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Chapter 5

Analytic Trigonometry

Educators

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Problem 1

Fill in the blank to complete the trigonometric identity.

$ \dfrac{\sin u}{\cos u} $= ________

Heather Z.
Numerade Educator

Problem 2

Fill in the blank to complete the trigonometric identity.

$ \dfrac{1}{\csc u} $= ________

Heather Z.
Numerade Educator

Problem 3

Fill in the blank to complete the trigonometric identity.

$ \dfrac{1}{\tan u} $= ________

Heather Z.
Numerade Educator

Problem 4

Fill in the blank to complete the trigonometric identity.

$ \dfrac{1}{\cos u} $= ________

Heather Z.
Numerade Educator

Problem 5

Fill in the blank to complete the trigonometric identity.

$ 1 + $ ________ =$ \csc^2 u $

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Alan G.
Numerade Educator

Problem 6

Fill in the blank to complete the trigonometric identity.

$ 1 + \tan^2 u $= ________

Heather Z.
Numerade Educator

Problem 7

Fill in the blank to complete the trigonometric identity.

$ \sin \left(\dfrac{\pi}{2} - u\right) $= ________

Heather Z.
Numerade Educator

Problem 8

Fill in the blank to complete the trigonometric identity.

$ \sec\left(\dfrac{\pi}{2} - u \right) $= ________

Heather Z.
Numerade Educator

Problem 9

Fill in the blank to complete the trigonometric identity.

$ \cos(-u) $= ________

Heather Z.
Numerade Educator

Problem 10

Fill in the blank to complete the trigonometric identity.

$ \tan(-u) $= ________

Heather Z.
Numerade Educator

Problem 11

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \sin x = \dfrac{1}{2} $, $ \cos x = \dfrac{\sqrt{3}}{2} $

Heather Z.
Numerade Educator

Problem 12

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \tan x = \dfrac{\sqrt{3}}{3} $, $ \cos x = - \dfrac{\sqrt{3}}{2} $

Heather Z.
Numerade Educator

Problem 13

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \sec \theta = \sqrt{2} $, $ \sin \theta = - \dfrac{\sqrt{3}}{2} $

Heather Z.
Numerade Educator

Problem 14

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \csc \theta = \dfrac{25}{7} $, $ \tan \theta = \dfrac{7}{24} $

Heather Z.
Numerade Educator

Problem 15

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \tan x = \dfrac{8}{15} $, $ \sec x = -\dfrac{17}{15} $

Heather Z.
Numerade Educator

Problem 16

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \cot \phi = -3 $, $ \sin \phi = \dfrac{\sqrt{10}}{10} $

Heather Z.
Numerade Educator

Problem 17

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \sec \phi = \dfrac{3}{2} $, $ \csc \phi = -\dfrac{3\sqrt{5}}{5} $

Heather Z.
Numerade Educator

Problem 18

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \cos\left(\dfrac{\pi}{2} - x\right) = \dfrac{3}{5} $, $ \cos x = \dfrac{4}{5} $

Heather Z.
Numerade Educator

Problem 19

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \sin(-x) = -\dfrac{1}{3} $, $ \tan x = -\dfrac{\sqrt{2}}{4} $

Heather Z.
Numerade Educator

Problem 20

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \sec x = 4 $, $ \sin x > 0 $

Heather Z.
Numerade Educator

Problem 21

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \tan \theta = 2 $, $ \sin \theta < 0 $

Heather Z.
Numerade Educator

Problem 22

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \csc \theta = -5 $, $ \cos \theta < 0 $

Heather Z.
Numerade Educator

Problem 23

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \sin \theta = -1 $, $ \cot \theta = 0 $

Heather Z.
Numerade Educator

Problem 24

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions.

$ \tan \theta $ is undefined, $ \sin \theta > 0 $

Heather Z.
Numerade Educator

Problem 25

In Exercises 25 - 30, match the trigonometric expression with one of the following.

