3
Radian
Measure and
the Unit Circle
Slide 1-1
Radian Measure and the Unit
Circle

3.1 Radian Measure

3.2 Applications of Radian Measure

3.3 The Unit Circle and Circular Functions

3.4 Linear and Angular Speed
Slide 1-2
3.1
Radian Measure
Radian Measure ▪ Converting Between Degrees and Radians ▪
Finding Function Values for Angles in Radians
Slide 1-3
3.1 Radian Measure
The circumference is equal to the 2𝜋 times the length of the radius.
2𝜋 is approximately 6.28, so the circumference is a little more than
6 radius lengths
Or, in terms of radian measure, a complete rotation (360 degrees) is
2𝜋 radians.
Slide 1-4
2 radians  360 degrees
Slide 1-5
Example 1: Convert each degree measure to radians
(a) 1080
(b)  1350
(c) 325.70
Example 2: Convert each radian measure to degrees.
(a)
11
12
(b) 
7
6
(c)  2.92
Slide 1-6
(page 97)
Example 3. Find each function value.
Slide 1-7
3.2 Applications of Radian Measure
𝑠
Central Angle Arc Length

0
360
2 r

S

 S  r Arc length formula
2 2 r
Central Angle Area of sector

0
360
 r2
AS

1 2
 2  AS   r Area of a sector
2  r
2
Slide 1-8
Example: Using Latitudes to Find the Distance
Between Two Cities (p.101) (page 101)

Erie, Pennsylvania is approximately due north of Columbia,
South Carolina. The latitude of Erie is 42° N, while that of
Columbia is 34° N. Find the north-south distance between the
two cities.
The radius of the earth is about 6400 km.
The central angle between Erie and Columbia is
42° – 34° = 8°. Convert 8° to radians:
Slide 1-9

Use s = rθ to find the north-south distance between the two
cities.
The north-south distance between Erie and Columbia is
about 890 km.
Slide 1-10
Example: Finding a Length Using s = rθ (page
102)

A rope is being wound around a drum with radius 0.327 m.
How much rope will be wound around the drum if the drum is
rotated through an angle of 132.6°?
The length of rope wound around the drum is the arc
length for a circle of radius 0.327 m and a central angle of
132.6°.
Slide 1-11
Convert 132.6° to radians:

Use s = rθ to find the arc length, which is the length of the
rope.
The length of the rope wound around the drum is about
0.757 m.
Slide 1-12
Example: Finding an Angle Measure Using s = rθ

(page 102)
Two gears are adjusted so that the smaller gear drives the
larger one. If the radii of the gears are 3.6 in. and 5.4 in., and
the smaller gear rotates through 150°, through how many
degrees will the larger gear rotate?
First find the radian measure of
the angle, and then find the arc
length on the smaller gear that
determines the motion of the
larger gear.
Slide 1-13

The arc length on the smaller gear is
An arc with length 3π cm on the larger gear corresponds
to an angle measure θ radians, where
The larger gear will rotate through 100°.
Slide 1-14
Example: Finding the Area of a Sector (page 103)

Find the area of a sector of a circle having radius 15.20 ft and
central angle 108.0°.
The area of the sector is about 217.8 sq ft.
Slide 1-15
3.3 The Unit Circle and Circular Functions
Slide 1-16
Example:
Slide 1-17
3.4 Linear and Angular Speed
Slide 1-18
Linear Speed

Suppose that a point P moves at a constant
speed along a circle of radius r and center O.
The measure of how fast the
position of P is changing is
the linear speed. If v
represents linear speed, then
or
where s is the length of the arc traced by point P
at time t.
Slide 1-19
Angular Speed

As point P moves along the circle, ray OP
rotates about the origin.
The measure of how fast
angle POB is changing is its
angular speed.
is the angular speed, θ is the measure of angle
POB (in radians) at time t.
Slide 1-20
Slide 1-21
Example

FINDING LINEAR SPEED AND
DISTANCE TRAVELED BY A SATELLITE
A satellite traveling in a
circular orbit 1600 km above
the surface of Earth takes 2
hr to make an orbit. The
radius of Earth is
approximately 6400 km.
(a) Approximate the linear speed of the satellite in
kilometers per hour.
The distance of the satellite from the center of Earth
is approximately r = 1600 + 6400 = 8000 km.
Slide 1-22
For one orbit, θ = 2π, and

Since it takes 2 hours to
complete an orbit, the linear
speed is
(b) Approximate the distance the satellite travels in
4.5 hr.
Slide 1-23