Download - Angles and Radian Measure
Angles and Radian Measure
Alg ll/Trig – Trig Ch C day 2
Notice that the graphs of sine, cosine and tangent repeat. Because these functions “repeat” we say they are __________
Sine repeats every ____. Sine has a ________of _____ Cosine repeats every ____. Cosine has a _______ of ____ Tangent repeats every ____. Tangent has a ________ of ____
Find the period for each of the following functions:
PERIODIC
360 PERIOD 360360 PERIOD 360180 PERIOD 180
y = sec x y = csc x y = cot x
360 360 180
Radian MeasureConsider an angle such that its vertex is at the center of a circle.Let r = _________ and S be the _________ of the arc created by the angle . We will no longer measure angles in __________. They will now be measured in ________ defined as such:
RADIUS LENGTHDEGREESRADIAN
S
r S
rS
In other words, is the number of _______that make up the ______ that creates.
This means that the radian measure of an angle is simply a __________ of the _________________ of the circle.
FRACTIONCIRCUMFERENCE
RADII ARC
1 Revolution:
360 2 radians
360¾ Revolution:270
3 3270 2 =
4 2radians
½ Revolution:
1180 2 =
2radians
180 ¼ Revolution: 90
190 2 =
4 2radians
Conversion Formulas:
Degrees Radians:
Multiply by ___________
Radians Degrees:
Multiply by ___________ 180
180
Change 32 into radians. Change in degrees
2
3
832
180 45
2 180
1203
What is the formula for the rate (or speed) at which an object is moving?
Linear SpeedWhen an object is moving at a constant speed in a ____________path with a radius of, r, the linear speed of the object
is given by _______________, which is just ________________.
Angular SpeedThe angular speed of the object is the measure of how ______ the _______ of _________ for the object _________ and is given
by _________, where is an angle measure in ________ and t is _______.
rate×time = distance distancerate =
time
CIRCULAR
S rort t
distance
time
FAST
ANGLE ROTATION CHANGES
t
RADIANS
TIME
Example 3: A weather satellite orbits the Earth at an altitude of approximately 22,200 miles above Earth’s surface. The radius of the Earth is 3960 miles. If the satellite observes a fixed region on Earth and has a period of revolution of 24 hours, what is the linear speed of the satellite? What is its angular speed?
LINEAR SPEED:S rort t
Time: 24 hours
Earth satellite
linear speed
angular speed
3960 + 22,200
Radius: 26,160 miles
26,160 26848 /
24miles hr
2
ANGULAR SPEED: t
2/
24 12radians hr
0
120135
150
210
225240
270
(1, 0)
( , )
( , )
( , )
( , )
( , )
( , )
( , )
( , )
(-1, 0)
( , )
( , )
( , )
(0 , 1)
(0, -1)
360
1 23 2
2 2 2 2
3 2 1 2
2 2 2 2
1 2 3 2
1 2 3 2
2 2 2 2
3 2 1 2 3 2
2 2
1 23 2
90
180
2
3 2
2 3
3 4
5 6
7 6
5 4
4 3
30
45
60
300315
330
( , )3 21 2
1 2
2 2
6
4
3
5 37 4
11 6