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Algebra 2

Prentice Hall, Basia Kennedy, Dan Ramirez

Chapter 14

Trigonometric Identities And Equations

Educators

HD

Problem 1

Verify each identity.
$$\cos \theta \cot \theta=\frac{1}{\sin \theta}-\sin \theta$$

Joshua E.
Numerade Educator

Problem 2

Verify each identity.
$$
\sin \theta \cot \theta=\cos \theta
$$

HD
Harrison D.
Numerade Educator

Problem 3

Verify each identity.
$$
\cos \theta \tan \theta=\sin \theta
$$

Joshua E.
Numerade Educator

Problem 4

Verify each identity.
$$
\sin \theta \sec \theta=\tan \theta
$$

HD
Harrison D.
Numerade Educator

Problem 5

Verify each identity.
$$
\cos \theta \sec \theta=1
$$

Joshua E.
Numerade Educator

Problem 6

Verify each identity.
$$
\tan \theta \cot \theta=1
$$

HD
Harrison D.
Numerade Educator

Problem 7

Verify each identity.
$$
\sin \theta \csc \theta=1
$$

Joshua E.
Numerade Educator

Problem 8

Verify each identity.
$$
\cot \theta=\frac{\csc \theta}{\sec \theta}
$$

HD
Harrison D.
Numerade Educator

Problem 9

Simplify each trigonometric expression.
$$
\tan \theta \cot \theta
$$

Joshua E.
Numerade Educator

Problem 10

Simplify each trigonometric expression.
$$
1-\cos ^{2} \theta
$$

HD
Harrison D.
Numerade Educator

Problem 11

Simplify each trigonometric expression.
$$
\sec ^{2} \theta-1
$$

Joshua E.
Numerade Educator

Problem 12

Simplify each trigonometric expression.
$$
1-\csc ^{2} \theta
$$

HD
Harrison D.
Numerade Educator

Problem 13

Simplify each trigonometric expression.
$$
\sec \theta \cot \theta
$$

Joshua E.
Numerade Educator

Problem 14

Simplify each trigonometric expression.
$$
\cos \theta \tan \theta
$$

HD
Harrison D.
Numerade Educator

Problem 15

Simplify each trigonometric expression.
$$
\sin \theta \cot \theta
$$

Joshua E.
Numerade Educator

Problem 16

Simplify each trigonometric expression.
$$
\sin \theta \csc \theta
$$

HD
Harrison D.
Numerade Educator

Problem 17

Simplify each trigonometric expression.
$$
\sec \theta \cos \theta \sin \theta
$$

Joshua E.
Numerade Educator

Problem 18

Simplify each trigonometric expression.
$$
\sin \theta \sec \theta \cot \theta
$$

HD
Harrison D.
Numerade Educator

Problem 19

Simplify each trigonometric expression.
$$
\sec ^{2} \theta-\tan ^{2} \theta
$$

Joshua E.
Numerade Educator

Problem 20

Simplify each trigonometric expression.
$$
\frac{\sin \theta}{\cos \theta \tan \theta}
$$

HD
Harrison D.
Numerade Educator

Problem 21

Simplify each trigonometric expression.
$$
\cos \theta+\sin \theta \tan \theta
$$

Joshua E.
Numerade Educator

Problem 22

Simplify each trigonometric expression.
$$
\csc \theta \cos \theta \tan \theta
$$

HD
Harrison D.
Numerade Educator

Problem 23

Simplify each trigonometric expression.
$$
\tan \theta(\cot \theta+\tan \theta)
$$

Joshua E.
Numerade Educator

Problem 24

Simplify each trigonometric expression.
$$
\sin ^{2} \theta+\cos ^{2} \theta+\tan ^{2} \theta
$$

HD
Harrison D.
Numerade Educator

Problem 25

Simplify each trigonometric expression.
$$
\cos ^{2} \theta \sec \theta \csc \theta
$$

Joshua E.
Numerade Educator

Problem 26

Simplify each trigonometric expression.
$$
\sin \theta\left(1+\cot ^{2} \theta\right)
$$

HD
Harrison D.
Numerade Educator

Problem 27

Simplify each trigonometric expression.
$$
\cot \theta \tan \theta-\sec ^{2} \theta
$$

Joshua E.
Numerade Educator

Problem 28

Simplify each trigonometric expression.
$$
\sin ^{2} \theta \csc \theta \sec \theta
$$

HD
Harrison D.
Numerade Educator

Problem 29

Simplify each trigonometric expression.
$$
\cos \theta\left(1+\tan ^{2} \theta\right)
$$

Joshua E.
Numerade Educator

Problem 30

Simplify each trigonometric expression.
$$
\frac{\tan \theta}{\sec \theta-\cos \theta}
$$

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Problem 31

Simplify each trigonometric expression.
$$
\sec \theta \cos \theta-\cos ^{2} \theta
$$

