Books(current) Courses (current) Earn 💰 Log in(current)

Chapter 5

Analytic Trigonometry

Educators


Problem 1

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

Check back soon!

Problem 2

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

Check back soon!

Problem 3

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

Check back soon!

Problem 4

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

Check back soon!

Problem 5

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

Check back soon!

Problem 6

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

Check back soon!

Problem 7

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

Check back soon!

Problem 8

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

Check back soon!

Problem 9

Use the given values to find the values (if possible) of all six trigonometric functions.
$\cos \left(\frac{\pi}{2}-x\right)=\frac{3}{5}, \quad \cos x=\frac{4}{5}$

Check back soon!

Problem 10

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

Check back soon!

Problem 11

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

Check back soon!

Problem 12

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

Check back soon!

Problem 13

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

Check back soon!

Problem 14

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

Check back soon!

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) $\sec ^{2} x+\tan ^{2} x$
$\sec x \cos x$

Check back soon!

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) $\sec ^{2} x+\tan ^{2} x$
$\cot ^{2} x-\csc ^{2} x$

Check back soon!

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) $\sec ^{2} x+\tan ^{2} x$
$\sec ^{4} x-\tan ^{4} x$

Check back soon!

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) $\sec ^{2} x+\tan ^{2} x$
$\cot x \sec x$

Check back soon!

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) $\sec ^{2} x+\tan ^{2} x$
$\frac{\sec ^{2} x-1}{\sin ^{2} x}$

Check back soon!

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) $\sec ^{2} x+\tan ^{2} x$
$\frac{\cos ^{2}[(\pi / 2)-x]}{\cos x}$

Check back soon!

Problem 21

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

Check back soon!

Problem 22

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

Check back soon!

Problem 23

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

Check back soon!

Problem 24

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

Check back soon!

Problem 25

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.
$1-2 \cos ^{2} x+\cos ^{4} x$

Check back soon!

Problem 26

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

Check back soon!

Problem 27

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

Check back soon!

Problem 28

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

Check back soon!

Problem 29

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

Check back soon!

Problem 30

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

Check back soon!

Problem 31

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

Check back soon!

Problem 32

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

Check back soon!

Problem 33

Perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.
$(\sin x+\cos x)^{2}$

Check back soon!

Problem 34

Perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.
$(2 \csc x+2)(2 \csc x-2)$

Check back soon!

Problem 35

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\cot \theta \sec \theta$

Check back soon!

Problem 36

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\tan (-x) \cos x$

Check back soon!

Problem 37

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\sin \phi(\csc \phi-\sin \phi)$

Check back soon!

Problem 38

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

Check back soon!

Problem 39

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\frac{1-\sin ^{2} x}{\csc ^{2} x-1}$

Check back soon!

Problem 40

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\frac{\tan \theta \cot \theta}{\sec \theta}$

Check back soon!

Problem 41

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

Check back soon!

Problem 42

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\frac{\cos ^{2} y}{1-\sin y}$

Check back soon!

Problem 43

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\sin \beta \tan \beta+\cos \beta$

Check back soon!

Problem 44

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
$\cot u \sin u+\tan u \cos u$

Check back soon!

Problem 45

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$\frac{1}{1+\cos x}+\frac{1}{1-\cos x}$

Check back soon!

Problem 46

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$\frac{1}{\sec x+1}-\frac{1}{\sec x-1}$

Check back soon!

Problem 47

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

Check back soon!

Problem 48

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.
$\frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}$

Check back soon!

Problem 49

Rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.
$\frac{\sin ^{2} y}{1-\cos y}$

Check back soon!

Problem 50

Rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.
$\frac{5}{\tan x+\sec x}$

Check back soon!

Problem 51

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.
$\cos x \cot x+\sin x$

Check back soon!

Problem 52

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

Check back soon!

Problem 53

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

Check back soon!

Problem 54

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

Check back soon!

Problem 55

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

Check back soon!

Problem 56

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

Check back soon!

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

Check back soon!

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

Check back soon!

Problem 59

Use a graphing utility to solve the equation for $\theta$ where $0 \leq \theta<2 \pi$
$\sin \theta=\sqrt{1-\cos ^{2} \theta}$

Check back soon!

Problem 60

Use a graphing utility to solve the equation for $\theta$ where $0 \leq \theta<2 \pi$
$\sec \theta=\sqrt{1+\tan ^{2} \theta}$

Check back soon!

Problem 61

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

Check back soon!

Problem 62

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

Check back soon!

Problem 63

Rewrite the expression as a single logarithm and simplify the result.
$\ln |\cot t|+\ln \left(1+\tan ^{2} t\right)$

Check back soon!

Problem 64

Rewrite the expression as a single logarithm and simplify the result.
$\ln \left(\cos ^{2} t\right)+\ln \left(1+\tan ^{2} t\right)$

Check back soon!

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.

Check back soon!

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 $\sin x \tan ^{2} x$

Check back soon!

Problem 67

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.

Check back soon!

Problem 68

Determine whether the statement is true or false. Justify your answer.
A cofunction identity can transform a tangent function into a cosecant function.

Check back soon!

Problem 69

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

Check back soon!

Problem 70

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

Check back soon!

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.

Check back soon!

Problem 72

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

Check back soon!

Problem 73

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

Check back soon!

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.

Check back soon!

Problem 75

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

Check back soon!

Problem 76

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

Check back soon!