4.1: Radian and Degree Measure

Objectives:•To use radian measure of an angle•To convert angle measures back and forth between radians and degrees•To find coterminal angles•To find arc length, linear speed, and angular speed

We are going to look at angles on the coordinate plane… An angle is determined by rotating a ray about its

endpoint Starting position: Initial side (does not move) Ending position: Terminal side (side that rotates) Standard Position: vertex at the origin; initial side

coincides with the positive x-axis Positive Angle: rotates counterclockwise (CCW) Negative Angle: rotates clockwise (CW)

Positive Angles

Negative Angle

1 complete rotation: 360⁰Angles are labeled with Greek letters: α (alpha), β (beta), and θ (theta)Angles that have the same initial and terminal

sides are called coterminal angles

RADIAN MEASURE (just another unit of measure!)

Two ways to measure an angle: radians and degrees For radians, use the central angle of a circle

s=rr

• s= arc length intercepted by angle• One radian is the measure of a

central angle, Ѳ, that intercepts an arc, s, equal to the length of the radius, r

• One complete rotation of a circle = 360°• Circumference of a circle: 2 r• The arc of a full circle = circumference

s= 2 rSince s= r, one full rotation in radians= 2 =360 °

, so just over 6 radians in a circle

28.62

(1 revolution)

½ a revolution =

¼ a revolution

1/6 a revolution=

1/8 a revolution=

3602

Quadrant 1Quadrant 2

Quadrant 3 Quadrant 4

20

2

2

3 2

2

3

Coterminal angles: same initial side and terminal side

Name a negative coterminal angle:

2

3

2

You can find an angle that is coterminal to a given angle by adding or subtracting

Find a positive and negative coterminal angle:

2

2

7.4

3

2.3

3.2

6.1

Finding Complementary and Supplementary Angles

Complement for :

Supplement for :

If >90, it has no complement. If > 180, it has no supplement.

2

Find the complement and the supplement of the angle, if possible.

:6.2

:3

2.1

Degree Measure

So………

Converting between degrees and radians:1. Degrees →radians: multiply degrees by

2. Radians → degrees: multiply radians by

180

2360

deg180

1

1801

rad

rad

180

180

Convert to Radians:

1. 320°

2. 45 °

3. -135 °

4. 270 °

5. 540 °

Convert to Radians:

4

5.4

5

6.3

3.2

2.1

Sketching Angles in Standard Position: Vertex is at origin, start at 0°

1. 2. 60°

4

3

3. 6

13

Finding Arc Length:

arc length = slength of radius = rcentral angle = (in radians)

and

rs r

s

Examples:

1. A circle has a radius of 4 inches. Find the length of the arc intercepted by and angle of 120°.

2. Find if the arc length is 32m and r = 7 m.

Linear and Angular Speed

The formula for the length of a circular arc can be used to analyze the motion of a particle moving at a constant speed along a circular path.

Linear speed: (how fast the particle moves)

Angular speed:(how fast the angle changes)

t

r

time

arclength

ttime

central

A neighborhood carnival has a Ferris Wheel whose radius is 30 ft. You measure the time it takes for one revolution to be 70 sec. What is the linear and angular speed of the Ferris Wheel?

2. A lawn roller with a 10-inch radius makes 1.2 revolutions per second.a.) Find the angular speed per second.b.) Find the speed of the tractor pulling the roller.