(a)$ \sec x $ (b) $ -1 $ (c) $ \cot x $
(d) $ 1 $ (e) $ -\tan x $ (d) $ \sin x $

$ \sec x \cos x $

Heather Z.
Numerade Educator

Problem 26

In Exercises 25 - 30, match the trigonometric expression with one of the following.

(a)$ \sec x $ (b) $ -1 $ (c) $ \cot x $
(d) $ 1 $ (e) $ -\tan x $ (d) $ \sin x $

$ \tan x \csc x $

Heather Z.
Numerade Educator

Problem 27

In Exercises 25 - 30, match the trigonometric expression with one of the following.

(a)$ \sec x $ (b) $ -1 $ (c) $ \cot x $
(d) $ 1 $ (e) $ -\tan x $ (d) $ \sin x $

$ \cot^2 x - \csc^2 x $

Heather Z.
Numerade Educator

Problem 28

In Exercises 25 - 30, match the trigonometric expression with one of the following.

(a)$ \sec x $ (b) $ -1 $ (c) $ \cot x $
(d) $ 1 $ (e) $ -\tan x $ (d) $ \sin x $

$ \left(1 - \cos^2 x\right)\left(\csc x\right) $

Heather Z.
Numerade Educator

Problem 29

In Exercises 25 - 30, match the trigonometric expression with one of the following.

(a)$ \sec x $ (b) $ -1 $ (c) $ \cot x $
(d) $ 1 $ (e) $ -\tan x $ (d) $ \sin x $

$ \dfrac{\sin(-x)}{\cos(-x)} $

Heather Z.
Numerade Educator

Problem 30

In Exercises 25 - 30, match the trigonometric expression with one of the following.

(a)$ \sec x $ (b) $ -1 $ (c) $ \cot x $
(d) $ 1 $ (e) $ -\tan x $ (d) $ \sin x $

$ \dfrac{\sin\left[(\pi/2) - x\right]}{\cos\left[(\pi/2) - x\right]} $

Heather Z.
Numerade Educator

Problem 31

In Exercises 31 - 36, match the trigonometric expression with one of the following.

(a)$ \csc x $ (b) $ \tan x $ (c) $ \sin^2 x $
(d) $ \sin x \tan x $ (e) $ \sec^2 x $ (d) $ \sec^2 x + \tan^2 x $

$ \sin x \sec x $

Heather Z.
Numerade Educator

Problem 32

In Exercises 31 - 36, match the trigonometric expression with one of the following.

(a)$ \csc x $ (b) $ \tan x $ (c) $ \sin^2 x $
(d) $ \sin x \tan x $ (e) $ \sec^2 x $ (d) $ \sec^2 x + \tan^2 x $

$ \cos^2 x\left(\sec^2 x - 1\right) $

Heather Z.
Numerade Educator

Problem 33

In Exercises 31 - 36, match the trigonometric expression with one of the following.

(a)$ \csc x $ (b) $ \tan x $ (c) $ \sin^2 x $
(d) $ \sin x \tan x $ (e) $ \sec^2 x $ (d) $ \sec^2 x + \tan^2 x $

$ \sec^4 x - \tan^4 x $

Heather Z.
Numerade Educator

Problem 34

In Exercises 31 - 36, match the trigonometric expression with one of the following.

(a)$ \csc x $ (b) $ \tan x $ (c) $ \sin^2 x $
(d) $ \sin x \tan x $ (e) $ \sec^2 x $ (d) $ \sec^2 x + \tan^2 x $

$ \cot x \sec x $

Heather Z.
Numerade Educator

Problem 35

In Exercises 31 - 36, match the trigonometric expression with one of the following.

(a)$ \csc x $ (b) $ \tan x $ (c) $ \sin^2 x $
(d) $ \sin x \tan x $ (e) $ \sec^2 x $ (d) $ \sec^2 x + \tan^2 x $

$ \dfrac{\sec^2 x - 1}{\sin^2 x} $

Heather Z.
Numerade Educator

Problem 36

In Exercises 31 - 36, match the trigonometric expression with one of the following.

