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13.3 – Radian Measures

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13.3 – Radian Measures

Find 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 mi4.diameter 3.4 ft

5.radius 5 mm6.diameter 6.3 cm

1. 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

- 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.

- 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.

This proportion can be used to convert to and from

Degrees to Radians.

Write a proportion.

Degrees°

180°

45°

180°

=

=

Write the cross-products.

r radians

radians

r radians

radians

45 • = 180 • r

45 •

180

r =

Divide each side by 45.

4

= 0.785Simplify.

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

An angle of 45° measures about 0.785 radians.

This proportion can be used to convert to and from

Degrees to Radians.

Write a proportion.

Degrees°

180°

-270°

180°

=

=

Write the cross-products.

r radians

radians

r radians

radians

-270 • = 180 • r

-270 •

180

r =

Divide each side by 45.

2

-4.71Simplify.

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

-3

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

Convert the following to radians

a.390ob. 54o c. 180o

13

13

Find the degree measure of .

6

6

13

13

=

Write a proportion.

d°

180

6

6

radians

• 180 = • dWrite the cross-product.

d = Divide each side by .

30

13 • 180

6 •

1

An angle of radians measures 390°.

= 390°Simplify.

90

.

180°

– radians • = – radians •

Multiply by

radians

1

3

3

3

3

2

2

2

2

An angle of – radians measures –270°.

180°

180°

radians

radians

Find the degree measure of an angle of – radians.

= –270°

3

3

=

radians

Simplify.

3

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

180°

An angle of 54° measures radians.

10

180°

180°

10

10

Find the radian measure of an angle of 54°.

radians

radians

radians

radians

radians • = 60° Convert to degrees.

Draw the angle.

Complete a 30°-60°-90° triangle.

3

3

3

3

3

1

2

180°

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

radians

1

2

Thus, cos

=

3

2

and sin

= .

Find the exact values of cos and sin .

The hypotenuse has length 1.

7

6

= 6 • Substitute 6 for r and for .

7

6

= 7Simplify.

22.0Use a calculator.

Use this circle to find length s to the nearest tenth.

s = rUse the formula.

The arc has length 22.0 in.

1

4

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

Another 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?

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

r = 6400 + 2900Add the radius of Earth and the distance from Earth’s surface to the satellite.

= 9300

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.

1

4

1

2

2

Step 3: Find s for = .

s = rUse the formula.

= 9300 • Substitute 9300 for r and for .

14608Simplify.

2

2

(continued)

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