Precalculus with Limits : A Graphing Approach

Ron Larson, Robert Hostetler, Bruce H. Edwards

Chapter 5

Analytic Trigonometry - all with Video Answers

Educators

SM

Section 1

Using Fundamental Identities

Problem 1

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sin x=\frac{1}{2}, \quad \cos x=\frac{\sqrt{3}}{2}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 2

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\csc \theta=2, \quad \tan \theta=\frac{\sqrt{3}}{3}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 3

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sec \theta=\sqrt{2}, \quad \sin \theta=-\frac{\sqrt{2}}{2}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 4

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan x=\frac{\sqrt{3}}{3}, \quad \cos x=-\frac{\sqrt{3}}{2}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 5

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan x=\frac{7}{24}, \quad \sec x=-\frac{25}{24}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 6

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\cot \phi=-5, \quad \sin \phi=\frac{\sqrt{26}}{26}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 7

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sec \phi=-\frac{17}{15}, \quad \sin \phi=\frac{8}{17}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 8

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\cos \left(\frac{\pi}{2}-x\right)=\frac{3}{5}, \quad \cos x=\frac{4}{5}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 9

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sin (-x)=-\frac{2}{3}, \quad \tan x=-\frac{2 \sqrt{5}}{5}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 10

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\csc (-x)=-5, \quad \cos x=\frac{\sqrt{24}}{5}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 11

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan \theta=2, \quad \sin \theta<0
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 12

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\sec \theta=-3, \quad \tan \theta<0
$$

Sheryl Ezze

Numerade Educator

Problem 13

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\csc \theta \text { is undefined, } \cos \theta<0
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 14

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions.
$$
\tan \theta \text { is undefined, } \quad \sin \theta>0
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 15

In Exercises 15–20, match the trigonometric expression with one of the following.
$$
\begin{array}{llll}{\text { (a) } \sec x} & {\text { (b) }-1} & {} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}
$$
$$
\sec x \cos x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 16

In Exercises 15–20, match the trigonometric expression with one of the following.
$$
\begin{array}{llll}{\text { (a) } \sec x} & {\text { (b) }-1} & {} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}
$$
$$
\tan x \csc x
$$

Sheryl Ezze

Numerade Educator

Problem 17

In Exercises 15–20, match the trigonometric expression with one of the following.
$$
\begin{array}{llll}{\text { (a) } \sec x} & {\text { (b) }-1} & {} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}
$$
$$
\cot ^{2} x-\csc ^{2} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 18

In Exercises 15–20, match the trigonometric expression with one of the following.
$$
\begin{array}{llll}{\text { (a) } \sec x} & {\text { (b) }-1} & {} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}
$$
$$
\left(1-\cos ^{2} x\right)(\csc x)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 19

In Exercises 15–20, match the trigonometric expression with one of the following.
$$
\begin{array}{llll}{\text { (a) } \sec x} & {\text { (b) }-1} & {} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}
$$
$$
\frac{\sin (-x)}{\cos (-x)}
$$

Sheryl Ezze

Numerade Educator

Problem 20

In Exercises 15–20, match the trigonometric expression with one of the following.
$$
\begin{array}{llll}{\text { (a) } \sec x} & {\text { (b) }-1} & {} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}
$$
$$
\frac{\sin [(\pi / 2)-x]}{\cos [(\pi / 2)-x]}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 21

In Exercises 21–26, match the trigonometric expression with one of the following.
$$
\sin x \sec x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 22

In Exercises 21–26, match the trigonometric expression with one of the following.
$$
\cos ^{2} x\left(\sec ^{2} x-1\right)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 23

In Exercises 21–26, match the trigonometric expression with one of the following.
$$
\sec ^{4} x-\tan ^{4} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 24

In Exercises 21–26, match the trigonometric expression with one of the following.
$$
\cot x \sec x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 25

In Exercises 21–26, match the trigonometric expression with one of the following.
$$
\frac{\sec ^{2} x-1}{\sin ^{2} x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 26

In Exercises 21–26, match the trigonometric expression with one of the following.
$$
\frac{\cos ^{2}[(\pi / 2)-x]}{\cos x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 27

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.
$$
\cot x \sin x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 28

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.
$$
\cos \beta \tan \beta
$$