(a)$ \csc x $ (b) $ \tan x $ (c) $ \sin^2 x $
(d) $ \sin x \tan x $ (e) $ \sec^2 x $ (d) $ \sec^2 x + \tan^2 x $

$ \dfrac{\cos^2\left[(\pi/2) - x\right]}{\cos x} $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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) $

Heather Z.
Numerade Educator

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\left(\csc \phi - \sin \phi\right) $

Heather Z.
Numerade Educator

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) $

Heather Z.
Numerade Educator

Problem 43

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{\cot x}{\csc x} $

Heather Z.
Numerade Educator

Problem 44

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{\csc \theta}{\sec \theta} $

Heather Z.
Numerade Educator

Problem 45

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{1 - \sin^2 x}{\csc^2 x - 1} $

Heather Z.
Numerade Educator

Problem 46

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{1}{\tan^2 x + 1} $

Heather Z.
Numerade Educator

Problem 47

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{\tan \theta \cot \theta}{\sec \theta} $

Heather Z.
Numerade Educator

Problem 48

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{\sin \theta \csc \theta}{\tan \theta} $

Heather Z.
Numerade Educator

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 \dfrac{\sin \alpha}{\tan \alpha} $

Heather Z.
Numerade Educator

Problem 50

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{\tan^2 \theta}{\sec^2 \theta} $

Heather Z.
Numerade Educator

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(\dfrac{\pi}{2} - x\right) \sec x $

Heather Z.
Numerade Educator

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(\dfrac{\pi}{2} - x\right) \cos x $

Heather Z.
Numerade Educator

Problem 53

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$ \dfrac{\cos^2y}{1 - \sin y} $

Heather Z.
Numerade Educator

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(1 + \tan^2 t) $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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.

$ \dfrac{\sec^2 x - 1}{\sec x - 1} $

Heather Z.
Numerade Educator

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.

$ \dfrac{\cos^2 x - 4}{\cos x - 2} $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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.

$ \left(\sin x + \cos x\right)^2 $

Heather Z.
Numerade Educator

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.

$ \left(\cot x + \csc x\right)\left(\cot x - \csc x\right) $

Heather Z.
Numerade Educator

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.

$ \left(2 \csc x + 2\right)\left(2 \csc x - 2\right) $

Heather Z.
Numerade Educator

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.

$ \left(3 - 3 \sin x\right)\left(3 + 3 \sin x\right) $

Heather Z.
Numerade Educator

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.

$ \dfrac{1}{1 + \cos x} + \dfrac{1}{1 - \cos x} $

Heather Z.
Numerade Educator

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.

$ \dfrac{1}{\sec x + 1} - \dfrac{1}{\sec x - 1} $

Heather Z.
Numerade Educator

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.

$ \dfrac{\cos x}{1 + \sin x} + \dfrac{1 + \sin x}{\cos x} $

Heather Z.
Numerade Educator

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.

$ \dfrac{\tan x}{1 + \sec x} + \dfrac{1 + \sec x}{\tan x} $

Heather Z.
Numerade Educator

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 + \dfrac{\cos x}{1 + \sin x} $

Heather Z.
Numerade Educator

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 - \dfrac{\sec^2 x}{\tan x} $

Heather Z.
Numerade Educator

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

$ \dfrac{\sin^2y}{1 - \cos y} $

Heather Z.
Numerade Educator

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

$ \dfrac{5}{\tan x + \sec x} $

Heather Z.
Numerade Educator

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

$ \dfrac{3}{\sec x - \tan x} $

Heather Z.
Numerade Educator

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

$ \dfrac{\tan^2 x}{\csc x + 1} $

Heather Z.
Numerade Educator

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(\dfrac{\pi}{2}\right) $, $ y_2 = \sin x $

Heather Z.
Numerade Educator

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 $, $ y_2 = \sin x \tan x $

Heather Z.
Numerade Educator

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 = \dfrac{\cos x}{1 - \sin x} $, $ y_2 = \dfrac{1 + \sin x}{\cos x} $

Heather Z.
Numerade Educator

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 $, $ y_2 = \tan^2 x + \tan^4 x $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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.

