13.3 – Radian Measures

Radian MeasureFind the circumference of a circle with the given radius or diameter. Round your answer to the nearest tenth.

1. radius 4 in. 2. diameter 70 m

3. radius 8 mi 4. diameter 3.4 ft

5. radius 5 mm 6. diameter 6.3 cm

Radian Measure1. C = 2 r = 2 (4 in.) 25.1 in.

2. C = d = (70 m) 219.9 m

3. C = 2 r = 2 (8 mi) 50.3 mi

4. C = d = (3.4 ft) 10.7 ft

5. C = 2 r = 2 (5 mm) 31.4 mm

6. C = d = (6.3 cm) 19.8 cm

Solutions

Vocabulary and Definitions

• A central angle of a circle is an angle with a vertex at the center of the circle.

• An intercepted arc is the arc that is “captured” by the central angle.

Vocabulary and Definitions

• When the central angle intercepts an arc that has the same length as a radius of the circle, the measure of the angle is defined as a radian.

r

r

Like degrees, radians measure the amount of rotation from the initial side to the terminal side of the angle.

The Unit CircleThe Unit Circle

The “Magic” ProportionThis proportion can be used to convert to and from Degrees to Radians.

Degrees°180° = r radians

radians

Example: Find the radian measure of angle of 45°.

Write a proportion.45°180° = r radians

radians

An angle of 45° measures about 0.785 radians.

Write the cross-products.45 • = 180 • r

Divide each side by 45.r = 45 •180

= 0.785 Simplify.4

The “Magic” ProportionThis proportion can be used to convert to and from Degrees to Radians.

Degrees°180° = r radians

radians

Example: Find the radian measure of angle of -270°.

Write a proportion.-270°180° = r radians

radians

An angle of -270° measures about -4.71 radians.

Write the cross-products.-270 • = 180 • r

Divide each side by 45.r = -270 •180

-4.71 Simplify.2-3

Let’s Try SomeConvert the following to radiansa. 390o b. 54o c. 180o

Example

= 390° Simplify.

Find the degree measure of .613

Write a proportion.613

radians= d°

180

• 180 = • d Write the cross-product.613

d = Divide each side by .13 • 1806 •1

30

An angle of radians measures 390°.613

Example Find the degree measure of an angle of – radians.

23

= –270°

An angle of – radians measures –270°.23

– radians • = – radians •23 180°

radians 23 180°

radians1

90Multiply by 180°

radians .

Radian Measure

Find the radian measure of an angle of 54°.

5 4° • radians = 54° • radians Multiply by radians.180° 180° 180°3

10

103 radians= Simplify.

An angle of 54° measures radians.103

Draw the angle.

Radian MeasureFind the exact values of cos and sin .radians3 radians3

radians • = 60° Convert to degrees.3180°radians

Complete a 30°-60°-90° triangle.

The hypotenuse has length 1.

radians3Thus, cos = 12

and sin radians3 = . 32

The shorter leg is the length of the hypotenuse, and the longer leg is 3 times the length of the shorter leg.

12

Radian MeasureUse this circle to find length s to the nearest tenth.

s = r Use the formula.

The arc has length 22.0 in.

= 7 Simplify.

22.0 Use a calculator.

= 6 • Substitute 6 for r and for .7 6

7 6

Radian MeasureAnother satellite completes one orbit around Earth every 4 h. The satellite orbits 2900 km above Earth’s surface. How far does the satellite travel in 1 h?

Since one complete rotation (orbit) takes 4 h, the satellite completes of a rotation in 1 h.

14

Step 1: Find the radius of the satellite’s orbit.

r = 6400 + 2900 Add the radius of Earth and the distance

from Earth’s surface to the satellite.

= 9300

Radian Measure(continued)

The satellite travels about 14,608 km in 1 h.

Step 2: Find the measure of the central angle the satellite travels through in 1 h.

= • 2 Multiply the fraction of the rotation by the number of radians in one complete rotation.

= • Simplify.

14

12

Step 3: Find s for = .

s = r Use the formula.

= 9300 • Substitute 9300 for r and for .

14608 Simplify.

2

2 2