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Precalculus with Limits (2010)

Ron Larson, David C. Falvo

Chapter 5

Analytic Trigonometry

Educators


Problem 1

$\frac{\sin u}{\cos u}=$ _____

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Problem 2

$\frac{1}{\csc u}=$ ______

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Problem 3

$\frac{1}{\tan u}=$ _____

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Problem 4

$\frac{1}{\cos u}=$ _____

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Problem 5

$1+$ _____ $=\csc ^{2} u$

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Problem 6

$1+\tan ^{2} u=$ _____

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Problem 7

$\sin \left(\frac{\pi}{2}-u\right)=$ _____

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Problem 8

$\sec \left(\frac{\pi}{2}-u\right)=$ _____

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Problem 9

$\cos (-u)=$ _____

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Problem 10

$\tan (-u)=$ _____

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Problem 11

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sin x=\frac{1}{2}, \quad \cos x=\frac{\sqrt{3}}{2}
$$

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Problem 12

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan x=\frac{\sqrt{3}}{3}, \quad \cos x=-\frac{\sqrt{3}}{2}
$$

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Problem 13

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sec \theta=\sqrt{2}, \quad \sin \theta=-\frac{\sqrt{2}}{2}
$$

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Problem 14

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\csc \theta=\frac{25}{7}, \quad \tan \theta=\frac{7}{24}
$$

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Problem 15

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan x=\frac{8}{15}, \quad \sec x=-\frac{17}{15}
$$

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Problem 16

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\cot \phi=-3, \quad \sin \phi=\frac{\sqrt{10}}{10}
$$

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Problem 17

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sec \phi=\frac{3}{2}, \quad \csc \phi=-\frac{3 \sqrt{5}}{5}
$$

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Problem 18

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\cos \left(\frac{\pi}{2}-x\right)=\frac{3}{5}, \quad \cos x=\frac{4}{5}
$$

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Problem 19

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sin (-x)=-\frac{1}{3}, \quad \tan x=-\frac{\sqrt{2}}{4}
$$

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Problem 20

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sec x=4, \quad \sin x>0
$$

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Problem 21

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan \theta=2, \quad \sin \theta<0
$$

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Problem 22

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\csc \theta=-5, \quad \cos \theta<0
$$

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Problem 23

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sin \theta=-1, \quad \cot \theta=0
$$

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Problem 24

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan \theta \text { is undefined, } \sin \theta>0
$$

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Problem 25

In Exercises 25–30, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$
$$
\sec x \cos x
$$

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Problem 26

In Exercises 25–30, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$
$$
\tan x \csc x
$$

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Problem 27

In Exercises 25–30, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$
$$
\cot ^{2} x-\csc ^{2} x
$$

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Problem 28

In Exercises 25–30, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$
$$
\left(1-\cos ^{2} x\right)(\csc x)
$$

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Problem 29

In Exercises 25–30, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$
$$
\frac{\sin (-x)}{\cos (-x)}
$$

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Problem 30

In Exercises 25–30, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$
$$
\frac{\sin [(\pi / 2)-x]}{\cos [(\pi / 2)-x]}
$$

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Problem 31

In Exercises 31–36, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$
$$
\sin x \sec x
$$

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Problem 32

In Exercises 31–36, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$
$$
\cos ^{2} x\left(\sec ^{2} x-1\right)
$$

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Problem 33

In Exercises 31–36, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$
$$
\sec ^{4} x-\tan ^{4} x
$$

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Problem 34

In Exercises 31–36, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$
$$
\cot x \sec x
$$

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Problem 35

In Exercises 31–36, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$
$$
\frac{\sec ^{2} x-1}{\sin ^{2} x}
$$

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Problem 36

In Exercises 31–36, match the trigonometric expression with one of the following.
$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$
$$
\frac{\cos ^{2}[(\pi / 2)-x]}{\cos x}
$$

