5.1 Angles

Find the radian measure of one-third of a full rotation.

Find the radian measure of three-fourths of a full rotation.

Convert each radian measure to degrees.

1. ⓐ $\frac{\pi}{6}$
2. ⓑ 3

Convert $- \frac{3 \pi}{4}$ radians to degrees.

Convert $15$ degrees to radians.

Convert 126° to radians.

Find an angle $\alpha$ that is coterminal with an angle measuring 870°, where $0° \leq \alpha < 360° .$

Find an angle $\beta$ that is coterminal with an angle measuring −300° such that $0° \leq \beta < 360° .$

Find an angle of measure $\theta$ that is coterminal with an angle of measure $- \frac{17 \pi}{6}$ where $0 \leq \theta < 2 \pi .$

Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

1. ⓐ In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
2. ⓑ Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.

Find the arc length along a circle of radius 10 units subtended by an angle of 215°.

In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.

A vintage vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.

A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.

A satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.

State what a positive or negative angle signifies, and explain how to draw each.

Explain the differences between linear speed and angular speed when describing motion along a circular path.

Find the area of the sector.

Find the area of the sector.

Convert angles in radians to degrees.

$\frac{\pi}{9}$ radians

Convert angles in radians to degrees.

$\frac{\pi}{3}$ radians

Convert angles in radians to degrees.

$- \frac{5 \pi}{12}$ radians

Convert angles in degrees to radians.

90°

Convert angles in degrees to radians.

−540°

Convert angles in degrees to radians.

180°

Convert angles in degrees to radians.

150°

Use to given information to find the length of a circular arc. Round to two decimal places.

Find the length of the arc of a circle of radius 5.02 miles subtended by the central angle of $\frac{\pi}{3} .$

Use to given information to find the length of a circular arc. Round to two decimal places.

Find the length of the arc of a circle of radius 10 centimeters subtended by the central angle of 50°.

Use to given information to find the length of a circular arc. Round to two decimal places.

Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63°.

Use the given information to find the area of the sector. Round to four decimal places.

A sector of a circle has a central angle of 30° and a radius of 20 cm.

Use the given information to find the area of the sector. Round to four decimal places.

A sector of a circle with radius of 0.7 inches and an angle of $\pi$ radians.

Find the angle between 0° and 360° that is coterminal to the given angle.

−110°

Find the angle between 0° and 360° that is coterminal to the given angle.

1400°

Find the angle between 0 and $2 \pi$ in radians that is coterminal to the given angle.

$\frac{10 \pi}{3}$

Find the angle between 0 and $2 \pi$ in radians that is coterminal to the given angle.

$\frac{44 \pi}{9}$

A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

A wheel of radius 14 inches is rotating 0.5 rad/s. What is the linear speed $v ,$ the angular speed in RPM, and the angular speed in deg/s?

When being burned in a writable CD-R drive, the angular speed of a CD varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4,800 RPM (revolutions per minute. Find the linear speed if the CD has diameter of 120millimeters.

Find the distance along an arc on the surface of Earth that subtends a central angle of 5 minutes $\left(1 \textrm{ }\text{minute} = \frac{1}{60} \textrm{ }\text{degree}\right)$. The radius of Earth is 3960 miles.

Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in $20$ minutes?

A city is located at 40 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city. Refer to Unit Circle: Sine and Cosine Functions for information on trigonometric functions.

Find the linear speed of the moon if the average distance between the earth and moon is 239,000 miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour.

A car travels 3 miles. Its tires make 2640 revolutions. What is the radius of a tire in inches?