Section 6.4

Radians, Arc Length, and Angular Speed

Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives

Find points on a unit circle determined by real numbers.

Convert between radian measure and degree measure; find coterminal, complementary, and supplementary angles.

Find the length of a arc of a circle; find the measure of a central angle of a circle.

Convert between linear speed and angular speed.

Unit Circle

The unit circle has radius r = 1 and circumference 2π. If a point A travels around the circle, it will travel a distance of 2π. If it travels halfway around the circle, it will travel a distance of π.

Unit Circle

If a point C travels 1/8 of the way around the circle, it will travel a distance of π/4.If a point C travels 1/6 of the way around the circle, it will travel a distance of π/3.

Example

On the unit circle, mark the point determined by each of the following real numbers.

Solution:

a) Think of 9π/4 as 2π + π/4.

a) 94

b) 76

Example (cont)

The point moves clockwise.Go π + π/6.

b) 76

Radian Measure

Measure, moving clockwise, an arc length 1 on the unit circle, to point T, draw a ray from the origin through T. The measure of the angle formed is 1 radian. One radian is about 57.3º.

Radian and Degree Measure

A rotation of 360º (1 revolution) has a measure of 2π radians. A half revolution is a rotation of 180º, or π radians.

Radian and Degree Measure

A quarter revolution is a rotation of 90º, or π/2 radians. A rotation of 270º, or 3π/2 radians.

Converting Between Degree Measure and Radian MeasureTo convert from degree to radian measure, multiply by

To convert from radian to degree measure, multiply by

radians

180º

180º

radians1

radians

180º

180º

radians

Example

Convert each of the following to radians.a) 120º b) –297.25º

Solution:

a) 120º120º radians

180º

120º

180º radians

23

radians, or about 2.09 radians

b) 297.25º 297.25º radians

180º

297.25

180 radians, or about 5.19 radians

Example

Convert each of the following to degrees.

Solution:

a) 34

radians b) 8.5 radians

a) 34

radians180º

radians

34

180º 135º

b) 8.5 radians180º

radians

8.5 180º

487.01º

Radian - Degree Equivalents

Arc Length and Central Angle

A unit circle with radius 1 is shown along with another circle with radius r ≠ 1.

The angle shown is a central angle of both circles.

Radian Measure

The radian measure of a rotation is the ratio of the distance s traveled by a point at a radius r from the center of rotation, to the length of the radius r:

When we are using the formula = s/r, must be in radians and s and r must be expressed in the same unit.

s

r

Example

Find the measure of a rotation in radians when a point 2 m from the center of rotation travels 4 m.

Solution:

s

r

4 m

2 m2

Example

Find the length of an arc of a circle of radius 5 cm associated with an angle of π/3 radians.

Solution:

or about 5.24 cm

s

r, or s r

s 5 cm 3,

Linear Speed in Terms of Angular Speed

The linear speed v of a point a distance r from the center of rotation is given by

v = r,where is the angular speed in radians per unit of time.

For the formula v = r, the units of distance for v and r must be the same, must be in radians per unit of time, and the units of time for v and must be the same.

Example

An earth satellite in circular orbit 1200 km high makes one complete revolution every 90 min. What is its linear speed? Use 6400 km for the length of a radius of the earth.

Find r : 6400 km + 1200 km = 7600 km

Now, use v = r :

The satellite’s linear speed is approximately 531 km/min.

Find : t

2

90 min

45 min

v 7600 km

45 min

760045 min

km

min531

km

min

Example

A 2010 Dodge Ram Crew Cab is traveling at a speed of 70 mph. Its tires have an outside diameter of 29.86 in. Find the angle through which a tire turns in 10 sec.Solution:Convert

Using v = r, we have

v 70mi

hr

1 hr

60 min1 min

60 sec5280 ft

1 mi102.667

ft

sec

1ft14.93 in

12 in14.93

1.24 ft12

r

102.667 ft/sec

1.24 ft

ft102.667 1.24ft

sec

Example (cont)

The angle, in radians, through which a tire, of a car traveling 70 mph, turns in 10 sec is 828 radians.

Then in 10 sec,

82.8010 sec

sec

102.667 ft/sec

1.24 ft

82.80

sec

t 828