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Precalculus

Cynthia Young

Chapter 5

Trigonometric Functions of Real Numbers - all with Video Answers

Educators

Section 1

Trigonometric Functions: The Unit Circle Approach

00:23

Problem 1

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=-\sin x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:12

Problem 2

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=\sin x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:15

Problem 3

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=\cos x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:17

Problem 4

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=-\cos x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:21

Problem 5

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=2 \sin x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:18

Problem 6

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=2 \cos x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:21

Problem 7

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=\sin \left(\frac{1}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:19

Problem 8

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=\cos \left(\frac{1}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:11

Problem 9

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=-2 \cos \left(\frac{1}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:16

Problem 10

Match the function with its graph (a-j). (GRAPH CAN'T COPY)
$$y=-2 \sin \left(\frac{1}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:24

Problem 11

State the amplitude and period of each function.
$$y=\frac{3}{2} \cos (3 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:27

Problem 12

State the amplitude and period of each function.
$$y=\frac{2}{3} \sin (4 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:24

Problem 13

State the amplitude and period of each function.
$$y=-\sin (5 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:18

Problem 14

State the amplitude and period of each function.
$$y=-\cos (7 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:30

Problem 15

State the amplitude and period of each function.
$$y=\frac{2}{3} \cos \left(\frac{3}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:25

Problem 16

State the amplitude and period of each function.
$$y=\frac{3}{2} \sin \left(\frac{2}{3} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:24

Problem 17

State the amplitude and period of each function.
$$y=-3 \cos (\pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:21

Problem 18

State the amplitude and period of each function.
$$y=-2 \sin (\pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:29

Problem 19

State the amplitude and period of each function.
$$y=5 \sin \left(\frac{\pi}{3} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:44

Problem 20

State the amplitude and period of each function.
$$y=4 \cos \left(\frac{\pi}{4} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:43

Problem 21

Graph the given function over one period.
$$y=8 \cos x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:43

Problem 22

Graph the given function over one period.
$$y=7 \sin x$$

Robert Leedy
Robert Leedy
Numerade Educator
01:09

Problem 23

Graph the given function over one period.
$$y=\sin (4 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
01:06

Problem 24

Graph the given function over one period.
$$y=\cos (3 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
01:20

Problem 25

Graph the given function over one period.
$$y=-3 \cos \left(\frac{1}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:58

Problem 26

Graph the given function over one period.
$$y=-2 \sin \left(\frac{1}{4} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:52

Problem 27

Graph the given function over one period.
$$y=-3 \sin (\pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:46

Problem 28

Graph the given function over one period.
$$y=-2 \cos (\pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:42

Problem 29

Graph the given function over one period.
$$y=5 \cos (2 \pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:37

Problem 30

Graph the given function over one period.
$$y=4 \sin (2 \pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:49

Problem 31

Graph the given function over one period.
$$y=-3 \sin \left(\frac{\pi}{4} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:42

Problem 32

Graph the given function over one period.
$$y=-4 \sin \left(\frac{\pi}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:46

Problem 33

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=-4 \cos \left(\frac{1}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
01:06

Problem 34

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=-5 \sin \left(\frac{1}{2} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
01:05

Problem 35

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=-\sin (6 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
01:05

Problem 36

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=-\cos (4 x)$$

Robert Leedy
Robert Leedy
Numerade Educator
01:02

Problem 37

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=3 \cos \left(\frac{\pi}{4} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:47

Problem 38

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=4 \sin \left(\frac{\pi}{4} x\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:48

Problem 39

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=\sin (4 \pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:44

Problem 40

Graph the given function over the interval $[-2 p, 2 p]$, where $p$ is the period of the function.
$$y=\cos (6 \pi x)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:25

Problem 41

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
00:32

Problem 42

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
00:30

Problem 43

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
00:26

Problem 44

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
00:51

Problem 45

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
00:20

Problem 46

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
01:02

Problem 47

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
00:42

Problem 48

Find the equation for each graph. (GRAPH CAN'T COPY)

Robert Leedy
Robert Leedy
Numerade Educator
00:45

Problem 49

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=2 \sin (\pi x-1)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:43

Problem 50

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=4 \cos (x+\pi)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:47