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Problem 37

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\cot \theta \sec \theta
$$

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Problem 38

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\cos \beta \tan \beta
$$

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Problem 39

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\tan (-x) \cos x
$$

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Problem 40

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\sin x \cot (-x)
$$

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Problem 41

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\sin \phi(\csc \phi-\sin \phi)
$$

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Problem 42

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\sec ^{2} x\left(1-\sin ^{2} x\right)
$$

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Problem 43

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{\cot x}{\csc x}
$$

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Problem 44

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{\csc \theta}{\sec \theta}
$$

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Problem 45

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{1-\sin ^{2} x}{\csc ^{2} x-1}
$$

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Problem 46

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{1}{\tan ^{2} x+1}
$$

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Problem 47

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{\tan \theta \cot \theta}{\sec \theta}
$$

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Problem 48

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{\sin \theta \csc \theta}{\tan \theta}
$$

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Problem 49

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\sec \alpha \cdot \frac{\sin \alpha}{\tan \alpha}
$$

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Problem 50

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{\tan ^{2} \theta}{\sec ^{2} \theta}
$$

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Problem 51

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\cos \left(\frac{\pi}{2}-x\right) \sec x
$$

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Problem 52

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\cot \left(\frac{\pi}{2}-x\right) \cos x
$$

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Problem 53

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\frac{\cos ^{2} y}{1-\sin y}
$$

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Problem 54

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\cos t\left(1+\tan ^{2} t\right)
$$

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Problem 55

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\sin \beta \tan \beta+\cos \beta
$$

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Problem 56

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\csc \phi \tan \phi+\sec \phi
$$

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Problem 57

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\cot u \sin u+\tan u \cos u
$$

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Problem 58

In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$$
\sin \theta \sec \theta+\cos \theta \csc \theta
$$

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Problem 59

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\tan ^{2} x-\tan ^{2} x \sin ^{2} x
$$

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Problem 60

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\sin ^{2} x \csc ^{2} x-\sin ^{2} x
$$

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Problem 61

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\sin ^{2} x \sec ^{2} x-\sin ^{2} x
$$

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Problem 62

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\cos ^{2} x+\cos ^{2} x \tan ^{2} x
$$

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Problem 63

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\frac{\sec ^{2} x-1}{\sec x-1}
$$

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Problem 64

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\frac{\cos ^{2} x-4}{\cos x-2}
$$

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Problem 65

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\tan ^{4} x+2 \tan ^{2} x+1
$$

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Problem 66

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
1-2 \cos ^{2} x+\cos ^{4} x
$$

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Problem 67

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\sin ^{4} x-\cos ^{4} x
$$

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Problem 68

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\sec ^{4} x-\tan ^{4} x
$$

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Problem 69

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\csc ^{3} x-\csc ^{2} x-\csc x+1
$$

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Problem 70

In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\sec ^{3} x-\sec ^{2} x-\sec x+1
$$

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Problem 71

In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
(\sin x+\cos x)^{2}
$$

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Problem 72

In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
(\cot x+\csc x)(\cot x-\csc x)
$$

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Problem 73

In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
(2 \csc x+2)(2 \csc x-2)
$$

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Problem 74

In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
(3-3 \sin x)(3+3 \sin x)
$$

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Problem 75

In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\frac{1}{1+\cos x}+\frac{1}{1-\cos x}
$$

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Problem 76

In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\frac{1}{\sec x+1}-\frac{1}{\sec x-1}
$$

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Problem 77

In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}
$$

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Problem 78

In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\frac{\tan x}{1+\sec x}+\frac{1+\sec x}{\tan x}
$$

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Problem 79

In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\tan x+\frac{\cos x}{1+\sin x}
$$

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Problem 80

In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$$
\tan x-\frac{\sec ^{2} x}{\tan x}
$$

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Problem 81

In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.
$$
\frac{\sin ^{2} y}{1-\cos y}
$$

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Problem 82

In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.
$$
\frac{5}{\tan x+\sec x}
$$

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Problem 83

In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.
$$
\frac{3}{\sec x-\tan x}
$$

