Algebra

Holliday, Luchin, Cuevas, Carter Marks, Day, Casey, Hayek

Chapter 14

Trigonometric Graphs and Identities - all with Video Answers

Educators

KB

RB

+ 6 more educators

Section 1

Graphing Trigonometric Functions

Problem 1

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=\frac{1}{2} \sin \theta
$$

Peggy R.

Peggy R.

Numerade Educator

Problem 2

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=2 \sin \theta
$$

Peggy R.

Numerade Educator

Problem 3

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=\frac{2}{3} \cos \theta
$$

Peggy R.

Numerade Educator

Problem 4

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=\frac{1}{4} \tan \theta
$$

Peggy R.

Numerade Educator

Problem 5

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=\csc 2 \theta
$$

Peggy R.

Numerade Educator

Problem 6

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=4 \sin 2 \theta
$$

Peggy R.

Numerade Educator

Problem 7

$$
y=4 \cos \frac{3}{4} \theta
$$

Peggy R.

Numerade Educator

Problem 8

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=\frac{1}{2} \sec 3 \theta
$$

Peggy R.

Numerade Educator

Problem 9

Find the amplitude, if it exists, and period of each function. Then graph each function.
$$
y=\frac{3}{4} \cos \frac{1}{2} \theta
$$

Raushan K.

Numerade Educator

Problem 10

Blotocy For Exercises 10 and $11,$ use the following information. In a certain wildlife refuge, the population of field mice can be modeled by $y=3000+1250 \sin \frac{\pi}{6} t,$ where $y$ represents the number of mice and $t$ represents the number of months past March 1 of a given year.

Determine the period of the function. What does this period represent?

Raushan K.

Numerade Educator

Problem 11

Blotocy For Exercises 10 and $11,$ use the following information. In a certain wildlife refuge, the population of field mice can be modeled by $y=3000+1250 \sin \frac{\pi}{6} t,$ where $y$ represents the number of mice and $t$ represents the number of months past March 1 of a given year.

What is the maximum number of mice, and when does this occur?

Raushan K.

Numerade Educator

Problem 12

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=3 \sin \theta$

Raushan K.

Numerade Educator

Problem 13

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=5 \cos \theta$

KB

Kay B.

Numerade Educator

Problem 14

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=2 \csc \theta$

Raushan K.

Numerade Educator

Problem 15

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=2 \tan \theta$

Raushan K.

Numerade Educator

Problem 16

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=\frac{1}{5} \sin \theta$

Raushan K.

Numerade Educator

Problem 17

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=\frac{1}{3} \sec \theta$

Raushan K.

Numerade Educator

Problem 18

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=\sin 4 \theta$

Raushan K.

Numerade Educator

Problem 19

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=\sin 2 \theta$

Supriya K.

Numerade Educator

Problem 20

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=\sec 3 \theta$

Raushan K.

Numerade Educator

Problem 21

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=\cot 5 \theta$

Raushan K.

Numerade Educator

Problem 22

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=4 \tan \frac{1}{3} \theta$

Raushan K.

Numerade Educator

Problem 23

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=2 \cot \frac{1}{2} \theta$

Raushan K.

Numerade Educator

Problem 24

MEDICINE For Exercises 24 and $25,$ use the following information.
If the amplitude of the sine function is $0.25,$ write the equations for tuning forks that resonate with a frequency of $64,256,$ and 512 Hertz.

AS

Adam S.

Numerade Educator

Problem 25

MEDICINE For Exercises 24 and $25,$ use the following information.
How do the periods of the tuning forks compare?

Raushan K.

Numerade Educator

Problem 26

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=6 \sin \frac{2}{3} \theta$

Raushan K.

Numerade Educator

Problem 27

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=3 \cos \frac{1}{2} \theta$

Raushan K.

Numerade Educator

Problem 28

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=3 \csc \frac{1}{2} \theta$

Raushan K.

Numerade Educator

Problem 29

Find the amplitude, if it exists, and period of each function. Then graph each function.
$y=\frac{1}{2} \cot 2 \theta$

Raushan K.

Numerade Educator

Problem 30

Find the amplitude, if it exists, and period of each function. Then graph each function.
$2 y=\tan \theta$

Raushan K.

Numerade Educator

Problem 31

Find the amplitude, if it exists, and period of each function. Then graph each function.
$\frac{3}{4} y=\frac{2}{3} \sin \frac{3}{5} \theta$

Raushan K.

Numerade Educator

Problem 32

Draw a graph of a sine function with an amplitude $\frac{3}{5}$ and a period of $90^{\circ} .$ Then write an equation for the function.

Raushan K.

Numerade Educator

Problem 33

Draw a graph of a cosine function with an amplitude of $\frac{7}{8}$ and a period of $\frac{2 \pi}{5}$ . Then write an equation for the function. Graph the functions $f(x)=\sin x$ and $g(x)=\cos x,$ where $x$ is measured in

Raushan K.

