Chapter 8, Analytic Trigonometry Video Solutions, Algebra and Trigonometry

Section 1

The Inverse sine, cosine, and Tangent Functions

00:42

Problem 1

What are the domain and the range of $y=\sin x ?$

Sarah L.

Numerade Educator

00:37

Problem 2

If the domain of a one-to-one function is $[3, \infty),$ the range of its inverse is _________

Sarah L.

Numerade Educator

00:47

Problem 3

True or False The graph of $y=\cos x$ is decreasing on the interval $[0, \pi]$

Sarah L.

Numerade Educator

02:30

Problem 4

$\tan \frac{\pi}{4}=\underline{;} \sin \frac{\pi}{3}=\frac{\quad}{3} \quad \sin \left(-\frac{\pi}{6}\right)=\underline{ }$
$\cos \pi=$_________

Tanishq G.

Numerade Educator

01:15

Problem 5

$y=\sin ^{-1} x$ if and only if________ where $-1 \leq x \leq 1$
and $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

Tanishq G.

Numerade Educator

00:51

Problem 6

$\cos ^{-1}(\cos x)=x$ for all numbers $x$ for which____________

Tanishq G.

Numerade Educator

01:03

Problem 7

True or False The domain of $y=\cos ^{-1} x$ is $-1 \leq x \leq 1$

Tanishq G.

Numerade Educator

02:14

Problem 8

True or False $\sin \left(\sin ^{-1} 0\right)=0$ and $\cos \left(\cos ^{-1} 0\right)=0$

Shaza H.

Numerade Educator

01:15

Problem 9

True or False $y=\tan ^{-1} x$ if and only if $x=\tan y$ where $-\infty<x<\infty$ and $-\frac{\pi}{2}<y<\frac{\pi}{2}$

Tanishq G.

Numerade Educator

00:55

Problem 10

Multiple Choice $\sin ^{-1}(\sin x)=x$ for all numbers $x$ for which
(a) $-\infty<x<\infty$
(b) $0 \leq x \leq \pi$
(c) $-1 \leq x \leq 1$
(d) $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

Tanishq G.

Numerade Educator

01:13

Problem 11

In Problems 11-26, find the exact value of each expression.
$\sin ^{-1} 0$

Tanishq G.

Numerade Educator

00:35

Problem 12

Find the exact value of each expression.
$\cos ^{-1} 1$

Ankit G.

Numerade Educator

00:51

Problem 13

Find the exact value of each expression.
$\sin ^{-1}(-1)$

Sarah L.

Numerade Educator

00:35

Problem 14

Find the exact value of each expression.
$\cos ^{-1}(-1)$

Ankit G.

Numerade Educator

00:35

Problem 15

Find the exact value of each expression.
$\tan ^{-1} 0$

Ankit G.

Numerade Educator

00:39

Problem 16

Find the exact value of each expression.
$\tan ^{-1}(-1)$

Ankit G.

Numerade Educator

00:48

Problem 17

Find the exact value of each expression.
$\sin ^{-1} \frac{\sqrt{2}}{2}$

Ankit G.

Numerade Educator

Problem 18

Find the exact value of each expression.
$\tan ^{-1} \frac{\sqrt{3}}{3}$

Check back soon!

00:47

Problem 19

Find the exact value of each expression.
$\tan ^{-1} \sqrt{3}$

Ankit G.

Numerade Educator

01:55

Problem 20

Find the exact value of each expression.
$\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

Kerri M.

Numerade Educator

00:30

Problem 21

Find the exact value of each expression.
$\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

Fuzail S.

Numerade Educator

00:48

Problem 22

Find the exact value of each expression.
$\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

Ankit G.

Numerade Educator

01:00

Problem 23

Find the exact value of each expression.
$\cos ^{-1} \frac{\sqrt{2}}{2}$

Ankit G.

Numerade Educator

01:03

Problem 24

Find the exact value of each expression.
$\cos ^{-1}\left(-\frac{1}{2}\right)$

Kerri M.