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Problem 84

In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.
$$
\frac{\tan ^{2} x}{\csc x+1}
$$

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Problem 85

In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .
$$
y_{1}=\cos \left(\frac{\pi}{2}-x\right), \quad y_{2}=\sin x
$$

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Problem 86

In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .
$$
y_{1}=\sec x-\cos x, \quad y_{2}=\sin x \tan x
$$

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Problem 87

In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .
$$
y_{1}=\frac{\cos x}{1-\sin x}, \quad y_{2}=\frac{1+\sin x}{\cos x}
$$

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Problem 88

In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .
$$
y_{1}=\sec ^{4} x-\sec ^{2} x, \quad y_{2}=\tan ^{2} x+\tan ^{4} x
$$

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Problem 89

In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.
$$
\cos x \cot x+\sin x
$$

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Problem 90

In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.
$$
\sec x \csc x-\tan x
$$

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Problem 91

In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.
$$
\frac{1}{\sin x}\left(\frac{1}{\cos x}-\cos x\right)
$$

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Problem 92

In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.
$$
\frac{1}{2}\left(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}\right)
$$

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Problem 93

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{9-x^{2}}, \quad x=3 \cos \theta
$$

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Problem 94

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{64-16 x^{2}}, \quad x=2 \cos \theta
$$

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Problem 95

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{16-x^{2}}, \quad x=4 \sin \theta
$$

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Problem 96

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{49-x^{2}}, \quad x=7 \sin \theta
$$

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Problem 97

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{x^{2}-9}, \quad x=3 \sec \theta
$$

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Problem 98

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{x^{2}-4}, \quad x=2 \sec \theta
$$

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Problem 99

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{x^{2}+25}, \quad x=5 \tan \theta
$$

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Problem 100

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{x^{2}+100}, \quad x=10 \tan \theta
$$

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Problem 101

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{4 x^{2}+9}, \quad 2 x=3 \tan \theta
$$

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Problem 102

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{9 x^{2}+25}, \quad 3 x=5 \tan \theta
$$

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Problem 103

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{2-x^{2}}, \quad x=\sqrt{2} \sin \theta
$$

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Problem 104

In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$
$$
\sqrt{10-x^{2}}, \quad x=\sqrt{10} \sin \theta
$$

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Problem 105

In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.
$$
3=\sqrt{9-x^{2}}, \quad x=3 \sin \theta
$$

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Problem 106

In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.
$$
3=\sqrt{36-x^{2}}, \quad x=6 \sin \theta
$$

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Problem 107

In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.
$$
2 \sqrt{2}=\sqrt{16-4 x^{2}}, \quad x=2 \cos \theta
$$

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Problem 108

In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.
$$
-5 \sqrt{3}=\sqrt{100-x^{2}}, \quad x=10 \cos \theta
$$

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Problem 109

In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\sin \theta=\sqrt{1-\cos ^{2} \theta}
$$

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Problem 110

In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\cos \theta=-\sqrt{1-\sin ^{2} \theta}
$$

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Problem 111

In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\sec \theta=\sqrt{1+\tan ^{2} \theta}
$$

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Problem 112

In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\csc \theta=\sqrt{1+\cot ^{2} \theta}
$$

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Problem 113

In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\cos x|-\ln |\sin x|
$$

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Problem 114

In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\sec x|+\ln |\sin x|
$$

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Problem 115

In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\sin x|+\ln |\cot x|
$$

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Problem 116

In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\tan x|+\ln |\csc x|
$$

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Problem 117

In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\cot t|+\ln \left(1+\tan ^{2} t\right)
$$

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Problem 118

In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.
$$
\ln \left(\cos ^{2} t\right)+\ln \left(1+\tan ^{2} t\right)
$$

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Problem 119

In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.
$\csc ^{2} \theta-\cot ^{2} \theta=1$
$\begin{array}{ll}{\text { (a) } \theta=132^{\circ}} & {\text { (b) } \theta=\frac{2 \pi}{7}}\end{array}$

