Warm Up

Covert to radians.

135 degrees 540 degrees 1

Example 3 –

Converting from Degrees to Radians

a.

Multiply by  rad / 180  .

b.

Multiply by  rad / 180  .

2

Applications

3

Applications

The

radian measure

formula,  =

s

/

r

, can be used to measure arc length along a circle.

4

Example 5 –

Finding Arc Length

A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240  , as shown in Figure 1.12.

Figure 1.12

5

Example 5 –

Solution

To use the formula

s

=

r

 , first convert 240  to radian measure.

6

Example 5 –

Solution

Then, using a radius of

r

= 4 inches, you can find the arc length to be

s

=

r

 cont’d Note that the units for

r

determine the units for

r

 because  is given in radian measure, which has no units.

7

Your Turn

A circle has a radius of 5 inches. Find the length of the arc intercepted by a central angle of 270  , as shown in Figure 1.12.

Figure 1.12

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Applications

The formula for the length of a circular arc can help you analyze the motion of a particle moving at a

constant speed

along a circular path.

9

Example

10

Linear speed

Recall the formula d = rt which holds for and object whose average speed is known.

If we want to know the rate at which an object is moving around a circle we can adapt this formula to be

s = vt

where s is the arc length, t is the time, and v is the angular velocity. 11

Example

An object is traveling around a circle with a radius of 5 cm. If in 20 minutes a central angle of 1/3 radians is swept out, what is the linear speed.

We are looking for linear speed. Find s.

s

=

r

 = 5(1/3)=5/3 Use s = vt where s = 5/3, t = 20 sec. Thus 5/3 = v(20) or v = 5/60. Since v is in cm/sec we have v = 5cm/12sec. 12

Angular speed

13

14

Applications

A

sector

of a circle is the region bounded by two radii of the circle and their intercepted arc (see Figure 1.15).

Figure 1.15

15

Applications

Find the area of a sector where r = 6m and theta = 120 degrees. 16