Numerade Educator

Problem 25

Find the exact value of each expression.
$\tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)$

Check back soon!

00:33

Problem 26

Find the exact value of each expression.
$\sin ^{-1} \frac{1}{2}$

Fuzail S.

Numerade Educator

00:38

Problem 27

In Problems $27-38$, use a calculator to find the approximate value of each expression rounded to two decimal places.
$\sin ^{-1} 0.1$

Tanishq G.

Numerade Educator

00:34

Problem 28

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\cos ^{-1} 0.6$

Tanishq G.

Numerade Educator

00:32

Problem 29

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\tan ^{-1} 5$

Tanishq G.

Numerade Educator

00:36

Problem 30

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\tan ^{-1} 0.2$

Tanishq G.

Numerade Educator

00:39

Problem 31

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\cos ^{-1} \frac{7}{8}$

Tanishq G.

Numerade Educator

00:44

Problem 32

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\sin ^{-1} \frac{1}{8}$

Tanishq G.

Numerade Educator

00:37

Problem 33

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\tan ^{-1}(-0.4)$

Tanishq G.

Numerade Educator

00:32

Problem 34

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\tan ^{-1}(-3)$

Tanishq G.

Numerade Educator

00:37

Problem 35

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\sin ^{-1}(-0.12)$

Tanishq G.

Numerade Educator

00:36

Problem 36

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\cos ^{-1}(-0.44)$

Tanishq G.

Numerade Educator

00:34

Problem 37

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\cos ^{-1} \frac{\sqrt{2}}{3}$

Tanishq G.

Numerade Educator

00:34

Problem 38

Use a calculator to find the approximate value of each expression rounded to two decimal places.
$\sin ^{-1} \frac{\sqrt{3}}{5}$

Tanishq G.

Numerade Educator

01:01

Problem 39

In Problems 39-58, find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\cos ^{-1}\left(\cos \frac{4 \pi}{5}\right)$

Tanishq G.

Numerade Educator

00:55

Problem 40

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\sin ^{-1}\left[\sin \left(-\frac{\pi}{10}\right)\right]$

Tanishq G.

Numerade Educator

00:51

Problem 41

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan ^{-1}\left[\tan \left(-\frac{3 \pi}{8}\right)\right]$

Tanishq G.

Numerade Educator

00:54

Problem 42

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\sin ^{-1}\left[\sin \left(-\frac{3 \pi}{7}\right)\right]$

Tanishq G.

Numerade Educator

01:56

Problem 43

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\sin ^{-1}\left(\sin \frac{9 \pi}{8}\right)$

Tanishq G.

Numerade Educator

01:33

Problem 44

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\cos ^{-1}\left[\cos \left(-\frac{5 \pi}{3}\right)\right]$

Tanishq G.

Numerade Educator

01:21

Problem 45

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan ^{-1}\left(\tan \frac{4 \pi}{5}\right)$

Tanishq G.

Numerade Educator

01:58

Problem 46

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan ^{-1}\left[\tan \left(-\frac{2 \pi}{3}\right)\right]$

Tanishq G.

Numerade Educator

00:58

Problem 47

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\cos ^{-1}\left[\cos \left(-\frac{\pi}{4}\right)\right]$

Tanishq G.

Numerade Educator

01:46

Problem 48

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\sin ^{-1}\left[\sin \left(-\frac{3 \pi}{4}\right)\right]$

Tanishq G.

Numerade Educator

00:55

Problem 49

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan ^{-1}\left[\tan \left(\frac{\pi}{2}\right)\right]$

Tanishq G.

Numerade Educator

01:52

Problem 50

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan ^{-1}\left[\tan \left(-\frac{3 \pi}{2}\right)\right]$

Tanishq G.

Numerade Educator

00:47

Problem 51

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\sin \left(\sin ^{-1} \frac{1}{4}\right)$

Tanishq G.

Numerade Educator

00:53

Problem 52

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\cos \left[\cos ^{-1}\left(-\frac{2}{3}\right)\right]$

Tanishq G.

Numerade Educator

00:45

Problem 52

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan \left(\tan ^{-1} 4\right)$

Tanishq G.

Numerade Educator

00:47

Problem 54

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan \left[\tan ^{-1}(-2)\right]$

Tanishq G.

Numerade Educator

01:11

Problem 55

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\cos \left(\cos ^{-1} 1.2\right)$

Tanishq G.

Numerade Educator

01:22

Problem 56

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\sin \left[\sin ^{-1}(-2)\right]$

Tanishq G.

Numerade Educator

00:50

Problem 57

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\tan \left(\tan ^{-1} \pi\right)$

Tanishq G.

Numerade Educator

01:15

Problem 58

Find the exact value, if any, of each composite function. If there is no value, state it is "not defined." Do not use a calculator.
$\sin \left[\sin ^{-1}(-1.5)\right]$

Tanishq G.

Numerade Educator

03:31

Problem 59

In Problems 59-66, find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=5 \sin x+2 ;-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

Tanishq G.

Numerade Educator

02:52

Problem 60

Find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=2 \tan x-3 ;-\frac{\pi}{2}<x<\frac{\pi}{2}$

Tanishq G.

Numerade Educator

03:01

Problem 61

Find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=-2 \cos (3 x) ; 0 \leq x \leq \frac{\pi}{3}$

Tanishq G.

Numerade Educator

02:55

Problem 62

Find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=3 \sin (2 x) ;-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$

Tanishq G.

Numerade Educator

02:26

Problem 63

Find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=-\tan (x+1)-3 ;-1-\frac{\pi}{2}<x<\frac{\pi}{2}-1$

Tanishq G.

Numerade Educator

02:23

Problem 64

Find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=\cos (x+2)+1 ;-2 \leq x \leq \pi-2$

Tanishq G.

Numerade Educator

02:42

Find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=3 \sin (2 x+1) ;-\frac{1}{2}-\frac{\pi}{4} \leq x \leq-\frac{1}{2}+\frac{\pi}{4}$

Tanishq G.

Numerade Educator

02:52

Problem 66

Find the inverse function $f^{-1}$ of each function $f$. Find the range of f and the domain and range of $f^{-1}$.
$f(x)=2 \cos (3 x+2) ;-\frac{2}{3} \leq x \leq-\frac{2}{3}+\frac{\pi}{3}$

Tanishq G.

Numerade Educator

00:59

Problem 67

In Problems 67-74, find the exact solution of each equation.
$4 \sin ^{-1} x=\pi$

Tanishq G.

Numerade Educator

00:35

Problem 68

Find the exact solution of each equation.
$2 \cos ^{-1} x=\pi$

Sarah L.

Numerade Educator

00:47

Problem 69

Find the exact solution of each equation.
$3 \cos ^{-1}(2 x)=2 \pi$

Sarah L.

Numerade Educator

00:50

Problem 70

Find the exact solution of each equation.
$-6 \sin ^{-1}(3 x)=\pi$

Sarah L.

Numerade Educator

00:38

Problem 71

Find the exact solution of each equation.
$3 \tan ^{-1} x=\pi$

Sarah L.

Numerade Educator

00:48

Problem 72

Find the exact solution of each equation.
$-4 \tan ^{-1} x=\pi$

Sarah L.

Numerade Educator

01:23

Problem 73

Find the exact solution of each equation.
$4 \cos ^{-1} x-2 \pi=2 \cos ^{-1} x$

Sarah L.

Numerade Educator

01:45

Problem 74

Find the exact solution of each equation.
$5 \sin ^{-1} x-2 \pi=2 \sin ^{-1} x-3 \pi$

Sarah L.

Numerade Educator

02:12

Problem 75

In Problems $75-80,$ use the following discussion. The formula
$$
D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right]
$$
Approximate the number of hours of daylight in Houston, Texas $\left(29^{\circ} 45^{\prime}\right.$ north latitude $)$, for the following dates:
(a) Summer solstice $\left(i=23.5^{\circ}\right)$
(b) Vernal equinox $\left(i=0^{\circ}\right)$
(c) July $4\left(i=22^{\circ} 48^{\prime}\right)$

Tanishq G.

Numerade Educator

Problem 76

Use the following discussion. The formula
$$
D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right]
$$
Approximate the number of hours of daylight in New York, New York $\left(40^{\circ} 45^{\prime}\right.$ north latitude $)$, for the following dates:
(a) Summer solstice $\left(i=23.5^{\circ}\right)$
(b) Vernal equinox $\left(i=0^{\circ}\right)$
(c) July $4\left(i=22^{\circ} 48^{\prime}\right)$

Check back soon!

02:08

Problem 77

Use the following discussion. The formula
$$
D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right]
$$
Approximate the number of hours of daylight in Honolulu, Hawaii $\left(21^{\circ} 18^{\prime}\right.$ north latitude $)$, for the following dates:
(a) Summer solstice $\left(i=23.5^{\circ}\right)$
(b) Vernal equinox $\left(i=0^{\circ}\right)$
(c) July $4\left(i=22^{\circ} 48^{\prime}\right)$

Tanishq G.

Numerade Educator

Problem 78

Use the following discussion. The formula
$$
D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right]
$$
Approximate the number of hours of daylight in Anchorage, Alaska $\left(61^{\circ} 10^{\prime}\right.$ north latitude $),$ for the following dates:
(a) Summer solstice $\left(i=23.5^{\circ}\right)$
(b) Vernal equinox $\left(i=0^{\circ}\right)$
(c) July $4\left(i=22^{\circ} 48^{\prime}\right)$

Check back soon!

02:45

Problem 79

Use the following discussion. The formula
$$
D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right]
$$
Approximate the number of hours of daylight at the Equator $\left(0^{\circ}\right.$ north latitude $)$ for the following dates:
(a) Summer solstice $\left(i=23.5^{\circ}\right)$
(b) Vernal equinox $\left(i=0^{\circ}\right)$
(c) July $4\left(i=22^{\circ} 48^{\prime}\right)$
(d) What do you conclude about the number of hours of daylight throughout the year for a location at the Equator?

Tanishq G.

Numerade Educator

Problem 80

Use the following discussion. The formula
$$
D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right]
$$
Approximate the number of hours of daylight for any location that is $66^{\circ} 30^{\prime}$ north latitude for the following dates:
(a) Summer solstice $\left(i=23.5^{\circ}\right)$
(b) Vernal equinox $\left(i=0^{\circ}\right)$
(c) July $4\left(i=22^{\circ} 48^{\prime}\right)$
(d) Thanks to the symmetry of the orbital path of Earth around the Sun, the number of hours of daylight on the winter solstice may be found by computing the number of hours of daylight on the summer solstice and subtracting this result from 24 hours. Compute the number of hours of daylight for this location on the winter solstice. What do you conclude about daylight for a location at $66^{\circ} 30^{\prime}$ north latitude?

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Problem 81

Being the First to See the Rising Sun Cadillac Mountain, elevation 1530 feet, is located in Acadia National Park, Maine, and is the highest peak on the east coast of the United States. It is said that a person standing on the summit will be the first person in the United States to see the rays of the rising Sun. How much sooner would a person atop Cadillac Mountain see the first rays than a person standing below, at sea level?

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Problem 82

Movie Theater Screens Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, $\theta$ is the viewing angle. Suppose that you sit $x$ feet from the screen. The viewing angle $\theta$ is given by the function
$$
\theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right)
$$
(a) What is your viewing angle if you sit 10 feet from the screen? 15 feet? 20 feet?
(b) If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row behind it, which row results in the largest viewing angle?
(c) Using a graphing utility, graph
$$
\theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right)
$$
What value of $x$ results in the largest viewing angle?

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Problem 83

Area under a Curve The area under the graph of $y=\frac{1}{1+x^{2}}$ and above the $x$ -axis between $x=a$ and $x=b$ is given by
$$
\tan ^{-1} b-\tan ^{-1} a
$$
(a) Find the exact area under the graph of $y=\frac{1}{1+x^{2}}$ and
above the $x$ -axis between $x=0$ and $x=\sqrt{3}$.
(b) Find the exact area under the graph of $y=\frac{1}{1+x^{2}}$ and above the $x$ -axis between $x=-\frac{\sqrt{3}}{3}$ and $x=1$

Check back soon!

01:57

Problem 84

Area under a Curve The area under the graph of $y=\frac{1}{\sqrt{1-x^{2}}}$ and above the $x$ -axis between $x=a$ and $x=b$ is given by
$$
\sin ^{-1} b-\sin ^{-1} a
$$
(a) Find the exact area under the graph of $y=\frac{1}{\sqrt{1-x^{2}}}$ and above the $x$ -axis between $x=0$ and $x=\frac{\sqrt{3}}{2}$.
(b) Find the exact area under the graph of $y=\frac{1}{\sqrt{1-x^{2}}}$ and above the $x$ -axis between $x=-\frac{1}{2}$ and $x=\frac{1}{2}$

Sarah L.

Numerade Educator

04:24

Problem 85

Shortest Distance from Chicago to Honolulu Find the shortest distance from Chicago, latitude $41^{\circ} 50^{\prime} \mathrm{N},$ longitude $87^{\circ} 37^{\prime} \mathrm{W}$ to Honolulu, latitude $21^{\circ} 18^{\prime} \mathrm{N},$ longitude $157^{\circ} 50^{\prime} \mathrm{W}$. Round your answer to the nearest mile.

Sarah L.

Numerade Educator

04:10

Problem 86

Shortest Distance from Honolulu to Melbourne, Australia Find the shortest distance from Honolulu to Melbourne, Australia, latitude $37^{\circ} 47^{\prime} \mathrm{S},$ longitude $144^{\circ} 58^{\prime} \mathrm{E} .$ Round your answer to the nearest mile.

Sarah L.

Numerade Educator

03:24

Problem 87

Solve:
$$
\cos \left(\sin ^{-1} x\right)=\tan \left(\cos ^{-1} \frac{4}{5}\right)
$$

Yujie W.

College of San Mateo

06:46

Problem 88

Find $u$ in terms of $x$ and $r$
$$
\tan \left(\cos ^{-1} \frac{x}{r}\right)=\sin \left(\tan ^{-1} u\right)
$$

Yujie W.

College of San Mateo

01:42

Problem 89

Solve: $|3 x-2|+5 \leq 9$

Tanishq G.

Numerade Educator

02:37

Problem 90

State why the graph of the function $f$ shown to the right is one-to-one. Then draw the graph of the inverse function $f^{-1}$.
Hint: The graph of $y=x$ is given.

Sheryl E.

Numerade Educator

01:40

Problem 91

The exponential function $f(x)=1+2^{x}$ is one-to-one. Find $f^{-1}$.

Shaza H.

Numerade Educator

05:25

Problem 92

Factor: $(2 x+1)^{-\frac{1}{2}}\left(x^{2}+3\right)^{-\frac{1}{2}}-\left(x^{2}+3\right)^{-\frac{3}{2}} x(2 x+1)^{\frac{1}{2}}$

Shaza H.

Numerade Educator

01:03

Problem 93

Solve: $e^{4 x}+7=10$

Tanishq G.

Numerade Educator

05:13

Problem 94

The diameter of each wheel of a bicycle is 20 inches. If the wheels are turning at 336 revolutions per minute, how fast is the bicycle moving? Express the answer in miles per hour, rounded to the nearest integer.

Yujie W.

College of San Mateo

00:59