Problem 51

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=-5 \cos (3 x+2)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:51

Problem 52

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=-7 \sin (4 x-3)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:52

Problem 53

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=6 \sin [-\pi(x+2)]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:28

Problem 54

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=3 \sin \left[-\frac{\pi}{2}(x-1)\right]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:47

Problem 55

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=3 \sin (2 x+\pi)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:43

Problem 56

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=-4 \cos (2 x-\pi)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:34

Problem 57

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=-\frac{1}{4} \cos \left(\frac{1}{4} x-\frac{\pi}{2}\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:40

Problem 58

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=\frac{1}{2} \sin \left(\frac{1}{3} x+\pi\right)$$

Robert Leedy
Robert Leedy
Numerade Educator
00:41

Problem 59

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=2 \cos \left[\frac{\pi}{2}(x-4)\right]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:55

Problem 60

State the amplitude, period, and phase shift (including direction) of the given function and graph.
$$y=-5 \sin [-\pi(x+1)]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:55

Problem 61

Sketch the graph of the function over the indicated interval.
$$y=\frac{1}{2}+\frac{3}{2} \cos (2 x+\pi),\left[-\frac{3 \pi}{2}, \frac{3 \pi}{2}\right]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:48

Problem 62

Sketch the graph of the function over the indicated interval.
$$y=\frac{1}{3}+\frac{2}{3} \sin (2 x-\pi),\left[-\frac{3 \pi}{2}, \frac{3 \pi}{2}\right]$$

Robert Leedy
Robert Leedy
Numerade Educator
01:23

Problem 63

Sketch the graph of the function over the indicated interval.
$$y=\frac{1}{2}-\frac{1}{2} \sin \left(\frac{1}{2} x-\frac{\pi}{4}\right),\left[-\frac{7 \pi}{2}, \frac{9 \pi}{2}\right]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:46

Problem 64

Sketch the graph of the function over the indicated interval.
$$y=-\frac{1}{2}+\frac{1}{2} \cos \left(\frac{1}{2} x+\frac{\pi}{4}\right),\left[-\frac{9 \pi}{2}, \frac{7 \pi}{2}\right]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:48

Problem 65

Sketch the graph of the function over the indicated interval.
$$y=-3+4 \sin [\pi(x-2)],[0,4]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:46

Problem 66

Sketch the graph of the function over the indicated interval.
$$y=4-3 \cos [\pi(x+1)],[-1,3]$$

Robert Leedy
Robert Leedy
Numerade Educator
00:11

Problem 67

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=2 x-\cos (\pi x), 0 \leq x \leq 4$$

Robert Leedy
Robert Leedy
Numerade Educator
00:10

Problem 68

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=3 x-2 \cos (\pi x), 0 \leq x \leq 4$$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 69

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\frac{1}{3} x+2 \cos (2 x), 0 \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:13

Problem 70

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\frac{1}{4} x+3 \cos \left(\frac{x}{2}\right), 0 \leq x \leq 4 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:12

Problem 71

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=x-\cos \left(\frac{3 \pi}{2} x\right), 0 \leq x \leq 6$$

Robert Leedy
Robert Leedy
Numerade Educator
00:11

Problem 72

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=-2 x+2 \sin \left(\frac{\pi}{2} x\right),-2 \leq x \leq 2$$

Robert Leedy
Robert Leedy
Numerade Educator
00:18

Problem 73

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\frac{1}{4} x-\frac{1}{2} \cos [\pi(x-1)], 2 \leq x \leq 6$$

Robert Leedy
Robert Leedy
Numerade Educator
00:17

Problem 74

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=-\frac{1}{3} x+\frac{1}{3} \sin \left[\frac{\pi}{6}(x+2)\right],-2 \leq x \leq 10$$

Robert Leedy
Robert Leedy
Numerade Educator
00:09

Problem 75

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\sin x-\cos x, 0 \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:13

Problem 76

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\cos x-\sin x, 0 \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:11

Problem 77

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=3 \cos x+\sin x, 0 \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 78

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=3 \sin x-\cos x, 0 \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:12

Problem 79

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=4 \cos x-\sin (2 x), 0 \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:12

