Algebra 2 and Trigonometry

Educators

Problem 1

Ryan and Rebecca each found the radian measure of a central angle by measuring the radius of the circle and the length of the intercepted arc. Ryan used inches and Rebecca used centimeters when making their measurements. If Ryan and Rebecca each measured accurately, will the measures that they obtain for the angle be equal? Justify your answer.

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Problem 2

If a wheel makes two complete revolutions, each spoke on the wheel turns through an angle of how many radians? Explain your answer.

Finian L.

Problem 3

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$30^{\circ}$

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Problem 4

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$90^{\circ}$

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Problem 5

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$45^{\circ}$

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Problem 6

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$120^{\circ}$

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Problem 7

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$160^{\circ}$

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Problem 8

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$135^{\circ}$

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Problem 9

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$225^{\circ}$

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Problem 10

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$240^{\circ}$

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Problem 11

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$270^{\circ}$

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Problem 12

In $3-12,$ find the radian measure of each angle whose degree measure is given.
$330^{\circ}$

Finian L.

Problem 13

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{\pi}{3}$

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Problem 14

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{\pi}{9}$

Finian L.

Problem 15

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{\pi}{10}$

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Problem 16

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{2 \pi}{5}$

Finian L.

Problem 17

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{10 \pi}{9}$

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Problem 18

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{3 \pi}{2}$

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Problem 19

In $13-22$ , find the degree measure of each angle whose radian measure is given.
3$\pi$

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Problem 20

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{11 \pi}{6}$

Finian L.

Problem 21

In $13-22$ , find the degree measure of each angle whose radian measure is given.
$\frac{7 \pi}{2}$

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Problem 22

In $13-22$ , find the degree measure of each angle whose radian measure is given.
1

Finian L.

Problem 23

In $23-27,$ for each angle with the given radian measure: a. Give the measure of the angle in degrees. b. Give the measure of the reference angle in radians c. Draw the angle in standard position and its] reference angle as an acute angle formed by the terminal side of the angle and the $x$ -axis.
$\frac{\pi}{3}$

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Problem 24

In $23-27,$ for each angle with the given radian measure: a. Give the measure of the angle in degrees. b. Give the measure of the reference angle in radians c. Draw the angle in standard position and its] reference angle as an acute angle formed by the terminal side of the angle and the $x$ -axis.
$\frac{7 \pi}{36}$

Finian L.

Problem 25

In $23-27,$ for each angle with the given radian measure: a. Give the measure of the angle in degrees. b. Give the measure of the reference angle in radians c. Draw the angle in standard position and its] reference angle as an acute angle formed by the terminal side of the angle and the $x$ -axis.
$\frac{10 \pi}{9}$

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Problem 26

In $23-27,$ for each angle with the given radian measure: a. Give the measure of the angle in degrees. b. Give the measure of the reference angle in radians c. Draw the angle in standard position and its] reference angle as an acute angle formed by the terminal side of the angle and the $x$ -axis.
$-\frac{7 \pi}{18}$

Finian L.

Problem 27

In $23-27,$ for each angle with the given radian measure: a. Give the measure of the angle in degrees. b. Give the measure of the reference angle in radians c. Draw the angle in standard position and its] reference angle as an acute angle formed by the terminal side of the angle and the $x$ -axis.
$\frac{25 \pi}{9}$

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Problem 28

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $s=6$ and $r=1,$ find $\theta$

Finian L.

Problem 29

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $\theta=4.5$ and $s=9,$ find $r$

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Problem 30

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $\theta=2.5$ and $r=10,$ find $s$

Finian L.

Problem 31

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $r=2$ and $\theta=1.6,$ find $s$

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Problem 32

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $r=2.5$ and $s=15,$ find $\theta$

Finian L.

Problem 33

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $s=16$ and $\theta=0.4,$ find $r$

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Problem 34

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $r=4.2$ and $s=21,$ find $\theta$

Finian L.

Problem 35

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $r=6$ and $\theta=\frac{2 \pi}{3},$ find $s$

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Problem 36

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $s=18$ and $\theta=\frac{6 \pi}{5},$ find $r$

Finian L.

Problem 37

In $28-37, \theta$ is the radian measure of a central angle that intercepts an arc of length $s$ in a circle with a radius of length $r .$
If $\theta=6 \pi$ and $r=1,$ find $s$

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Problem 38

Circle $O$ has a radius of 1.7 inches. What is the length, in inches, of an arc intercepted by a central angle whose measure is 2 radians?

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Problem 39

In a circle whose radius measures 5 feet, a central angle intercepts an are of length 12 feet. Find the radian measure of the central angle.

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Problem 40

The central angle of circle $O$ has a measure of 4.2 radians and it intercepts an arc whose length is 6.3 meters. What is the length, in meters, of the radius of the circle?

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Problem 41

Complete the following table, expressing degree measures in radian measure in terms of $\pi$
$$\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Degrees } 30^{\circ} & {45^{\circ}} & {60^{\circ}} & {90^{\circ}} & {180^{\circ}} & {270^{\circ}} & {360^{\circ}} \\ \hline \text { Radians } & {} & {} & {} \\ \hline\end{array}$$

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Problem 42

The pendulum of a clock makes an angle of 2.5 radians as its tip travels 18 feet. What is the length of the pendulum?

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Problem 43

A wheel whose radius measures 16 inches is rotated. If a point on the circumference of the wheel moves through an arc of 12 feet, what is the measure, in radians, of the angle through which a spoke of the wheel travels?

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Problem 44

The wheels on a bicycle have a radius of 40 centimeters. The wheels on a cart have a radius of 10 centimeters. The wheels of the bicycle and the wheels of the cart all make one complete revolution.
a. Do the wheels of the bicycle rotate through the same angle as the wheels of the cart? Justify your answer.
b. Does the bicvcle travel the same distance as the cart? Justify vour answer.

Finian L.
Latitude represents the measure of a central angle with vertex at the center of the earth, its initial side passing through a point on the equator, and its terminal side passing through the given location. (See the figure.) Cities A and $\mathrm{B}$ are on a north-south line. City $\mathrm{A}$ is located at $30^{\circ} \mathrm{N}$ and City $\mathrm{B}$ is located at $52^{\circ} \mathrm{N}$ . If the radius of the earth is
approximately $6,400$ kilometers, find $d$ , the distance between the two cities along the circumference of the earth. Assume that the earth is a perfect sphere.