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Problem 120

In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.
$\tan ^{2} \theta+1=\sec ^{2} \theta$
(a) $\theta=346^{\circ} \quad$ (b) $\theta=3.1$

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Problem 121

In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.
$\cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta$
$\begin{array}{ll}{\text { (a) } \theta=80^{\circ}} & {\text { (b) } \theta=0.8}\end{array}$

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Problem 122

In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.
$\sin (-\theta)=-\sin \theta$
$\begin{array}{lll}{\text { (a) } \theta=250^{\circ}} & {\text { (b) } \theta=\frac{1}{2}}\end{array}$

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Problem 123

The forces acting on an object weighing $W$ units on an inclined plane positioned at an angle of
$\theta$ with the horizontal (see figure) are modeled by
$$\mu W \cos \theta=W \sin \theta$$
where $\mu$ is the coefficient of friction. Solve the equation for $\mu$ and simplify the result.

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Problem 124

The rate of change of the function $f(x)=-x+\tan x$ is given by the expression $-1+\sec ^{2} x .$ Show that that expression can also be written as tan' $x$ .

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Problem 125

The rate of change of the function $f(x)=\sec x+\cos x$ is given by the expression $\sec x \tan x-\sin x .$ Show that this expression can also be written as $\sin x \tan ^{2} x .$

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Problem 126

The rate of change of the function $f(x)=-\csc x-\sin x$ is given by the expression $\csc x \cot x-\cos x .$ Show that this expression can also be written as $\cos x \cot ^{2} x$

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Problem 127

In Exercises 127 and 128, determine whether the statement is true or false. Justify your answer.
The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative.

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Problem 128

In Exercises 127 and 128, determine whether the statement is true or false. Justify your answer.
A cofunction identity can be used to transform a tangent function so that it can be represented by a
cosecant function.

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Problem 129

In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.
As $x \rightarrow \frac{\pi^{-}}{2}, \sin x \rightarrow \qquad$ and $\csc x \rightarrow$

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Problem 130

In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.
As $x \rightarrow 0^{+}, \cos x \rightarrow \qquad$ and and $\sec x \rightarrow$

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Problem 131

In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.
As $x \rightarrow \frac{\pi}{2}, \tan x \rightarrow \qquad$ and $\cot x \rightarrow$

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Problem 132

In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.
As $x \rightarrow \pi^{+}, \sin x \rightarrow \qquad$ and $\csc x \rightarrow$

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Problem 133

In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.
$$
\cos \theta=\sqrt{1-\sin ^{2} \theta}
$$

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Problem 134

In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.
$$
\cot \theta=\sqrt{\csc ^{2} \theta+1}
$$

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Problem 135

In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.
$\frac{(\sin k \theta)}{(\cos k \theta)}=\tan \theta, \quad k$ is a constant.

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Problem 136

In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.
$$
\frac{1}{(5 \cos \theta)}=5 \sec \theta
$$

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Problem 137

In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.
$$
\sin \theta \csc \theta=1
$$

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Problem 138

In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.
$$
\csc ^{2} \theta=1
$$

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Problem 139

Use the trigonometric substitution $u=a \sin \theta,$ where $-\pi / 2<\theta<\pi / 2$ and $a>0,$ to simplify the expression $\sqrt{a^{2}-u^{2}}$

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Problem 140

Use the trigonometric substitution $u=a \tan \theta,$ where $-\pi / 2<\theta<\pi / 2$ and $a>0,$ to simplify the expression $\sqrt{a^{2}+u^{2}}$

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Problem 141

Use the trigonometric substitution $u=a \sec \theta,$ where $0<\theta<\pi / 2$ and $a>0,$ to simplify the expression $\sqrt{u^{2}-a^{2}}$

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Problem 142

(a) Use the definitions of sine and cosine to derive the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$
(b) Use the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$ to derive the other Pythagorean identities, $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $1+\cot ^{2} \theta=\csc ^{2} \theta$ Discuss how to remember these identities and other fundamental identities.

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