Problem 80

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\frac{1}{2} \sin x+2 \cos (4 x),-\pi \leq x \leq \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:16

Problem 81

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=2 \sin [\pi(x-1)]-2 \cos [\pi(x+1)],-1 \leq x \leq 2$$

Robert Leedy
Robert Leedy
Numerade Educator
00:17

Problem 82

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\sin \left[\frac{\pi}{4}(x+2)\right]+3 \cos \left[\frac{3 \pi}{3}(x-1)\right], 1 \leq x \leq 5$$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 83

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\cos \left(\frac{x}{2}\right)+\cos (2 x), 0 \leq x \leq 4 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:11

Problem 84

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\sin (2 x)+\sin (3 x),-\pi \leq x \leq \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:11

Problem 85

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=\sin \left(\frac{x}{2}\right)+\sin (2 x), 0 \leq x \leq 4 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:16

Problem 86

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=-\sin \left(\frac{\pi}{4} x\right)-3 \sin \left(\frac{5 \pi}{4} x\right), 0 \leq x \leq 4$$

Robert Leedy
Robert Leedy
Numerade Educator
00:16

Problem 87

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=-\frac{1}{3} \sin \left(\frac{\pi}{6} x\right)+\frac{2}{3} \sin \left(\frac{5 \pi}{6} x\right), 0 \leq x \leq 3$$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 88

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=8 \cos x-6 \cos \left(\frac{1}{2} x\right),-2 \pi \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:16

Problem 89

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=-\frac{1}{4} \cos \left(\frac{\pi}{6} x\right)-\frac{1}{2} \cos \left(\frac{\pi}{3} x\right), 0 \leq x \leq 12$$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 90

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=2 \cos \left(\frac{3}{2} x\right)-\cos \left(\frac{1}{2} x\right),-2 \pi \leq x \leq 2 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 91

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=2 \sin \left(\frac{x}{2}\right)-\cos (2 x), 0 \leq x \leq 4 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:12

Problem 92

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=2 \cos \left(\frac{x}{2}\right)+\sin (2 x), 0 \leq x \leq 4 \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:18

Problem 93

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=2 \sin [\pi(x-1)]+3 \sin \left[2 \pi\left(x+\frac{1}{2}\right)\right],-2 \leq x \leq 2$$

Robert Leedy
Robert Leedy
Numerade Educator
00:19

Problem 94

Add the ordinates of the individual functions to graph each summed function on the indicated interval.
$$y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)-2 \cos \left(x-\frac{\pi}{6}\right),-\pi \leq x \leq \pi$$

Robert Leedy
Robert Leedy
Numerade Educator
00:24

Problem 95

Refer to the following:
An analysis of demand $d$ for widgets manufactured by WidgetsRUs (measured in thousands of units per week) indicates that demand can be modeled by the graph below, where $t$ is time in months since January 2010 (note that $t=0$ corresponds to January 2010 ). (GRAPH CAN'T COPY)
Business. Find the amplitude of the graph.

Robert Leedy
Robert Leedy
Numerade Educator
00:18

Problem 96

Refer to the following:
An analysis of demand $d$ for widgets manufactured by WidgetsRUs (measured in thousands of units per week) indicates that demand can be modeled by the graph below, where $t$ is time in months since January 2010 (note that $t=0$ corresponds to January 2010 ). (GRAPH CAN'T COPY)
Business. Find the period of the graph.

Robert Leedy
Robert Leedy
Numerade Educator
00:34

Problem 97

Refer to the following:
Researchers have been monitoring oxygen levels (milligrams per liter) in the water of a lake and have found that the oxygen levels fluctuate with an eight-week period. The following tables illustrate data from eight weeks.
Environment. Find the amplitude of the oxygen level fluctuations.
$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|}\hline \text { Week: } t & \begin{array}{c}0 \text { (initial } \\\text { measurement } )\end{array} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\hline \begin{array}{l}\text { Oxygen } \\\text { levels: } \\\mathrm{mg} / \mathrm{L}\end{array} & 7 & 7.7 & 8 & 7.7 & 7 & 6.3 & 6 & 6.3 & 7\\\hline\end{array}$$