Joshua E.
Numerade Educator

Problem 32

Simplify each trigonometric expression.
$$
\sin \theta \csc \theta-\cos ^{2} \theta
$$

HD
Harrison D.
Numerade Educator

Problem 33

Simplify each trigonometric expression.
$$
\csc \theta-\cos \theta \cot \theta
$$

Joshua E.
Numerade Educator

Problem 34

Simplify each trigonometric expression.
$$
\cos \theta+\sin \theta \tan \theta
$$

HD
Harrison D.
Numerade Educator

Problem 35

Simplify each trigonometric expression.
$$
\sec \theta\left(1+\cot ^{2} \theta\right)
$$

Joshua E.
Numerade Educator

Problem 36

Simplify each trigonometric expression.
$$
\csc ^{2} \theta\left(1-\cos ^{2} \theta\right)
$$

HD
Harrison D.
Numerade Educator

Problem 37

Simplify each trigonometric expression.
$$
\frac{\cos \theta \csc \theta}{\cot \theta}
$$

Joshua E.
Numerade Educator

Problem 38

Simplify each trigonometric expression.
$$
\frac{\sin ^{2} \theta \csc \theta \sec \theta}{\tan \theta}
$$

HD
Harrison D.
Numerade Educator

Problem 39

Express the first trigonometric function in terms of the second.
$$
\sin \theta, \cos \theta
$$

Joshua E.
Numerade Educator

Problem 40

Express the first trigonometric function in terms of the second.
$$
\tan \theta, \cos \theta
$$

HD
Harrison D.
Numerade Educator

Problem 41

Express the first trigonometric function in terms of the second.
$$
\cot \theta, \sin \theta
$$

Joshua E.
Numerade Educator

Problem 42

Express the first trigonometric function in terms of the second.
$$
\csc \theta, \cot \theta
$$

HD
Harrison D.
Numerade Educator

Problem 43

Express the first trigonometric function in terms of the second.
$$
\cot \theta, \csc \theta
$$

Joshua E.
Numerade Educator

Problem 44

Express the first trigonometric function in terms of the second.
$$
\sec \theta, \tan \theta
$$

HD
Harrison D.
Numerade Educator

Problem 45

Verify each identity.
$$
\sin ^{2} \theta \tan ^{2} \theta=\tan ^{2} \theta-\sin ^{2} \theta
$$

Joshua E.
Numerade Educator

Problem 46

Verify each identity.
$$
\sec \theta-\sin \theta \tan \theta=\cos \theta
$$

HD
Harrison D.
Numerade Educator

Problem 47

Verify each identity.
$$
\sin \theta \cos \theta(\tan \theta+\cot \theta)=1
$$

Joshua E.
Numerade Educator

Problem 48

Verify each identity.
$$
\frac{1-\sin \theta}{\cos \theta}=\frac{\cos \theta}{1+\sin \theta}
$$

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Problem 49

Verify each identity.
$$
\frac{\sec \theta}{\cot \theta+\tan \theta}=\sin \theta
$$

Joshua E.
Numerade Educator

Problem 50

Verify each identity.
$$
(\cot \theta+1)^{2}=\csc ^{2} \theta+2 \cot \theta
$$

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Problem 51

Verify each identity.
Express $\cos \theta \csc \theta \cot \theta$ in terms of $\sin \theta$

Joshua E.
Numerade Educator

Problem 52

Verify each identity.
Express $\frac{\cos \theta}{\sec \theta+\tan \theta}$ in terms of $\sin \theta$

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Problem 53

Verify each identity.
Open-Ended Create a trigonometric identity. (Hint: Start with a simple
trigonometric expression and work backward.)

Joshua E.
Numerade Educator

Problem 54

Verify each identity.
Writing Describe the similarities and differences in solving an equation and in
verifying an identity.

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Problem 55

Verify each identity.
$$
1+\sec \theta=\frac{1+\cos \theta}{\cos \theta}
$$

Joshua E.
Numerade Educator

Problem 56

Verify each identity.
$$
\frac{1+\tan \theta}{\tan \theta}=\cot \theta+1
$$

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Problem 57

Verify each identity.
$$
\frac{\cot \theta \sin \theta}{\sec \theta}+\frac{\tan \theta \cos \theta}{\csc \theta}=1
$$

Joshua E.
Numerade Educator

Problem 58

Verify each identity.
$$
\sin ^{2} \theta \tan ^{2} \theta+\cos ^{2} \theta \tan ^{2} \theta=\sec ^{2} \theta-1
$$

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Problem 59

Simplify each trigonometric expression.
$$
\frac{\cot ^{2} \theta-\csc ^{2} \theta}{\tan ^{2} \theta-\sec ^{2} \theta}
$$

Joshua E.
Numerade Educator

Problem 60

Simplify each trigonometric expression.
$$
(1-\sin \theta)(1+\sin \theta) \csc ^{2} \theta+1
$$

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Problem 61

Physics When a ray of light passes from one medium into a second, the angle of incidence $\theta_{1}$ and the angle of refraction $\theta_{2}$ are related by Snell's law: $n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2},$ where $n_{1}$ is the index of refraction of the first medium and $n_{2}$ is the index of refraction of the second medium. How are $\theta_{1}$ and $\theta_{2}$ related if $n_{2}>n_{1} ?$ If $n_{2}<n_{1} ?$ If $n_{2}=n_{1} ?$

Joshua E.
Numerade Educator

Problem 62

Which expression is NOT equal to the other three expressions?
$\begin{array}{llll}{\text { A. } \frac{2}{\tan \theta}} & {\text { B. } \frac{\cot \theta}{\frac{1}{2}}} & {\text { C. } \frac{\sin \theta}{\frac{1}{2} \cos \theta}} & {\text { D. } \frac{2 \cos \theta}{\sin \theta}}\end{array}$

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Problem 63

Which equation is NOT true?
$\begin{array}{ll}{\mathbf{F} \cos ^{2} \theta=1-\sin ^{2} \theta} & {\text { G. } \cot ^{2} \theta=\csc ^{2} \theta-1} \\ {\text { H. } \sin ^{2} \theta=\cos ^{2} \theta-1} & {\text { I. } \tan ^{2} \theta=\sec ^{2} \theta-1}\end{array}$

Joshua E.
Numerade Educator

Problem 64

Which expressions are equivalent?
1. $(\sin \theta)(\csc \theta-\sin \theta)$ II. $\sin ^{2} \theta-1$ III. $\cos ^{2} \theta$
A. I and II only
B. II and ll only
C.l and III only
D. $1,11,$ and $\| 1$

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Problem 65

$\begin{array}{ll}{\text { How can you express } \csc ^{2} \theta-2 \cot ^{2} \theta \text { in terms of } \sin \theta \text { and } \cos \theta ?} \\ {\mathrm{F} \cdot \frac{1-2 \cos ^{2} \theta}{\sin ^{2} \theta}} & {\text { G. } \frac{1-2 \sin ^{2} \theta}{\sin ^{2} \theta}} \\ {\mathrm{H} \cdot \sin ^{2} \theta-2 \cos ^{2} \theta} & {\text { J. } \frac{1}{\sin ^{2} \theta}-\frac{2}{\tan ^{2} \theta}}\end{array}$

Joshua E.
Numerade Educator

Problem 66

Which expression is equivalent to $\frac{\tan \theta}{\cos \theta-\sec \theta} ?$
$\begin{array}{llll}{\text { A. } \csc \theta} & {\text { B. } \sec \theta} & {\text { C. }-\csc \theta} & {\text { D. } \tan ^{2} \theta}\end{array}$

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Problem 67

Show that $(\sec \theta+1)(\sec \theta-1)=\tan ^{2} \theta$ is an identity.

Joshua E.
Numerade Educator

Problem 68

Show that $\frac{\cos x}{1-\sin ^{2} x}=\sec x$ is an identity.

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Problem 69

Graph each function in the interval from 0 to 2$\pi$.
$$
y=\csc (-\theta)
$$

Joshua E.
Numerade Educator

Problem 70

Graph each function in the interval from 0 to 2$\pi$.
$$
y=-\cot \theta
$$

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Problem 71

Graph each function in the interval from 0 to 2$\pi$.
$$
y=-\sec 0.5 \theta
$$

Joshua E.
Numerade Educator

Problem 72

Graph each function in the interval from 0 to 2$\pi$.
$$
y=-\sec (0.5 \theta+2)
$$

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Problem 73

Graph each function in the interval from 0 to 2$\pi$.
$$
y=\cot \frac{\theta}{5}
$$

Joshua E.
Numerade Educator

Problem 74

Graph each function in the interval from 0 to 2$\pi$.
$$
y=\pi \sec \theta
$$

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Problem 75

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
395^{\circ}
$$

Joshua E.
Numerade Educator

Problem 76

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
405^{\circ}
$$

HD
Harrison D.
Numerade Educator

Problem 77

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-225^{\circ}
$$

Joshua E.
Numerade Educator

Problem 78

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-149^{\circ}
$$

HD
Harrison D.
Numerade Educator

Problem 79

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
627^{\circ}
$$

Joshua E.
Numerade Educator

Problem 80

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-281^{\circ}
$$

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Problem 81

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
493^{\circ}
$$

Joshua E.
Numerade Educator

Problem 82

Find the measure of an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-609^{\circ}
$$

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Problem 83

Make a box-and-whisker plot for each set of values.
300$\quad 345 \quad 333 \quad 295 \quad 302 \quad 321$

Joshua E.
Numerade Educator

Problem 84

Make a box-and-whisker plot for each set of values.
3248$\quad 87 \quad 43 \quad 62 \quad 15 \quad 49 \quad 51 \quad 47 \quad 36 \quad 50 \quad 109 \quad 64$

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