$ \dfrac{1}{\sin x}\left(\dfrac{1}{\cos x} - \cos x\right) $

Heather Z.
Numerade Educator

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.

$ \dfrac{1}{2}\left(\dfrac{1 + \sin \theta}{\cos \theta} + \dfrac{\cos \theta}{1 + \sin \theta}\right)$

Heather Z.
Numerade Educator

Problem 93

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{9 - x^2} $, $ x = 3 \cos \theta $

Heather Z.
Numerade Educator

Problem 94

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{64 - 16x^2} $, $ x = 2 \cos \theta $

Heather Z.
Numerade Educator

Problem 95

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{16 - x^2} $, $ x = 4 \sin \theta $

Heather Z.
Numerade Educator

Problem 96

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{49 - x^2} $, $ x = 7 \sin \theta $

Heather Z.
Numerade Educator

Problem 97

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{x^2 - 9} $, $ x = 3 \sec \theta $

Heather Z.
Numerade Educator

Problem 98

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{x^2 - 4} $, $ x = 2 \sec \theta $

Heather Z.
Numerade Educator

Problem 99

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{x^2 + 25} $, $ x = 5 \tan \theta $

Heather Z.
Numerade Educator

Problem 100

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{x^2 + 100} $, $ x = 10 \tan \theta $

Heather Z.
Numerade Educator

Problem 101

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{4x^2 + 9} $, $ 2x = 3 \tan \theta $

Heather Z.
Numerade Educator

Problem 102

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{9x^2 + 25} $, $ 3x = 5 \tan \theta $

Heather Z.
Numerade Educator

Problem 103

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{2 - x^2} $, $ x = \sqrt{2} \sin \theta $

Heather Z.
Numerade Educator

Problem 104

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of $ \theta $, where $ 0 < \theta < \pi/2 $.

$ \sqrt{10 - x^2} $, $ x = \sqrt{10} \sin \theta $

Heather Z.
Numerade Educator

Problem 105

In Exercises 105 - 108, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of where $ -\pi/2 < \theta < \pi/2 $.Then find $ \sin \theta $ and $ \cos \theta $.

$ 3 = \sqrt{9 - x^2} $, $ x = 3 \sin \theta $

Heather Z.
Numerade Educator

Problem 106

In Exercises 105 - 108, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of where $ -\pi/2 < \theta < \pi/2 $.Then find $ \sin \theta $ and $ \cos \theta $.

$ 3 = \sqrt{36 - x^2} $, $ x = 6 \sin \theta $

Heather Z.
Numerade Educator

Problem 107

In Exercises 105 - 108, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of where $ -\pi/2 < \theta < \pi/2 $.Then find $ \sin \theta $ and $ \cos \theta $.

$ 2 \sqrt{2} = \sqrt{16 - 4x^2} $, $ x = 2 \cos \theta $

Heather Z.
Numerade Educator

Problem 108

In Exercises 105 - 108, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of where $ -\pi/2 < \theta < \pi/2 $.Then find $ \sin \theta $ and $ \cos \theta $.

$ -5 \sqrt{3} = \sqrt{100 - x^2} $, $ x = 10 \cos \theta $

Heather Z.
Numerade Educator

Problem 109

In Exercises 109 - 112, use a graphing utility to solve the equation for $ \theta $, where $ 0 \le \theta < 2\pi $.

$ \sin \theta = \sqrt{1 - \cos^2 \theta} $

Heather Z.
Numerade Educator

Problem 110

In Exercises 109 - 112, use a graphing utility to solve the equation for $ \theta $, where $ 0 \le \theta < 2\pi $.

$ \cos \theta = -\sqrt{1 - \sin^2 \theta} $

Heather Z.
Numerade Educator

Problem 111

In Exercises 109 - 112, use a graphing utility to solve the equation for $ \theta $, where $ 0 \le \theta < 2\pi $.