Robert Leedy
Robert Leedy
Numerade Educator
00:25

Problem 98

Refer to the following:
Researchers have been monitoring oxygen levels (milligrams per liter) in the water of a lake and have found that the oxygen levels fluctuate with an eight-week period. The following tables illustrate data from eight weeks.
Environment. Find the amplitude of the oxygen level fluctuations.
$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|}\hline \text { Week: } t & \begin{array}{c}0 \text { (initial } \\\text { measurement } )\end{array} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\hline \begin{array}{l}\text { Oxygen } \\\text { levels: } \\\mathrm{mg} / \mathrm{L}\end{array} & 7 & 8.4 & 9 & 8.4 & 7 & 5.6 & 5 & 5.6&7\\\hline\end{array}$$

Robert Leedy
Robert Leedy
Numerade Educator
00:48

Problem 99

Refer to the following:
A weight hanging on a spring will oscillate up and down about its equilibrium position after it is pulled down and released. This is an example of simple harmonic motion. This motion would continue forever if there were not any friction or air resistance. Simple harmonic motion can be described with the function $y=A \cos (t \sqrt{\frac{k}{m}}),$ where $|A|$ is the amplitude, $t$ is the time in seconds, $m$ is the mass of the weight, and $k$ is a constant particular to the spring. (FIGURE CAN'T COPY)
Simple Harmonic Motion. If the height of the spring is measured in centimeters and the mass in grams, then what are the amplitude and mass if $y=4 \cos \left(\frac{t \sqrt{k}}{2}\right) ?$

Robert Leedy
Robert Leedy
Numerade Educator
01:28

Problem 100

Refer to the following:
A weight hanging on a spring will oscillate up and down about its equilibrium position after it is pulled down and released. This is an example of simple harmonic motion. This motion would continue forever if there were not any friction or air resistance. Simple harmonic motion can be described with the function $y=A \cos (t \sqrt{\frac{k}{m}}),$ where $|A|$ is the amplitude, $t$ is the time in seconds, $m$ is the mass of the weight, and $k$ is a constant particular to the spring. (FIGURE CAN'T COPY)
Simple Harmonic Motion. If a spring is measured in centimeters and the mass in grams, then what are the amplitude and mass if $y=3 \cos (3 t \sqrt{k}) ?$

Robert Leedy
Robert Leedy
Numerade Educator
00:40

Problem 101

Refer to the following:
A weight hanging on a spring will oscillate up and down about its equilibrium position after it is pulled down and released. This is an example of simple harmonic motion. This motion would continue forever if there were not any friction or air resistance. Simple harmonic motion can be described with the function $y=A \cos (t \sqrt{\frac{k}{m}}),$ where $|A|$ is the amplitude, $t$ is the time in seconds, $m$ is the mass of the weight, and $k$ is a constant particular to the spring. (FIGURE CAN'T COPY)
Frequency of Oscillations. The frequency of the oscillations in cycles per second is determined by $f=\frac{1}{p},$ where $p$ is the period. What is the frequency for the oscillation modeled by $y=3 \cos \left(\frac{t}{2}\right) ?$

Robert Leedy
Robert Leedy
Numerade Educator
00:32

Problem 102

Refer to the following:
A weight hanging on a spring will oscillate up and down about its equilibrium position after it is pulled down and released. This is an example of simple harmonic motion. This motion would continue forever if there were not any friction or air resistance. Simple harmonic motion can be described with the function $y=A \cos (t \sqrt{\frac{k}{m}}),$ where $|A|$ is the amplitude, $t$ is the time in seconds, $m$ is the mass of the weight, and $k$ is a constant particular to the spring. (FIGURE CAN'T COPY)
Frequency of Oscillations. The frequency of the oscillations
$f$ is given by $f=\frac{1}{p},$ where $p$ is the period. What is the frequency of oscillation modeled by $y=3.5 \cos (3 t) ?$

Robert Leedy
Robert Leedy
Numerade Educator
00:52

Problem 103

Sound Waves. A pure tone created by a vibrating tuning fork shows up as a sine wave on an oscilloscope's screen. A tuning fork vibrating at 256 hertz (Hz) gives the tone middle $\mathrm{C}$ and can have the equation $y=0.005 \sin [(2 \pi)(256 t)]$ where the amplitude is in centimeters (cm) and the time $t$ in seconds. What are the amplitude and frequency of the wave where the frequency is $\frac{1}{p}$ in cycles per second? Note: 1 hertz $=1$ cycle per second.

