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
Jan 02, 2016
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.