Numerade Educator

Problem 34

Graph the functions $f(x)=\sin x$ and $g(x)=\cos x,$ where $x$ is measured in radian, for $x$ between 0 and 2$\pi .$ Identify the points of intersection of the two graphs.

Raushan K.

Numerade Educator

Problem 35

Identify all asymptotes to the graph of $g(x)=\sec x$

Raushan K.

Numerade Educator

Problem 36

BOATING For Exercises $36-38,$ use the following information.
Write an equation for the motion of the buoy. Assume that it is at equilibrium at $t=0$ and that it is on the way up from the normal water level.

Raushan K.

Numerade Educator

Problem 37

BOATING For Exercises $36-38,$ use the following information.
Draw a graph showing the height of the buoy as a function of time.

WS

William S.

Numerade Educator

Problem 38

BOATING For Exercises $36-38,$ use the following information.
What is the height of the buoy after 12 seconds?

Michael K.

Numerade Educator

Problem 39

OPEN ENDED Write a trigonometric function that has an amplitude of 3 and a period of $\pi$ . Graph the function.

Raushan K.

Numerade Educator

Problem 40

REASONING Explain what it means to say that the period of a function is $180^{\circ} .$

Raushan K.

Numerade Educator

Problem 41

CHALENGE A function is called even if the graphs of $y=f(x)$ and $y=f(-x)$ are exactly the same. Which of the six trigoometric function are even? Justify your answer with a graph of each function.

Raushan K.

Numerade Educator

Problem 42

FIND THE ERROR Dante and Jamile graphed $y=3 \cos \frac{2}{3} \theta$ . Who is correct? Explain your reasoning.

Raushan K.

Numerade Educator

Problem 43

Whiting in Math Use the information on page 822 to explain how you can predict the behavior of tides. Explain why certain tidal characteristics follow the patterns seen in the graph of the sine function.

Raushan K.

Numerade Educator

Problem 44

ACT/SAT Identify the equation of the graphed function.
$$
\begin{array}{ll}{\mathbf{A} y=\frac{1}{2} \sin 4 \theta} & {\mathbf{C} y=2 \sin \frac{1}{4} \theta} \\ {\mathbf{B} \quad y=\frac{1}{4} \sin 2 \theta} & {\mathbf{D} y=4 \sin \frac{1}{2} \theta}\end{array}
$$

Raushan K.

Numerade Educator

Problem 45

REVIEW Refer to the figure below. If $\tan x=\frac{10}{24},$ what are $\sin x$ and $\cos x ?$
$$
\begin{array}{l}{F \sin x=\frac{26}{10} \text { and } \cos x=\frac{24}{26}} \\ {G \sin x=\frac{10}{26} \text { and } \cos x=\frac{24}{26}} \\ {H \sin x=\frac{26}{10} \text { and } \cos x=\frac{26}{24}} \\ {J \sin x=\frac{26}{10} \text { and } \cos x=\frac{24}{26}}\end{array}
$$

Mason B.

Numerade Educator

Problem 46

Solve each equation.
$$
x=\sin ^{-1} 1
$$

Raushan K.

Numerade Educator

Problem 47

Solve each equation.
$$
\operatorname{Arcsin}(-1)=y
$$

Raushan K.

Numerade Educator

Problem 48

Solve each equation.
$$
\operatorname{Arccos} \frac{\sqrt{2}}{2}=x
$$

Raushan K.

Numerade Educator

Problem 49

Find the exact value of each function.
$$
\sin 390^{\circ}
$$

Raushan K.

Numerade Educator

Problem 50

Find the exact value of each function.
$$
\sin \left(-315^{\circ}\right)
$$

Uma K.

Numerade Educator

Problem 51

Find the exact value of each function.
$$
\cos 405^{\circ}
$$

Raushan K.

Numerade Educator

Problem 52

PROBABILITY There are 8 girls and 8 boys on the Faculty Advisory Board. Three are juniors. Find the probability of selecting a boy or a girl from the committee who is not a junion

Kirsten B.

Numerade Educator

Problem 53

Find the first five terms of the sequence in which $a_{1}=3, a_{n+1}=2 a_{n}+5$

Raushan K.

Numerade Educator

Problem 54

PREREQUISITE SKILL Graph each pair of functions on the same set of axes.
$$
y=x^{2}, y=3 x^{2}
$$

Raushan K.

Numerade Educator

Problem 55

PREREQUISITE SKILL Graph each pair of functions on the same set of axes.
$$
y=3 x^{2}, y=3 x^{2}-4
$$

Raushan K.

Numerade Educator

Problem 56

PREREQUISITE SKILL Graph each pair of functions on the same set of axes.
$$
y=2 x^{2}, y=2(x+1)^{2}
$$

Raushan K.

Numerade Educator