Robert Leedy
Robert Leedy
Numerade Educator
00:37

Problem 104

Sound Waves. A pure tone created by a vibrating tuning fork shows up as a sine wave on an oscilloscope's screen. A tuning fork vibrating at 288 hertz gives the tone $D$ and can have the equation $y=0.005 \sin [(2 \pi)(288 t)],$ where the amplitude is in centimeters $(\mathrm{cm})$ and the time $t$ in seconds. What are the amplitude and frequency of the wave where the frequency is $\frac{1}{p}$ in cycles per second?

Robert Leedy
Robert Leedy
Numerade Educator
00:43

Problem 105

Sound Waves. If a sound wave is represented by $y=0.008 \sin (750 \pi t) \mathrm{cm},$ what are its amplitude and frequency? See Exercise 103

Robert Leedy
Robert Leedy
Numerade Educator
00:35

Problem 106

Sound Waves. If a sound wave is represented by $y=0.006 \cos (1000 \pi t) \mathrm{cm},$ what are its amplitude and frequency? See Exercise 103

Robert Leedy
Robert Leedy
Numerade Educator
00:26

Problem 107

Refer to the following:
When an airplane flies faster than the speed of sound, the sound waves that are formed take on a cone shape, and where the cone hits the ground, a sonic boom is heard. If $\theta$ is the angle of the vertex of the cone, then $\sin \left(\frac{\theta}{2}\right)=\frac{330 \mathrm{m} / \mathrm{sec}}{V}=\frac{1}{M},$ where $V$ is the speed of the plane and $M$ is the Mach number. (FIGURE CAN'T COPY)
Sonic Booms. What is the speed of the plane if the plane is flying at Mach $2 ?$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 108

Refer to the following:
When an airplane flies faster than the speed of sound, the sound waves that are formed take on a cone shape, and where the cone hits the ground, a sonic boom is heard. If $\theta$ is the angle of the vertex of the cone, then $\sin \left(\frac{\theta}{2}\right)=\frac{330 \mathrm{m} / \mathrm{sec}}{V}=\frac{1}{M},$ where $V$ is the speed of the plane and $M$ is the Mach number. (FIGURE CAN'T COPY)
Sonic Booms. What is the Mach number if the plane is flying at 990 meters per second?

Robert Leedy
Robert Leedy
Numerade Educator
00:27

Problem 109

Refer to the following:
When an airplane flies faster than the speed of sound, the sound waves that are formed take on a cone shape, and where the cone hits the ground, a sonic boom is heard. If $\theta$ is the angle of the vertex of the cone, then $\sin \left(\frac{\theta}{2}\right)=\frac{330 \mathrm{m} / \mathrm{sec}}{V}=\frac{1}{M},$ where $V$ is the speed of the plane and $M$ is the Mach number. (FIGURE CAN'T COPY)
Sonic Booms. What is the speed of the plane if the cone angle is $60^{\circ} ?$

Robert Leedy
Robert Leedy
Numerade Educator
00:38

Problem 110

Refer to the following:
When an airplane flies faster than the speed of sound, the sound waves that are formed take on a cone shape, and where the cone hits the ground, a sonic boom is heard. If $\theta$ is the angle of the vertex of the cone, then $\sin \left(\frac{\theta}{2}\right)=\frac{330 \mathrm{m} / \mathrm{sec}}{V}=\frac{1}{M},$ where $V$ is the speed of the plane and $M$ is the Mach number. (FIGURE CAN'T COPY)
Sonic Booms. What is the speed of the plane if the cone angle is $30^{\circ} ?$

Robert Leedy
Robert Leedy
Numerade Educator
00:36

Problem 111

Refer to the following:
With the advent of summer come fireflies. They are intriguing because they emit a flashing luminescence that beckons their mate to them. It is known that the speed and intensity of the flashing are related to the temperature- -the higher the temperature, the quicker and more intense the flashing becomes. If you ever watch a single firefly, you will see that the intensity of the flashing is periodic with time. The intensity of light emitted is measured in candelas per square meter (of firefly). To give an idea of this unit of measure, the intensity of a picture on a typical TV screen is about 450 candelas per square meter. The measurement for the intensity of the light emitted by a typical firefly at its brightest moment is about 50 candelas per square meter. Assume that a typical cycle of this flashing is 4 seconds and that the intensity is essentially zero candelas at the beginning and ending of a cycle.
Bioluminescence in Fireflies. Find an equation that describes this flashing. What is the intensity of the flashing at 4 minutes?

Robert Leedy
Robert Leedy
Numerade Educator
00:15

Problem 112

Refer to the following:
With the advent of summer come fireflies. They are intriguing because they emit a flashing luminescence that beckons their mate to them. It is known that the speed and intensity of the flashing are related to the temperature- -the higher the temperature, the quicker and more intense the flashing becomes. If you ever watch a single firefly, you will see that the intensity of the flashing is periodic with time. The intensity of light emitted is measured in candelas per square meter (of firefly). To give an idea of this unit of measure, the intensity of a picture on a typical TV screen is about 450 candelas per square meter. The measurement for the intensity of the light emitted by a typical firefly at its brightest moment is about 50 candelas per square meter. Assume that a typical cycle of this flashing is 4 seconds and that the intensity is essentially zero candelas at the beginning and ending of a cycle.
Bioluminescence in Fireflies. Graph the equation from Exercise 111 for a period of 30 seconds.

Robert Leedy
Robert Leedy
Numerade Educator
00:24

Problem 113

Explain the mistake that is made. (GRAPH CAN'T COPY)
Graph the function $y=-2 \cos x$
Solution:
Find the amplitude.
The graph of $y=-2 \cos x$ is 0 similar to the graph of $y=\cos x$ with amplitude 2 $\frac{1}{-2 \pi}$

Robert Leedy
Robert Leedy
Numerade Educator
00:31

Problem 114

Explain the mistake that is made. (GRAPH CAN'T COP)
Graph the function $y=-\sin (2 x)$
Solution:
Make a table with values.
$$\begin{array}{|l|l|l|}\hline x & y=-\sin (2 x) & (x, y) \\\hline 0 & y=-\sin 0=0 & (0,0) \\\hline \frac{\pi}{2} & y=-\sin \pi=0 & (0,0) \\\hline \pi & y=-\sin (2 \pi)=0 & (0,0) \\\hline \frac{3 \pi}{2} & y=-\sin (3 \pi)=0 & (0,0) \\\hline 2 \pi & y=-\sin (4 \pi)=0 &(0,0)\\\hline\end{array}$$
Graph the function by plotting these points and connecting them with a
sinusoidal curve. This is incorrect. What mistake was made?

Robert Leedy
Robert Leedy
Numerade Educator
00:28

Problem 115

Determine whether each statement is true or false. ( $A$ and $B$ are positive real numbers.)
The graph of $y=-A \cos (B x)$ is the graph of $y=A \cos (B x)$ reflected about the $x$ -axis.

Robert Leedy
Robert Leedy
Numerade Educator
00:13

Problem 116

Determine whether each statement is true or false. ( $A$ and $B$ are positive real numbers.)
The graph of $y=A \sin (-B x)$ is the graph of $y=A \sin (B x)$ reflected about the $x$ -axis.

Robert Leedy
Robert Leedy
Numerade Educator
00:24

Problem 117

Determine whether each statement is true or false. ( $A$ and $B$ are positive real numbers.)
The graph of $y=-A \cos (-B x)$ is the graph of $y=A \cos (B x)$

Robert Leedy
Robert Leedy
Numerade Educator
00:19

Problem 118

Determine whether each statement is true or false. ( $A$ and $B$ are positive real numbers.)
The graph of $y=-A \sin (-B x)$ is the graph of $y=A \sin (B x)$

Robert Leedy
Robert Leedy
Numerade Educator
00:14

Problem 119

In Exercises $119-122, A$ and $B$ are positive real numbers.
Find the $y$ -intercept of the function $y=A \cos (B x)$

Robert Leedy
Robert Leedy
Numerade Educator
00:13

Problem 120

A and B are positive real numbers.
Find the $y$ -intercept of the function $y=A \sin (B x)$

Robert Leedy
Robert Leedy
Numerade Educator
00:38

Problem 121

A and B are positive real numbers.
Find the $x$ -intercepts of the function $y=A \sin (B x)$

Robert Leedy
Robert Leedy
Numerade Educator
00:43

Problem 122

A and B are positive real numbers.
Find the $x$ -intercepts of the function $y=A \cos (B x)$

Robert Leedy
Robert Leedy
Numerade Educator
00:46

Problem 123

Find the $y$ -intercept of $y=-A \sin \left(B x+\frac{\pi}{6}\right)$

Robert Leedy
Robert Leedy
Numerade Educator
00:26

Problem 124

Find the $y$ -intercept of $y=A \cos (B x-\pi)+C$

Robert Leedy
Robert Leedy
Numerade Educator
00:43

Problem 125

Find the $x$ -intercept(s) of $y=A \sin (B x)+A$

Robert Leedy
Robert Leedy
Numerade Educator
00:37

Problem 126

Find an expression involving $C$ and $A$ that describes the values of $C$ for which the graph of $y=A \cos (B x)+C$ does not cross the $x$ -axis. (Assume that $A>0 .$ )

Robert Leedy
Robert Leedy
Numerade Educator
00:42

Problem 127

What is the range of $y=2 A \sin (B x+C)-\frac{A}{2} ?$

Robert Leedy
Robert Leedy
Numerade Educator
00:13

Problem 128

Can the $y$ -coordinate of a point on the graph of $y=A \sin (B x)+3 A \cos \left(\frac{B}{2} x\right)$ exceed $4 A ?$ Explain.
(Assume that $A>0 .)$

Robert Leedy
Robert Leedy
Numerade Educator
00:11

Problem 129

Use a graphing calculator to graph $Y_{1}=5 \sin x$ and $Y_{2}=\sin (5 x) .$ Is the following statement true based on what you see? $y=\sin (c x)$ has the same graph as $y=c \sin x$

Robert Leedy
Robert Leedy
Numerade Educator
00:10

Problem 130

Use a graphing calculator to graph $Y_{1}=3 \cos x$ and $Y_{2}=\cos (3 x) .$ Is the following statement true based on what you see? $y=\cos (c x)$ has the same graph as $y=c \cos x$

Robert Leedy
Robert Leedy
Numerade Educator
00:17

Problem 131

Use a graphing calculator to graph $Y_{1}=\sin x$ and $Y_{2}=\cos \left(x-\frac{\pi}{2}\right) .$ What do you notice?

Robert Leedy
Robert Leedy
Numerade Educator
00:17

Problem 132

Use a graphing calculator to graph $Y_{1}=\cos x$ and $Y_{2}=\sin \left(x+\frac{\pi}{2}\right) .$ What do you notice?

Robert Leedy
Robert Leedy
Numerade Educator
00:28

Problem 133

Use a graphing calculator to graph $Y_{1}=\cos x$ and $Y_{2}=\cos (x+c),$ where
a. $c=\frac{\pi}{3},$ and explain the relationship between $Y_{2}$ and $Y_{1}$
b. $c=-\frac{\pi}{3},$ and explain the relationship between $Y_{2}$ and $Y_{1}$

Robert Leedy
Robert Leedy
Numerade Educator
00:29

Problem 134

Use a graphing calculator to graph $Y_{1}=\sin x$ and $Y_{2}=\sin (x+c),$ where
a. $c=\frac{\pi}{3},$ and explain the relationship between $Y_{2}$ and $Y_{1}$
b. $c=-\frac{\pi}{3},$ and explain the relationship between $Y_{2}$ and $Y_{1}$

Robert Leedy
Robert Leedy
Numerade Educator
00:32

Problem 135

Refer to the following:
Damped oscillatory motion, or damped oscillation, occurs when things in oscillatory motion experience friction or resistance. The friction causes the amplitude to decrease as a function of time. Mathematically, we can use a negative exponential function to damp the oscillations in the form of
$$f(t)=e^{-t} \sin t$$
Damped Oscillation. Graph the functions $Y_{1}=e^{-t}$ $Y_{2}=\sin t,$ and $Y_{3}=e^{-t} \sin t$ in the same viewing window (let $t$ range from 0 to $2 \pi$ ). What happens as $t$ increases?

Robert Leedy
Robert Leedy
Numerade Educator
00:34

Problem 136

Refer to the following:
Damped oscillatory motion, or damped oscillation, occurs when things in oscillatory motion experience friction or resistance. The friction causes the amplitude to decrease as a function of time. Mathematically, we can use a negative exponential function to damp the oscillations in the form of $$f(t)=e^{-t} \sin t$$
Damped Oscillation. Graph the functions $Y_{1}=e^{-t}$ $Y_{2}=\sin t,$ and $Y_{3}=e^{-t} \sin t$ in the same viewing window (let $t$ range from 0 to $2 \pi$ ). What happens as $t$ increases?

Robert Leedy
Robert Leedy
Numerade Educator
00:45

Problem 137

Use a graphing calculator to graph $Y_{1}=\sin x$ and $Y_{2}=\sin x+c,$ where
a. $c=1,$ and explain the relationship between $Y_{2}$ and $Y_{1}$
b. $c=-1,$ and explain the relationship between $Y_{2}$ and $Y_{1}$

Robert Leedy
Robert Leedy
Numerade Educator
00:40

Problem 138

Use a graphing calculator to graph $Y_{1}=\cos x$ and $Y_{2}=\cos x+c,$ where
a. $c=\frac{1}{2},$ and explain the relationship between $Y_{2}$ and $Y_{1}$
b. $c=-\frac{1}{2},$ and explain the relationship between $Y_{2}$ and $Y_{1}$

Robert Leedy
Robert Leedy
Numerade Educator
00:41

Problem 139

What is the amplitude of the function $y=3 \cos x+4 \sin x ?$ Use a graphing calculator to graph $Y_{1}=3 \cos x, Y_{2}=4 \sin x$ and $Y_{3}=3 \cos x+4 \sin x$ in the same viewing window.

Robert Leedy
Robert Leedy
Numerade Educator
00:34

Problem 140

What is the amplitude of the function $y=\sqrt{3} \cos x-\sin x ?$ Use a graphing calculator to graph $Y_{1}=\sqrt{3} \cos x, Y_{2}=\sin x$
and $Y_{3}=\sqrt{3} \cos x-\sin x$ in the same viewing window.

Robert Leedy
Robert Leedy
Numerade Educator
00:50

Problem 141

First shade the area corresponding to the definite integral and then use the information below to find the exact value of the area.
$$\begin{array}{|l|c|c|}\hline \text { Function } & \sin x & \cos x \\\hline \text { ANTIDERIVATIVE } & \cos x & \sin x \\\hline \end{array}$$
$$\int_{0}^{\pi} \sin x d x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:48

Problem 142

First shade the area corresponding to the definite integral and then use the information below to find the exact value of the area.
$$\begin{array}{|l|c|c|}\hline \text { Function } & \sin x & \cos x \\\hline \text { ANTIDERIVATIVE } & \cos x & \sin x \\\hline \end{array}$$
$$\int_{-\pi / 2}^{\pi / 2} \cos x d x$$

Robert Leedy
Robert Leedy
Numerade Educator
00:57

Problem 143

First shade the area corresponding to the definite integral and then use the information below to find the exact value of the area.
$$\begin{array}{|l|c|c|}\hline \text { Function } & \sin x & \cos x \\\hline \text { ANTIDERIVATIVE } & \cos x & \sin x \\\hline \end{array}$$
$$\int_{0}^{\pi / 2} \cos x d x$$

Robert Leedy
Robert Leedy
Numerade Educator
01:11

Problem 144

First shade the area corresponding to the definite integral and then use the information below to find the exact value of the area.
$$\begin{array}{|l|c|c|}\hline \text { Function } & \sin x & \cos x \\\hline \text { ANTIDERIVATIVE } & \cos x & \sin x \\\hline \end{array}$$
$$\int_{0}^{\pi / 2} \sin x d x$$

Robert Leedy
Robert Leedy
Numerade Educator