4-1 : Angles and Their MeasuresWhat you’ll learn about

■ The Problem of Angular Measure

■ Degrees and Radians■ Circular Arc Length■ Angular and Linear Motion

. . . and whyAngles are the domain elements of the

trigonometric functions.

Getting StartedDarla watches Larry walk around a circle. The circle’s radius is 1 meter. Darla stands at the center and Ben begins walking counterclockwise. Consider a coordinate grid, with Darla standing at the origin and Ben starting at point (1,0).

Consider Ben’s location after certain distances

How far will Ben walk

before returning to

the point (1,0)?

Consider Ben’s location after certain distances

Where will Ben be after

walking exactly π meters?

Consider Ben’s location after certain distances

Where will Ben be after

walking exactly π/2

meters?

Consider Ben’s location after certain distances

Where will Ben be after

walking exactly 3π meters?

Consider Ben’s location after certain distances

Where will Ben be after

walking exactly 3 meters?

Radian Measure • Definition of RadianOne radian is the measure of a central angle that intercepts an arc equal in length to the radius of the circle. (picture) Algebraically, this means that

where is measured in radians.

Where is Ben after he has

walked exactly π/4

meters? How about after

3π/4?

π/43π/4

2π/4 = π/2

At some point Ben has walked exactly 9π/4 meters.

Find 2 other distances Ben could

have walked around the

circle to end up there.

How far will Ben walk when he reaches the point (0, -1)?

As Ben continues to walk, he will reach (0, -1) again.

Give another distance that Ben

could walk to reach

(0, -1).

As Ben continues to walk, he will reach (0, -1) again.

Describe a method you could use to generate a

large number of these

distances.

Ben runs 100 meters along the circle.What quadrant is he in at the end of this 100-meter

run? 9155.152

100

2478.942*15

75222.52478.94100

• Note: Two angles are coterminal if they have the same initial and terminal sides.

• For instance, the angles 0 and 2 are coterminal, as are the angles and .

• You can find an angle that is coterminal to a given angle by adding or subtracting 2 (one revolution)

Example 1

For the positive angle subtract 2 to obtain a coterminal angle

Example 2

ExampleConverting from Degrees to Radians

ExampleConverting from Radians to Degrees

If you have a calculator with a “radian-to-degree” conversion key, try using it to verify the result shown in part c

On your own:1. What quadrant will Austin be in if he walks

around the unit circle for 45 radians?

2. Convert from degrees to radians:a) 30° b) 350° c) -52°

3. Convert from radians to degrees:a) π/5 b) 5π/7 c) -33

1st Quadrant

π/6 35π/18 -13π/45

36° ≈ 128.6° -5940/π °

Arc Length

Example A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of as shown in the figure

On your own:Find the perimeter of a 45° slice of a large (10 in radius) pizza.

s = 7.9P =10 + 10 + sP = 27.9 in

v = wr = s/ts = θrw = θ/t

Variables to know:θ = angle (in radians)r = radius (distance)s = arc length (distance)t = time v = linear velocity (distance/time)ω = angular velocity (radians/time)

Q: The second hand of a clock is 10.2 centimeters long, as shown in Figure. Find the linear speed of the tip of this second hand as it passes around the clock face.

Q: A Ferris wheel with a 50-foot radius makes 1.5 revolutions per minute.a. Find the angular speed of the Ferris wheel in radians per minute.b. Find the linear speed of the Ferris wheel.

In groups, practice on the worksheet problems