# Precalculus with Limits

## Educators

ST
EB
JH

### Problem 1

Fill in the blank to complete the trigonometric identity.
$\frac{\sin u}{\cos u}=$________

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### Problem 2

Fill in the blank to complete the trigonometric identity.
$\frac{1}{\csc u}=$________

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### Problem 3

Fill in the blank to complete the trigonometric identity.
$\frac{1}{\tan u}=$________

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### Problem 4

Fill in the blank to complete the trigonometric identity.
$\sec \left(\frac{\pi}{2}-u\right)=$________

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### Problem 5

Fill in the blank to complete the trigonometric identity.
$1+$ ________ $=\csc ^{2} u$

ST
Shivani T.

### Problem 6

Fill in the blank to complete the trigonometric identity.
$\cot (-u)=$ ________

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### Problem 7

Use the given values to find the values (if possible) of all six trigonometric functions.
$\sin x=\frac{1}{2}, \cos x=\frac{\sqrt{3}}{2}$

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### Problem 8

Use the given values to find the values (if possible) of all six trigonometric functions.
$\csc \theta=\frac{25}{7}, \tan \theta=\frac{7}{24}$

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### Problem 9

Use the given values to find the values (if possible) of 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 10

Use the given values to find the values (if possible) of all six trigonometric functions.
$\sin (-x)=-\frac{1}{3}, \quad \tan x=-\frac{\sqrt{2}}{4}$

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### Problem 11

Use the given values to find the values (if possible) of all six trigonometric functions.
$\sec x=4, \quad \sin x>0$

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### Problem 12

Use the given values to find the values (if possible) of all six trigonometric functions.
$\csc \theta=-5, \quad \cos \theta<0$

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### Problem 13

Use the given values to find the values (if possible) of all six trigonometric functions.
$\sin \theta=-1, \quad \cot \theta=0$

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### Problem 14

Use the given values to find the values (if possible) of all six trigonometric functions.
$\tan \theta$ is undefined, $\sin \theta>0$

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### Problem 15

Match the trigonometric expression with one of the following.
(a) $\csc x \quad$ (b) $-1 \quad$ (c) 1
(d) $\sin x \tan x \quad$ (e) $\sec ^{2} x$ $\quad$(f) $\sec ^{2} x+\tan ^{2} x$
$\sec x \cos x$

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### Problem 16

Match the trigonometric expression with one of the following.
(a) $\csc x \quad$ (b) $-1 \quad$ (c) 1
(d) $\sin x \tan x \quad$ (e) $\sec ^{2} x$ $\quad$(f) $\sec ^{2} x+\tan ^{2} x$
$\cot ^{2} x-\csc ^{2} x$

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### Problem 17

Match the trigonometric expression with one of the following.
(a) $\csc x \quad$ (b) $-1 \quad$ (c) 1
(d) $\sin x \tan x \quad$ (e) $\sec ^{2} x$ $\quad$(f) $\sec ^{2} x+\tan ^{2} x$
$\sec ^{4} x-\tan ^{4} x$

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### Problem 18

Match the trigonometric expression with one of the following.
(a) $\csc x \quad$ (b) $-1 \quad$ (c) 1
(d) $\sin x \tan x \quad$ (e) $\sec ^{2} x$ $\quad$(f) $\sec ^{2} x+\tan ^{2} x$
$\cot x \sec x$

EB
Erika B.

### Problem 19

Match the trigonometric expression with one of the following.
(a) $\csc x \quad$ (b) $-1 \quad$ (c) 1
(d) $\sin x \tan x \quad$ (e) $\sec ^{2} x$ $\quad$(f) $\sec ^{2} x+\tan ^{2} x$
$\frac{\sec ^{2} x-1}{\sin ^{2} x}$

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### Problem 20

Match the trigonometric expression with one of the following.
(a) $\csc x \quad$ (b) $-1 \quad$ (c) 1
(d) $\sin x \tan x \quad$ (e) $\sec ^{2} x$ $\quad$(f) $\sec ^{2} x+\tan ^{2} x$
$\frac{\cos ^{2}[(\pi / 2)-x]}{\cos x}$

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### Problem 21

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 22

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 23

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 24

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$\frac{\cos x-2}{\cos ^{2} x-4}$

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### Problem 25

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 26

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 27

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$\cot ^{3} x+\cot ^{2} x+\cot x+1$

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### Problem 28

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 29

Factor the trigonometric expression. There is more than one correct form of each answer.
$3 \sin ^{2} x-5 \sin x-2$

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### Problem 30

Factor the trigonometric expression. There is more than one correct form of each answer.
$6 \cos ^{2} x+5 \cos x-6$

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### Problem 31

Factor the trigonometric expression. There is more than one correct form of each answer.
$\cot ^{2} x+\csc x-1$

JH
Jacquelyn H.

### Problem 32

Factor the trigonometric expression. There is more than one correct form of each answer.
$\sin ^{2} x+3 \cos x+3$

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### Problem 33

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 34

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 35

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 36

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 37

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 38

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 39

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 40

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 41

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 42

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 43

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 44

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 45

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 46

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 47

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 48

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 49

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 50

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 51

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 52

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 53

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi 2 .$
$\sqrt{9-x^{2}}, \quad x=3 \cos \theta$

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### Problem 54

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi 2 .$
$\sqrt{49-x^{2}}, \quad x=7 \sin \theta$

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### Problem 55

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi 2 .$
$\sqrt{x^{2}-4}, \quad x=2 \sec \theta$

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### Problem 56

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $0<\theta<\pi 2 .$
$\sqrt{9 x^{2}+25}, \quad 3 x=5 \tan \theta$

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### Problem 57

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 58

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 59

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 60

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 61

Rewrite the expression as a single logarithm and simplify the result.
$\ln |\sin x|+\ln |\cot x|$

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### Problem 62

Rewrite the expression as a single logarithm and simplify the result.
$\ln |\cos x|-\ln |\sin x|$

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### Problem 63

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 64

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 65

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 66

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 67

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 68

A cofunction identity can transform a tangent function into a cosecant function.

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### Problem 69

Fill in the blanks.
$\operatorname{As} x \rightarrow\left(\frac{\pi}{2}\right)^{-}, \tan x \rightarrow$ ____ and $\cot x \rightarrow$ ____ .

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### Problem 70

Fill in the blanks.
As $x \rightarrow \pi^{+}, \quad \sin x \rightarrow$ ____ and $\csc x \rightarrow$ ____ .

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### Problem 71

Determine whether 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 72

Determine whether the equation is an identity, and give a reason for your answer.
$\sin \theta \csc \theta=1$

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### Problem 73

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 74

Explain how to use the figure to derive the Pythagorean identities
$\sin ^{2} \theta+\cos ^{2} \theta=1$ $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $1+\cot ^{2} \theta=\csc ^{2} \theta$
Discuss how to remember these identities and other fundamental trigonometric identities.

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### Problem 75

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

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### Problem 76

Rewrite the following expression in terms of $\sin \theta$ and $\cos \theta .$
$\frac{\sec \theta(1+\tan \theta)}{\sec \theta+\csc \theta}$

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