$ \sec \theta = \sqrt{1 + \tan^2 \theta} $

Heather Z.
Numerade Educator

Problem 112

In Exercises 109 - 112, use a graphing utility to solve the equation for $ \theta $, where $ 0 \le \theta < 2\pi $.

$ \csc \theta = \sqrt{1 + \cot^2 \theta} $

Heather Z.
Numerade Educator

Problem 113

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result.

$ \ln\mid\cos x\mid - \ln\mid\sin x\mid $

Heather Z.
Numerade Educator

Problem 114

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result.

$ \ln\mid\sec x\mid + \ln\mid\sin x\mid $

Heather Z.
Numerade Educator

Problem 115

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result.

$ \ln\mid\sin x\mid + \ln\mid\cot x\mid $

Heather Z.
Numerade Educator

Problem 116

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result.

$ \ln\mid\tan x\mid + \ln\mid\csc x\mid $

Heather Z.
Numerade Educator

Problem 117

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result.

$ \ln\mid\cot t\mid + \ln\left(1 + \tan^2 t\right) $

Heather Z.
Numerade Educator

Problem 118

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result.

$ ln\left(\cos^2t\right) + \ln\left(1 + \tan^2 t\right) $

Heather Z.
Numerade Educator

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 $

(a) $ \theta = 132^\circ $

(b) $ \theta = \dfrac{2\pi}{7} $

Heather Z.
Numerade Educator

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 $

(b) $ \theta = 3.1 $

Heather Z.
Numerade Educator

Problem 121

In Exercises 119 - 122, use a calculator to demonstrate the identity for each value of $ \theta $.

$ \cos\left(\dfrac{\pi}{2} - \theta\right) = \sin \theta $

(a) $ \theta = 80^circ $

(b) $ \theta = 0.8 $

Heather Z.
Numerade Educator

Problem 122

In Exercises 119 - 122, use a calculator to demonstrate the identity for each value of $ \theta $.

$ \sin(-\theta) = - \sin \theta $

(a) $ \theta = 250^circ $

(b) $ \theta = \dfrac{1}{2} $

Heather Z.
Numerade Educator

Problem 123

The forces acting on an object weighing 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 $ the coefficient of friction. Solve the equation for $ \mu $ and simplify the result.

ag
Alan G.
Numerade Educator

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 this expression can also be written as $ \tan^2 x $.

Heather Z.
Numerade Educator

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 $.

Heather Z.
Numerade Educator

Problem 126

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

Heather Z.
Numerade Educator

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.

Heather Z.
Numerade Educator

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.

Heather Z.
Numerade Educator

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 \dfrac{\pi^-}{2} $, $ \sin x \rightarrow \square $ and $ \csc x \rightarrow \square $.

.

Heather Z.
Numerade Educator

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 \square $ and $ \sec x \rightarrow \square $.

Heather Z.
Numerade Educator

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 \dfrac{\pi^-}{2} $, $ \tan x \rightarrow \square $ and $ \cot x \rightarrow \square $.

Heather Z.
Numerade Educator

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 \square $ and $ \csc x \rightarrow \square $.

Heather Z.
Numerade Educator

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} $

Heather Z.
Numerade Educator

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} $

Heather Z.
Numerade Educator

Problem 135

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer.

$ \dfrac{\left(\sin k\theta\right)}{\left(\cos k\theta\right)} = \tan \theta $, $ k $ is a constant.

Heather Z.
Numerade Educator

Problem 136

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer.

$ \dfrac{1}{\left(5 \cos \theta\right)} = 5 \sec \theta $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

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 $

Heather Z.
Numerade Educator

Problem 139

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer.

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} $.

Heather Z.
Numerade Educator

Problem 140

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer.

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} $.

Heather Z.
Numerade Educator

Problem 141

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer.

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

Heather Z.
Numerade Educator

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.

Heather Z.
Numerade Educator