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# 13.3 – Radian Measures - PowerPoint PPT Presentation

13.3 – Radian Measures. Radian Measure. 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 mi 4. diameter 3.4 ft 5. radius 5 mm 6. diameter 6.3 cm. Radian Measure.

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Presentation Transcript

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 mi 4. diameter 3.4 ft

5. radius 5 mm 6. 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

Write a proportion.

Degrees°

180°

45°

180°

=

=

Write the cross-products.

45 • = 180 • r

45 •

180

r =

Divide each side by 45.

4

= 0.785 Simplify.

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

This proportion can be used to convert to and from

Write a proportion.

Degrees°

180°

-270°

180°

=

=

Write the cross-products.

-270 • = 180 • r

-270 •

180

r =

Divide each side by 45.

2

-4.71 Simplify.

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

-3

a. 390o b. 54o c. 180o

13

Find the degree measure of .

6

6

13

13

=

Write a proportion.

180

6

6

• 180 = • dWrite the cross-product.

d = Divide each side by .

30

13 • 180

6 •

1

An angle of radians measures 390°.

Example

= 390° Simplify.

.

180°

Multiply by

1

3

3

3

3

2

2

2

2

An angle of – radians measures –270°.

180°

180°

Example

Find the degree measure of an angle of – radians.

= –270°

3

=

Simplify.

3

180°

An angle of 54° measures radians.

10

180°

180°

10

10

Find the radian measure of an angle of 54°.

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.

1

2

Thus, cos

=

3

2

and sin

= .

Find the exact values of cos and sin .

The hypotenuse has length 1.

6

= 6 • Substitute 6 for r and for .

7

6

= 7 Simplify.

22.0 Use a calculator.

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

s = rUse the formula.

The arc has length 22.0 in.

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 + 2900 Add 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 .

14608 Simplify.

2

2