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### Section 13.3

Radian Measure

Central Angles

Central Angle

Example

8

60°

4

A central angle has its vertex at the center of the circle

This is similar to the angles formed on the Unit Circle

The cos(60°) on the Unit Circle is ½

Find cos(60°) if the radius is 8.

Simplifying gives us ½

Radian Measurements

intercepted arc

central angle

r

1 radian

r

Any central angle will have its rays contact the circle.

The portion of the circle on the interior of the angle is an intercepted arc

If the intercepted arc has a length equal to the radius of the circle, the angle has a measurement of 1 radian

Converting Measurements

The circumference of a circle is 2πr making 2π radians in any circle

We can change from degrees to radians using the following proportion:

To convert degrees to radians, multiply by

To convert radians to degrees, multiply by

Practice: Degrees to Radians

Write each measure in radians. Write each answer in terms of π and as a decimal rounded to the nearest hundredth.

1) –300° 2) 150° 3) –90°

4) –60° 5) 160° 6) 20°

(5π)/6

2.62

–5π/3

– 5.24

–π/2

–1.57

–π/3

–1.05

8π/9

2.79

π/9

0.35

Practice: Radians to Degrees

Write each measure in degrees. Round the answer to the nearest degree, if necessary.

7) 3π 8) 11π/10 9) –2π/3

10) –3 11) 1.5712) 4.71

198°

540°

–120°

–172°

90°

270°

Homework

π

2

π

3

2π

3

3π

4

π

4

5π

6

π

6

0

π

2π

7π

6

11π

6

5π

4

7π

4

4π

3

5π

3

3π

2

For homework, use the conversion formulas to fill in the radian measurements on your Unit Circle diagram.

Arc Length

- We can use proportions to find the length of an arc contained inside of a central angle.
- The length of an arc compared to the entire circle is proportional to the central angle to the circle
- Simplifying this gives
s = rθ

Find the width of the parachute that is kept 22 ft from the person and fan out at 83°.

1) Convert 83° to radians.

θ ≈ 1.45 radians

2) s = rθ

= 22(1.45)

= 31.9 ft

Practice: Arc Length

13)

14)

15)

c

m

t

π

3

2π

3

9 ft

11π

6

3 cm

5 m

m ≈ 51.84

t = π

c ≈ 10.47

a

16)

17)

18)

11 cm

4π

3

6 in

2 m

z

5π

4

3π

4

w

a ≈ 25.13

w ≈ 4.71

z ≈ 43.20

Uses

Any application that involves orbit or circular travel can use radians and arc length

Some examples are length of a curve in a road for civil engineering, satellite or planetary orbits, safe fan speeds, and etc.

Colorado Springs is about 3082 miles form earth’s rotation axis. What is our rotational speed in miles per hour?

- Find the θ we travel each hour.
θ = (1/24) 2π = π/12

- Calculate s = rθ for the given situation
s = 3082(π/12) ≈806.87 mph

Two More Problems

A freeway on-ramp has a radius of 150 ft and the driver completes a –305° arc while using the ramp. What is the distance that the driver covers?

about 798.5 ft

A ceiling fan’s outer tip is 2.5 feet from the center of rotation. The fan completes an average of 5 rotations each second. What is the speed of the blade tip in miles per hour?

about 53.55 mph

Completed Work

We did exercises 1 – 12, 20 – 25, 27, and 47 starting on page 715.

For extra practice do 26, and 28 – 42.

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