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