosine.
(1, 2)
sin 4
15. Use a half-angle formula to simplify the expression
.
1 cos 4
16. Write 4 cos 2 sin 4 as a sum or difference.
u
x
17. Write sin 3 sin 4 as a product.
In Exercises 18 – 21, find all solutions of the equation in the interval [0, 2␲冈.
18. tan2 x tan x 0
19. sin 2 cos 0
20. 4 cos x 3 0
21. csc2 x csc x 2 0
2
22. Use a graphing utility to approximate the solutions of the equation 3 cos x x 0
accurate to three decimal places.
Figure for 23
Air
θ
23. Use the figure to find the exact values of sin 2u, cos 2u, and tan 2u.
24. The index of refraction n of a transparent material is the ratio of the speed of light in a
vacuum to the speed of light in the material. For the triangular glass prism in the figure,
n 1.5 and 60
. Find the angle for the glass prism if
n
sin共兾2 兾2兲
.
sin共兾2兲
α
ht
Lig
Prism
Figure for 24
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Analytic Trigonometry
Proofs in Mathematics
Sum and Difference Formulas
(p. 380)
sin共u v兲 sin u cos v cos u sin v
sin共u v兲 sin u cos v cos u sin v
cos共u v兲 cos u cos v sin u sin v
cos共u v兲 cos u cos v sin u sin v
tan共u v兲 tan u tan v
1 tan u tan v
tan共u v兲 tan u tan v
1 tan u tan v
Proof
You can use the figures at the right for the proofs of the formulas for cos共u ± v兲.
In the top figure, let A be the point 共1, 0兲 and then use u and v to locate the points
B 共x1, y1兲, C 共x2, y2兲, and D 共x3, y3兲 on the unit circle. So, x i2 y i2 1
for i 1, 2, and 3. For convenience, assume that 0 < v < u < 2. In the
bottom figure, note that arcs AC and BD have the same length. So, line segments
AC and BD are also equal in length, which implies that
y
B = (x1, y1 )
C = (x2, y2 )
u−v
v
A = (1, 0)
x
u
冪共x2 1兲 共 y2 0兲 冪共x3 x1兲 共 y3 y1兲
2
2
2
2
x22 2x2 1 y22 x32 2x1x3 x12 y32 2y1 y3 y12
D = (x3, y3 )
共x22 y22兲 1 2x2 共x32 y32兲 共x12 y12兲 2x1x3 2y1y3
1 1 2x2 1 1 2x1 x3 2y1 y3
x2 x3 x1 y3 y1.
Finally, by substituting the values x2 cos共u v兲, x3 cos u, x1 cos v,
y3 sin u, and y1 sin v, you obtain cos共u v兲 cos u cos v sin u sin v.
The formula for cos共u v兲 can be established by considering
u v u 共v兲 and using the formula just derived to obtain
y
C = (x2, y2 )
cos共u v兲 cos关u 共v兲兴 cos u cos共v兲 sin u sin 共v兲
A = (1, 0)
x
cos u cos v sin u sin v.
You can use the sum and difference formulas for sine and cosine to prove the
formulas for tan共u ± v兲.
tan共u ± v兲 sin共u ± v兲
cos共u ± v兲
Quotient identity
sin u cos v ± cos u sin v
cos u cos v sin u sin v
Sum and difference formulas
sin u cos v ± cos u sin v
cos u cos v
cos u cos v sin u sin v
cos u cos v
Divide numerator and denominator
by cos u cos v.
B = (x1, y1 )
D = (x3, y3 )
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Proofs in Mathematics
sin u cos v
cos u sin v
±
cos u cos v cos u cos v
cos u cos v
sin u sin v
cos u cos v cos u cos v
Write as separate fractions.
sin v
sin u
±
cos u cos v
sin u sin v
1
cos u cos v
Product of fractions
tan u ± tan v
1 tan u tan v
Double-Angle Formulas
tan 2u Quotient identity
(p. 387)
sin 2u 2 sin u cos u
Trigonometry
and Astronomy
cos 2u cos2 u sin2 u
2 cos2 u 1
2 tan u
1 tan2 u
1 2 sin2 u
Proof
To prove all three formulas, let v u in the corresponding sum formulas.
sin 2u sin共u u兲 sin u cos u cos u sin u 2 sin u cos u
cos 2u cos共u u兲 cos u cos u sin u sin u cos2 u sin2 u
tan 2u tan共u u兲 tan u tan u
2 tan u
1 tan u tan u 1 tan2 u
Power-Reducing Formulas
sin2 u 1 cos 2u
2
(p. 389)
cos2 u 1 cos 2u
2
tan2 u 1 cos 2u
1 cos 2u
Proof
To prove the first formula, solve for sin2 u in the double-angle formula
cos 2u 1 2 sin2 u, as follows.
cos 2u 1 2 sin2 u
2 sin2 u 1 cos 2u
sin2 u 1 cos 2u
2
405
Write double-angle formula.
Subtract cos 2u from and add 2 sin2 u to each side.
Divide each side by 2.
Trigonometry was used by
early astronomers to calculate
measurements in the universe.
Trigonometry was used to
calculate the circumference
of Earth and the distance from
Earth to the moon. Another
major accomplishment in
astronomy using trigonometry
was computing distances to
stars.
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Analytic Trigonometry
In a similar way you can prove the second formula, by solving for cos2 u in the
double-angle formula
cos 2u 2 cos2 u 1.
To prove the third formula, use a quotient identity, as follows.
tan2 u sin2 u
cos2 u
1 cos 2u
2
1 cos 2u
2
1 cos 2u
1 cos 2u
Sum-to-Product Formulas
sin u sin v 2 sin
(p. 392)
冢u 2 v冣 cos冢u 2 v冣
sin u sin v 2 cos
冢u 2 v冣 sin冢u 2 v冣
cos u cos v 2 cos
冢u 2 v冣 cos冢u 2 v冣
cos u cos v 2 sin
冢u 2 v冣 sin冢u 2 v冣
Proof
To prove the first formula, let x u v and y u v. Then substitute
u 共x y兲兾2 and v 共x y兲兾2 in the product-to-sum formula.
1
sin u cos v 关sin共u v兲 sin共u v兲兴
2
sin
2 sin
冢x 2 y冣 cos冢x 2 y冣 12 共sin x sin y兲
冢x 2 y冣 cos冢x 2 y冣 sin x sin y
The other sum-to-product formulas can be proved in a similar manner.