Sheryl Ezze

Numerade Educator

Problem 29

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.
$$
\sin \phi(\csc \phi-\sin \phi)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 30

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.
$$
\sec ^{2} x\left(1-\sin ^{2} x\right)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 31

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.
$$
\frac{\csc x}{\cot x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 32

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.
$$
\frac{\sec \theta}{\csc \theta}
$$

Sheryl Ezze

Numerade Educator

Problem 33

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.
$$
\sec \alpha \cdot \frac{\sin \alpha}{\tan \alpha}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 34

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.
$$
\frac{\tan ^{2} \theta}{\sec ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 35

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.
$$
\sin \left(\frac{\pi}{2}-x\right) \csc x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 36

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.
$$
\cot \left(\frac{\pi}{2}-x\right) \cos x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 37

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.
$$
\frac{\cos ^{2} y}{1-\sin y}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 38

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.
$$
\frac{1}{\cot ^{2} x+1}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 39

In Exercises 39–44, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically.
$$
\sin \theta+\cos \theta \cot \theta=\csc \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 40

In Exercises 39–44, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically.
$$
(\sec \theta-\tan \theta)(\csc \theta+1)=\cot \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 41

In Exercises 39–44, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically.
$$
\frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 42

In Exercises 39–44, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically.
$$
\frac{1+\csc \theta}{\cot \theta+\cos \theta}=\sec \theta
$$

SM

Snehali Marimuthu

Numerade Educator

Problem 43

In Exercises 39–44, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically.
$$
\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=2 \csc \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 44

In Exercises 39–44, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically.
$$
\frac{\sin \theta+\cos \theta}{\sin \theta}-\frac{\cos \theta-\sin \theta}{\cos \theta}=\sec \theta \csc \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 45

In Exercises 45–50, verify the identity algebraically. Use a graphing utility to check your result graphically.
$$
\csc \theta \tan \theta=\sec \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 46

In Exercises 45–50, verify the identity algebraically. Use a graphing utility to check your result graphically.
$$
\sin \theta \csc \theta-\sin ^{2} \theta=\cos ^{2} \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 47

In Exercises 45–50, verify the identity algebraically. Use a graphing utility to check your result graphically.
$$
1-\frac{\sin ^{2} \theta}{1-\cos \theta}=-\cos \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 48

In Exercises 45–50, verify the identity algebraically. Use a graphing utility to check your result graphically.
$$
\frac{\tan \theta}{1+\sec \theta}+\frac{1+\sec \theta}{\tan \theta}=2 \csc \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 49

In Exercises 45–50, verify the identity algebraically. Use a graphing utility to check your result graphically.
$$
\frac{\cot (-\theta)}{\csc \theta}=-\cos \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 50

In Exercises 45–50, verify the identity algebraically. Use a graphing utility to check your result graphically.
$$
\frac{\csc \left(\frac{\pi}{2}-\theta\right)}{\tan (-\theta)}=-\csc \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 51

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\cot ^{2} x-\cot ^{2} x \cos ^{2} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 52

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\sec ^{2} x \tan ^{2} x+\sec ^{2} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 53

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\frac{\cos ^{2} x-4}{\cos x-2}
$$

Sheryl Ezze

Numerade Educator

Problem 54

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\frac{\csc ^{2} x-1}{\csc x-1}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 55

In Exercises $51-60,$ factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$\tan ^{4} x+2 \tan ^{2} x+1$

Sheryl Ezze

Numerade Educator

Problem 56

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
1-2 \sin ^{2} x+\sin ^{4} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 57

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\sin ^{4} x-\cos ^{4} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 58

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\sec ^{4} x-\tan ^{4} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 59

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\csc ^{3} x-\csc ^{2} x-\csc x+1
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 60

In Exercises 51–60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically.
$$
\sec ^{3} x-\sec ^{2} x-\sec x+1
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 61

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
(\sin x+\cos x)^{2}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 62

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
(\tan x+\sec x)(\tan x-\sec x)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 63

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
(\csc x+1)(\csc x-1)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 64

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
(5-5 \sin x)(5+5 \sin x)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 65

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
\frac{1}{1+\cos x}+\frac{1}{1-\cos x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 66

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
\frac{1}{\sec x+1}-\frac{1}{\sec x-1}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 67

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
\tan x-\frac{\sec ^{2} x}{\tan x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 68

In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify.
$$
\frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 69

In Exercises 69–72, rewrite the expression so that it is not in fractional form.
$$
\frac{\sin ^{2} y}{1-\cos y}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 70

In Exercises 69–72, rewrite the expression so that it is not in fractional form.
$$
\frac{5}{\tan x+\sec x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 71

In Exercises 69–72, rewrite the expression so that it is not in fractional form.
$$
\frac{3}{\sec x-\tan x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 72

In Exercises 69–72, rewrite the expression so that it is not in fractional form.
$$
\frac{\tan ^{2} x}{\csc x+1}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 73

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 $y_{1}$ and $y_{2} .$
$$
y_{1}=\cos \left(\frac{\pi}{2}-x\right), \quad y_{2}=\sin x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 74

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 $y_{1}$ and $y_{2} .$
$$
y_{1}=\cos x+\sin x \tan x, \quad y_{2}=\sec x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 75

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 $y_{1}$ and $y_{2} .$
$$
y_{1}=\frac{\cos x}{1-\sin x}, \quad y_{2}=\frac{1+\sin x}{\cos x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 76

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 $y_{1}$ and $y_{2} .$
$$
y_{1}=\sec ^{4} x-\sec ^{2} x, \quad y_{2}=\tan ^{2} x+\tan ^{4} x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 77

In Exercises 77–80, use a graphing utility to determine which of the six trigonometric functions is equal to the expression.
$$
\cos x \cot x+\sin x
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 78

In Exercises 77–80, use a graphing utility to determine which of the six trigonometric functions is equal to the expression.
$$
\sin x(\cot x+\tan x)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 79

In Exercises 77–80, use a graphing utility to determine which of the six trigonometric functions is equal to the expression.
$$
\sec x-\frac{\cos x}{1+\sin x}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 80

In Exercises 77–80, use a graphing utility to determine which of the six trigonometric functions is equal to the expression.
$$
\frac{1}{2}\left(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}\right)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 81

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{25-x^{2}}, \quad x=5 \sin \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 82

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{64-16 x^{2}}, \quad x=2 \cos \theta
$$

Diana Jimenez

Numerade Educator

Problem 83

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{x^{2}-9}, \quad x=3 \sec \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 84

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{x^{2}+100}, \quad x=10 \tan \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 85

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{9-x^{2}}, \quad x=3 \sin \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 86

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{4-x^{2}}, \quad x=2 \cos \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 87

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{4 x^{2}+9}, \quad 2 x=3 \tan \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 88

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{9 x^{2}+4}, \quad 3 x=2 \tan \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 89

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{16 x^{2}-9}, \quad 4 x=3 \sec \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 90

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{9 x^{2}-25}, \quad 3 x=5 \sec \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 91

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{2-x^{2}}, \quad x=\sqrt{2} \sin \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 92

In Exercises $81-92,$ use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi / 2 .$
$$
\sqrt{5-x^{2}}, \quad x=\sqrt{5} \cos \theta
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 93

In Exercises $93-96$ , use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\sin \theta=\sqrt{1-\cos ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 94

In Exercises $93-96$ , use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\cos \theta=-\sqrt{1-\sin ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 95

In Exercises $93-96$ , use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\sec \theta=\sqrt{1+\tan ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 96

In Exercises $93-96$ , use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$
$$
\tan \theta=\sqrt{\sec ^{2} \theta-1}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 97

In Exercises 97–100, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\cos \theta|-\ln |\sin \theta|
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 98

In Exercises 97–100, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\csc \theta|+\ln |\tan \theta|
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 99

In Exercises 97–100, rewrite the expression as a single logarithm and simplify the result.
$$
\ln (1+\sin x)-\ln |\sec x|
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 100

In Exercises 97–100, rewrite the expression as a single logarithm and simplify the result.
$$
\ln |\cot t|+\ln \left(1+\tan ^{2} t\right)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 101

In Exercises $101-106$ , show that the identity is not true for all values of $\theta$ . (There are many correct answers.)
$$
\cos \theta=\sqrt{1-\sin ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 102

In Exercises $101-106$ , show that the identity is not true for all values of $\theta$ . (There are many correct answers.)
$$
\tan \theta=\sqrt{\sec ^{2} \theta-1}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 103

In Exercises $101-106$ , show that the identity is not true for all values of $\theta$ . (There are many correct answers.)
$$
\sin \theta=\sqrt{1-\cos ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 104

In Exercises $101-106$ , show that the identity is not true for all values of $\theta$ . (There are many correct answers.)
$$
\sec \theta=\sqrt{1+\tan ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 105

In Exercises $101-106$ , show that the identity is not true for all values of $\theta$ . (There are many correct answers.)
$$
\csc \theta=\sqrt{1+\cot ^{2} \theta}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 106

In Exercises $101-106$ , show that the identity is not true for all values of $\theta$ . (There are many correct answers.)
$$
\cot \theta=\sqrt{\csc ^{2} \theta-1}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 107

In Exercises $107-110$ , use the table feature of a graphing utility to demonstrate the identity for each value of $\theta$ .
$\csc ^{2} \theta-\cot ^{2} \theta=1,\left(\text { a) } \theta=132^{\circ}(\text { b }) \theta=\frac{2 \pi}{7}\right.$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 108

In Exercises $107-110$ , use the table feature of a graphing utility to demonstrate the identity for each value of $\theta$ .
$$
\tan ^{2} \theta+1=\sec ^{2} \theta,\left(\text { a) } \theta=346^{\circ} \text { (b) } \theta=3.1\right.
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 109

In Exercises $107-110$ , use the table feature of a graphing utility to demonstrate the identity for each value of $\theta$ .
$$
\cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta,\left(\text { a) } \theta=80^{\circ}(\text { b) } \theta=0.8\right.
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 110

In Exercises $107-110$ , use the table feature of a graphing utility to demonstrate the identity for each value of $\theta$ .
$$
\sin (-\theta)=-\sin \theta, \quad\left(\text { a) } \quad \theta=250^{\circ} \text { (b) } \theta=\frac{1}{2}\right.
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 111

Rate of Change 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 $\cos x \cot ^{2} x$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 112

Rate of Change 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 Zimmers

Heather Zimmers

Numerade Educator

Problem 113

True or False? In Exercises 113 and 114, determine whether the statement is true or false. Justify your answer.
$$
\sin \theta \csc \theta=1
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 114

True or False? In Exercises 113 and 114, determine whether the statement is true or false. Justify your answer.
$$
\cos \theta \sec \phi=1
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 115

In Exercises $115-118$ , fill in the blanks. (Note: $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right, and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.
As $x \rightarrow \frac{\pi^{-}}{2}, \sin x \rightarrow \quad$ and $\csc x \rightarrow$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 116

In Exercises $115-118$ , fill in the blanks. (Note: $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right, and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.
As $x \rightarrow 0^{+}, \cos x \rightarrow \quad$ and $\sec x \rightarrow$
As $x \rightarrow 0^{+}, \cos x \rightarrow \quad$ and $\sec x \rightarrow$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 117

In Exercises $115-118$ , fill in the blanks. (Note: $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right, and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.
As $x \rightarrow \frac{\pi^{-}}{2}, \tan x \rightarrow \quad$ and $\cot x \rightarrow$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 118

Fill in the blanks. (Note: $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right, and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.)

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 119

Write each of the other trigonometric functions of $\theta$ in terms of $\sin \theta .$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 120

Write each of the other trigonometric functions of $\theta$ in terms of $\cos \theta .$

Vysakh M

Numerade Educator

Problem 121

Use the definitions of sine and cosine to derive the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 122

Writing 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.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 123

In Exercises 123–126, sketch the graph of the function. (Include two full periods.)
$$f(x)=\frac{1}{2} \sin \pi x$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 124

In Exercises 123–126, sketch the graph of the function. (Include two full periods.)
$$
f(x)=-2 \tan \frac{\pi x}{2}
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 125

In Exercises 123–126, sketch the graph of the function. (Include two full periods.)
$$
f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right)
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 126

In Exercises 123–126, sketch the graph of the function. (Include two full periods.)
$$
f(x)=\frac{3}{2} \cos (x-\pi)+3
$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator