Precalculus

David Cohen, Theodore B. Lee, David Sklar

Chapter 6

The Trigonometric Functions - all with Video Answers

Educators

Section 1

Radian Measure

Problem 1

Use the definition $\theta=s / r$ to determine the $r a-$ dian measure of each angle.

(FIGURE CAN'T COPY)

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 2

Use the definition $\theta=s / r$ to determine the $r a-$ dian measure of each angle.

(FIGURE CAN'T COPY)

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 3

Use the definition $\theta=s / r$ to determine the $r a-$ dian measure of each angle.

(FIGURE CAN'T COPY)

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 4

Use the definition $\theta=s / r$ to determine the $r a-$ dian measure of each angle.

(FIGURE CAN'T COPY)

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 5

Convert to radian measure. Express your answers both in terms of $\pi$and as decimal approximations rounded to two decimal places.
(a) $45^{\circ}$
(b) $90^{\circ}$
(c) $135^{\circ}$

Katelyn Chen

Numerade Educator

Problem 6

Convert to radian measure. Express your answers both in terms of $\pi$and as decimal approximations rounded to two decimal places.
(a) $30^{\circ}$
(b) $150^{\circ}$
(c) $300^{\circ}$

Katelyn Chen

Numerade Educator

Problem 7

Convert to radian measure. Express your answers both in terms of $\pi$and as decimal approximations rounded to two decimal places.
(a) $0^{\circ}$
(b) $360^{\circ}$
(c) $450^{\circ}$

Katelyn Chen

Numerade Educator

Problem 8

Convert to radian measure. Express your answers both in terms of $\pi$and as decimal approximations rounded to two decimal places.
(a) $36^{\circ}$
(b) $35^{\circ}$
(c) $720^{\circ}$

Katelyn Chen

Numerade Educator

Problem 9

Convert the radian measures to degrees.
(a) $\pi / 12$
(b) $\pi / 6$
(c) $\pi / 4$

Katelyn Chen

Numerade Educator

Problem 10

Convert the radian measures to degrees.
(a) $3 \pi$
(b) $3 \pi / 2$
(c) $2 \pi$

Katelyn Chen

Numerade Educator

Problem 11

Convert the radian measures to degrees.
(a) $\pi / 3$
(b) $5 \pi / 3$
(c) $4 \pi$

Katelyn Chen

Numerade Educator

Problem 12

Convert the radian measures to degrees.
(a) $5 \pi / 6$
(b) $11 \pi / 6$
(c) 0

Katelyn Chen

Numerade Educator

Problem 13

Convert the radian measures to degrees. Bound the answers to two decimal places.
$$\begin{array}{lll}
\text { (a) } 2 & \text { (b) } 3 & \text { (c) } \pi^{2}
\end{array}$$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 14

$$\begin{array}{lll}
\text { (a) } 2 & \text { (b) } 3 & \text { (c) } \pi^{2}
\end{array}$$(a) 1.32
(b) 0.96
(c) $1 / \pi$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 15

Suppose that the radian measure of an angle is $3 / 2 .$ Without using a calculator or tables, determine if this angle is larger or smaller than a right angle. Hint: What is the radian measure of a right angle?

Katelyn Chen

Numerade Educator

Problem 16

Two angles in a triangle have radian measure $\pi / 5$ and $\pi / 6$ What is the radian measure of the third angle?

Katelyn Chen

Numerade Educator

Problem 17

Refer to the following figure, which shows all of the angles from $0^{\circ}$ to $360^{\circ}$ that are multiples of $30^{\circ}$ or $45^{\circ}$.
(FIGURE CAN'T COPY)
In the figure, relabel the angles in Quadrants I and II using radian measure.

Katelyn Chen

Numerade Educator

Problem 18

Refer to the following figure, which shows all of the angles from $0^{\circ}$ to $360^{\circ}$ that are multiples of $30^{\circ}$ or $45^{\circ}$.
(FIGURE CAN'T COPY)
In the figure, relabel the angles in Quadrants III and IV using radian measure.

Katelyn Chen

Numerade Educator

Problem 19

Find the arc length s in each case.
(FIGURE CAN'T COPY)

Julian Wong

Numerade Educator

Problem 20

Find the arc length s in each case.
(FIGURE CAN'T COPY)

Julian Wong

Numerade Educator

Problem 21

Find the arc length s in each case.
(FIGURE CAN'T COPY)

Julian Wong

Numerade Educator

Problem 22

Find the arc length s in each case.
(FIGURE CAN'T COPY)

Julian Wong

Numerade Educator

Problem 23

Find the area of the sector determined by the given radius r and central angle $\theta .$ Express the answer both in terms of $\pi$and as a decimal approximation rounded to two decimal places.

(a) $r=6 \mathrm{cm} ; \theta=2 \pi / 3$
(b) $r=5 \mathrm{m} ; \theta=80^{\circ}$
(c) $r=24 \mathrm{m} ; \theta=\pi / 20$
(d) $r=1.8 \mathrm{cm} ; \theta=144^{\circ}$

Katelyn Chen

Numerade Educator

Problem 24

Find the area of the sector determined by the given radius r and central angle $\theta .$ Express the answer both in terms of $\pi$and as a decimal approximation rounded to two decimal places.

(a) $r=4 \mathrm{cm} ; \theta=\pi / 10$
(b) $r=16 \mathrm{m} ; \theta=5^{\circ}$
(c) $r=21 \mathrm{ft} ; \theta=11 \pi / 6$
(d) $r=4.2$ in.; $\theta=170^{\circ}$

Katelyn Chen

Numerade Educator

Problem 25

In a circle of radius $1 \mathrm{cm},$ the area of a certain sector is $\pi / 5 \mathrm{cm}^{2} .$ Find the radian measure of the central angle. Express the answer in terms of $\pi$ rather than as a decimal approximation.

Katelyn Chen

Numerade Educator

Problem 26

In a circle of radius $3 \mathrm{m},$ the area of a certain sector is
$20 \mathrm{m}^{2} .$ Find the degree measure of the central angle. Round the answer to two decimal places.

Katelyn Chen

Numerade Educator

Problem 27

Find (a) the perimeter of the sector; and (b) the area of the sector. In each case, use a calculator to evaluate the answer and round to two decimal places.
(FIGURE CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 28

Find (a) the perimeter of the sector; and (b) the area of the sector. In each case, use a calculator to evaluate the answer and round to two decimal places.
(FIGURE CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 29

You are given the rate of rotation of a wheel as well as its radius. In each case, determine the following: (a) the angular speed, in units of radians/sec; (b) the linear speed, in units of cm/sec. of a point on the circumference of the wheel; and (c) the linear speed, in cm/sec, of a point halfway
between the center of the wheel and the circumference.
6 revolutions/sec; $r=12 \mathrm{cm}$

Katelyn Chen

Numerade Educator

Problem 30

You are given the rate of rotation of a wheel as well as its radius. In each case, determine the following: (a) the angular speed, in units of radians/sec; (b) the linear speed, in units of cm/sec. of a point on the circumference of the wheel; and (c) the linear speed, in cm/sec, of a point halfway
between the center of the wheel and the circumference.
15 revolutions/sec; $r=20 \mathrm{cm}$

Katelyn Chen

Numerade Educator

Problem 31

You are given the rate of rotation of a wheel as well as its radius. In each case, determine the following: (a) the angular speed, in units of radians/sec; (b) the linear speed, in units of cm/sec. of a point on the circumference of the wheel; and (c) the linear speed, in cm/sec, of a point halfway
between the center of the wheel and the circumference.
$$1080 \%=25 \mathrm{cm}$$

Katelyn Chen

Numerade Educator

Problem 32

You are given the rate of rotation of a wheel as well as its radius. In each case, determine the following: (a) the angular speed, in units of radians/sec; (b) the linear speed, in units of cm/sec. of a point on the circumference of the wheel; and (c) the linear speed, in cm/sec, of a point halfway
between the center of the wheel and the circumference.
$$2160^{\circ} / \mathrm{sec} ; r=60 \mathrm{cm}$$

Katelyn Chen

Numerade Educator

Problem 33

You are given the rate of rotation of a wheel as well as its radius. In each case, determine the following: (a) the angular speed, in units of radians/sec; (b) the linear speed, in units of cm/sec. of a point on the circumference of the wheel; and (c) the linear speed, in cm/sec, of a point halfway
between the center of the wheel and the circumference.
$$500 \mathrm{rpm} ; r=45 \mathrm{cm}$$

Katelyn Chen

Numerade Educator

Problem 34

You are given the rate of rotation of a wheel as well as its radius. In each case, determine the following: (a) the angular speed, in units of radians/sec; (b) the linear speed, in units of cm/sec. of a point on the circumference of the wheel; and (c) the linear speed, in cm/sec, of a point halfway
between the center of the wheel and the circumference.
$$1250 \mathrm{rpm} ; r=10 \mathrm{cm}$$

Katelyn Chen

Numerade Educator

Problem 35

For this problem, assume that the earth is a sphere with a radius of 3960 miles and a rotation rate of 1 revolution per 24 hours.
(a) Find the angular speed. Express your answer in units of radians/sec, and round to two significant digits.
(b) Find the linear speed of a point on the equator. Express the answer in units of miles per hour, and round to the nearest 10 mph.

Katelyn Chen

Numerade Educator

Problem 36

A wheel 3 ft in diameter makes $x$ revolutions. Find $x$ given that the distance traveled by a point on the circumference of the wheel is 22619 ft. (Round your answer to the nearest whole number.)

Katelyn Chen

Numerade Educator

Problem 37

Suppose that you have two sticks and a piece of wire, each of length $1 \mathrm{ft}$, fastened at the ends to form an equilateral triangle; see Figure A. If side $\overline{B C}$ is bent out to form an arc of a circle with center $A$, then the angle at $A$ will decrease from $60^{\circ}$ to something less. See Figure B. What is the measure of this new angle at $A$ in both radians and degrees?
(FIGURES CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 38

(a) When a clock reads 4: 00 , what is the radian measure of the (smaller) angle between the hour hand and the minute hand?
(b) When a clock reads 5: 30 , what is the radian measure of the (smaller) angle between the hour hand and the minute hand?

Sirat Shah

Numerade Educator

Problem 39

Are there any real numbers $x$ with the property that $x$ degrees equals $x$ radians? If so, find them; if not, explain why not.

Katelyn Chen

Numerade Educator

Problem 40

Are there any real numbers $x$ with the property that $x$ degrees equals $2 x$ radians? If so, find them; if not, explain why not.

Katelyn Chen

Numerade Educator

Problem 41

Suppose that a belt drives two wheels of radii rand $R,$ as indicated in the figure.
(FIGURE CAN'T COPY)
If $r=6 \mathrm{cm}, R=10 \mathrm{cm},$ and the angular speed of the larger wheel is 100 rpm, determine each of the following:
(a) the angular speed of the larger wheel in radians per minute;
(b) the linear speed of a point on the circumference of the larger wheel;
(c) the angular speed of the smaller wheel in radians per minute. Hint: Because of the belt, the linear speed of a point on the circumference of the larger wheel is equal to the linear speed of a point on the circumference of the smaller wheel.
(d) The angular speed of the smaller wheel in rpm.

Katelyn Chen

Numerade Educator

Problem 42

Suppose that a belt drives two wheels of radii rand $R,$ as indicated in the figure.
(FIGURE CAN'T COPY)
Follow Exercise $41,$ assuming that $r=5 \mathrm{cm}, R=15 \mathrm{cm}$ and the angular speed of the larger wheel is 1800 rpm.

Katelyn Chen

Numerade Educator

Problem 43

The latitude of a point P on the surface of the Earth is specified by means of the angle $\theta$ in the figure. For instance, the latitude of Paris, France, is $48^{\circ} 52^{\prime}$ N. The letter $N$ is used here to indicate that the location is north of, rather than south of, the equator. (Recall that the notation $52^{\prime}$ indicates $52 / 60$ of one degree.) In Exercises $43-48,$ use the arc length formula (and your calculator to determine the distance PE from the given location $P$ to the equator. Assume that the Earth is a sphere with radius $O P=O E=3960$ miles. Round each answer to the nearest 10 miles.
(FIGURE CAN'T COPY)
Point Barrow, Alaska: $71^{\circ} 23^{\prime} \mathrm{N}$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 44

The latitude of a point P on the surface of the Earth is specified by means of the angle $\theta$ in the figure. For instance, the latitude of Paris, France, is $48^{\circ} 52^{\prime}$ N. The letter $N$ is used here to indicate that the location is north of, rather than south of, the equator. (Recall that the notation $52^{\prime}$ indicates $52 / 60$ of one degree.) In Exercises $43-48,$ use the arc length formula (and your calculator to determine the distance PE from the given location $P$ to the equator. Assume that the Earth is a sphere with radius $O P=O E=3960$ miles. Round each answer to the nearest 10 miles.
(FIGURE CAN'T COPY)
$$\text { singapore: } 1^{\circ} 17^{\prime} \mathrm{N}$$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 45

The latitude of a point P on the surface of the Earth is specified by means of the angle $\theta$ in the figure. For instance, the latitude of Paris, France, is $48^{\circ} 52^{\prime}$ N. The letter $N$ is used here to indicate that the location is north of, rather than south of, the equator. (Recall that the notation $52^{\prime}$ indicates $52 / 60$ of one degree.) In Exercises $43-48,$ use the arc length formula (and your calculator to determine the distance PE from the given location $P$ to the equator. Assume that the Earth is a sphere with radius $O P=O E=3960$ miles. Round each answer to the nearest 10 miles.
(FIGURE CAN'T COPY)
$$\text { Honolulu: } 21^{\circ} 19^{\prime} \mathrm{N}$$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 46

The latitude of a point P on the surface of the Earth is specified by means of the angle $\theta$ in the figure. For instance, the latitude of Paris, France, is $48^{\circ} 52^{\prime}$ N. The letter $N$ is used here to indicate that the location is north of, rather than south of, the equator. (Recall that the notation $52^{\prime}$ indicates $52 / 60$ of one degree.) In Exercises $43-48,$ use the arc length formula (and your calculator to determine the distance PE from the given location $P$ to the equator. Assume that the Earth is a sphere with radius $O P=O E=3960$ miles. Round each answer to the nearest 10 miles.
(FIGURE CAN'T COPY)
$$\text { Lagos: } 6^{\circ} 27^{\prime} \mathrm{N}$$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 47

The latitude of a point P on the surface of the Earth is specified by means of the angle $\theta$ in the figure. For instance, the latitude of Paris, France, is $48^{\circ} 52^{\prime}$ N. The letter $N$ is used here to indicate that the location is north of, rather than south of, the equator. (Recall that the notation $52^{\prime}$ indicates $52 / 60$ of one degree.) In Exercises $43-48,$ use the arc length formula (and your calculator to determine the distance PE from the given location $P$ to the equator. Assume that the Earth is a sphere with radius $O P=O E=3960$ miles. Round each answer to the nearest 10 miles.
(FIGURE CAN'T COPY)
Washington, D.C: $38^{\circ} 54^{\prime} \mathrm{N}$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 48

The latitude of a point P on the surface of the Earth is specified by means of the angle $\theta$ in the figure. For instance, the latitude of Paris, France, is $48^{\circ} 52^{\prime}$ N. The letter $N$ is used here to indicate that the location is north of, rather than south of, the equator. (Recall that the notation $52^{\prime}$ indicates $52 / 60$ of one degree.) In Exercises $43-48,$ use the arc length formula (and your calculator to determine the distance PE from the given location $P$ to the equator. Assume that the Earth is a sphere with radius $O P=O E=3960$ miles. Round each answer to the nearest 10 miles.
(FIGURE CAN'T COPY)
Fairbanks, Alaska: $64^{\circ} 51^{\prime} \mathrm{N}$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 49

Provide geometric results that you will need in working Exercises $51-54 .$ For Exercise $50,$ you need to know that a segment of a circle is the region bounded by an arc of the circle and its chord. In the accompanying figure, the red region is a segment. (The white region also is a segment.)
(FIGURE CAN'T COPY)
Show that the area of an equilateral triangle of side $s$ is given by
$\frac{\sqrt{3}}{4}$
Hint: Draw an altitude and use the Pythagorean theorem. (This exercise does not require any knowledge of radian measure. $)$

AG

Ankit Gupta

Numerade Educator

Problem 50

Provide geometric results that you will need in working Exercises $51-54 .$ For Exercise $50,$ you need to know that a segment of a circle is the region bounded by an arc of the circle and its chord. In the accompanying figure, the red region is a segment. (The white region also is a segment.)
(FIGURE CAN'T COPY)
In the accompanying figure, $\triangle A B C$ is equilateral and $s$ denotes the length of each side. The arc in the figure is a portion of a circle with center $A$ and radius $A B=s .$ Use the result in Exercise 49 and a formula from this section to show that the area of the shaded segment in the figure is given by
$$
s^{2}\left(\frac{2 \pi-3 \sqrt{3}}{12}\right)
$$

Anurag Kumar

Numerade Educator

Problem 51

Many of the window designs used in gothic architecture involve circles, sectors, and segments of circles. The equilateral arch in Figure $A$ is an example of this. Figure $B$ shows how the arch is designed. Starting with the equilateral triangle $A B C,$ circular arc $A C$ is drawn with center $B$and radius $A B$. Similarly, circular arc $\widehat{B C}$ is drawn with center $A$ and radius $A B$
(FIGURES CAN'T COPY)
(a) Let $s$ denote the length of a side of the equilateral triangle in Figure $\mathrm{B}$. Express the perimeter of the equilateral arch in terms of $s$
(b) Express the area of the equilateral arch in terms of $s$

Ethan Somes

Numerade Educator

Problem 52

Closely related to the equilateral arch in Exercise 51 is the equilateral curved triangle shown in Figure C. Just as with the equilateral arch in Exercise $51,$ the design begins with the equilateral triangle $A B C$. Circular arcs are then constructed on each side of the triangle, following the method explained in Exercise 51
(FIGURE CAN'T COPY)
(a) In Figure C, let $s$ denote the length of a side of the equilateral triangle $A B C .$ Express the perimeter and the area of the equilateral curved triangle in terms of $s$
(b) Show that the area of the equilateral triangle $A B C$ is approximately $61 \%$ of the area of the equilateral curved triangle $A B C$.

Lourence Gonhovi

Lourence Gonhovi

Numerade Educator

Problem 53

Figure D shows one of the gothic window designs used in Wells Cathedral in England. (The cathedral was constructed in the mid-thirteenth century.) Figure E indicates how the design is formed. We start with the equilateral $\triangle A B C$ and construct the equilateral arch $A B C .$ Next, the midpoints of the three sides of $\triangle A B C$ are joined to create four smaller equilateral triangles. The two equilateral triangles $A D F$ and $F E C$ are then used to construct the two smaller equilateral arches shown in Figure E. And finally,the equilateral triangle $D B E$ is used to construct the equilateral curved triangle within the top half of the figure.
(a) Let $s$ denote the length of a side of the equilateral triangle $A B C$. Express the area of each of the equilateral arches $A B C$ and $A D F$ in terms of $s$. Also, find the ratio of the area of arch $A D F$ to arch $A B C .$
(b) Express the area of the equilateral curved triangle $D B E$ in terms of $s$
(c) Express the area of the curved figure $D E F$ in terms of $s$. (By "the curved figure $D E F$ " we mean the region bounded by the circular arcs $\overline{D E}, \widehat{E F},$ and $\overline{F D} .$ )
(d) Express the area of the curved figure $B E C$ in terms of $s$
(FIGURE CAN'T COPY)

Anurag Kumar

Numerade Educator

Problem 54

Figure $F$ shows a gothic window design from the cathedral at Reims, France. (The cathedral was constructed during the years $1211-1311 .$ ) Figure G indicates how the design is formed. Triangle $A B C$ is equilateral and arch $A B C$ is the corresponding equilateral arch. Equilateral triangle GID, congruent to triangle $A B C,$ is constructed such that $D$ is the midpoint of $\overline{A B},$ and $\overline{G I}$ is parallel to
$\overline{A B} .$ The points $E$ and $F$ are the midpoints of segments $\overline{D G}$ and $D I$, respectively. The two small arches at the bottom of the figure are equilateral arches corresponding to the equilateral triangles $G H E$ and $H I F .$ For the circle in the figure, the center is $D$ and the radius is $A D(=E D=F D)$
(a) Let $s$ denote the length of a side in each of the two equilateral triangles $A B C$ and $G I D .$ Find the area and the perimeter (in terms of $s$ ) of the curved figure AJBCA. Hint: For the area, subtract the area of
the semicircle $A J B$ from the area of the equilateral $\operatorname{arch} A B C$
(b) Express (in terms of $s$ ) the area and the perimeter of equilateral arch GHE.
(c) Show that the area of the curved figure $E H F$ is $s^{2}(2 \sqrt{3}-\pi) / 8$
(d) Express (in terms of $s$ ) the perimeter and the area of the curved figure $F B I$
(FIGURE CAN'T COPY)

Anurag Kumar

Numerade Educator

Problem 55

The accompanying figure shows a circular sector with radius $r \mathrm{cm}$ and central angle $\theta$ (radian measure). The perimeter of the sector is $12 \mathrm{cm} .$

(FIGURE CAN'T COPY)
(a) Express $r$ as a function of $\theta$
(b) Express the area $A$ of the sector as a function of $\theta .$ Is this a quadratic function?
(c) Express $\theta$ as a function of $r$
(d) Express the area $A$ of the sector as a function of $r .$ Is this a quadratic function?
(e) For which value of $r$ is the area $A$ a maximum? What is the corresponding value of $\theta$ in this case?

Chris Trentman

Numerade Educator

Problem 56

The following figure shows a semicircle of radius 1 unit and two adjacent sectors, $A O C$ and $C O B$
(a) Show that the product $P$ of the areas of the two sectors is given by
$$
P=\frac{\pi \theta}{4}-\frac{\theta^{2}}{4}
$$
Is this a quadratic function?
(b) For what value of $\theta$ is $P$ a maximum?
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator