13-2 General Angles and Radian Measure

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Transcript 13-2 General Angles and Radian Measure

13-2 General Angles and Radian Measure

What is pi?

A delicious dessert? Yes, but not quite… Def: The ratio of a circle’s circumference to it’s diameter.

What is a radian?

Def: Standard unit of angle measure, equal to the arc length of the corresponding angle on the unit circle.

What is the Unit Circle you ask??

It’s just what it sounds like, the Unit Circle is a circle that has a radius of One Unit. The Unit Circle gives us some very important principles in Trigonometry.

What is a radian?

We always start with an angle created from the x-axis-also called the standard position.

The ending position (or terminal side) of the angle gives us the radian measure.

θ

r How to convert from degrees to radians:

deg

ree

180 

radians

Standard position

Ex 1: Convert 120° to radians

deg

ree

180 

radians

 120 180 

radians

 π( 2 2 3  3 ) ( 

radians

radians

deg

ree

180 deg 180

Ex 2: Convert to degrees

4 

rees radians

   4  180( deg deg

rees

180

rees

 ) ( )180 45 4 deg

rees

180   4  1 

Arc Length and Area of a Sector

Arc Length:

s

r

 1

Area of Sector:

2

r

2 

(θ is in radians) θ r Arc Length (s)

Surface Area = Area of the sector

Ex 3: You cut yourself a slice of pie, from an apple pie that has a radius of 3 in. The slice that you cut has a central angle of 60°. What is the surface area of the pie you are about to consume? How much crust are you eating?

1 2

Crust = Arc Length

s

r

2 

r

  1 2   3 2 ( 3 (  3  ) 3  ) 3  2   r= 3 θ   3

So…What does the a radian LOOK like?

A circle has 360°, how many radians is that?

360 180 

radians

 π( ) (

radians

 2  

radians

If 360° is 2π, then how many radians is 180° do you suppose?

 2 

To find the rest of the angles, we simply “slice” up the pi… Ex 4: Draw an angle with the given measure in standard position.

3  4 3   2

Ex 5: Draw an angle with

4

the given measure in

standard position.

 2  3  2  3 2 

810° Ex 6: Draw an angle with the given measure in standard position.

This is obviously larger then 360°, so what will this angle look like?

810° 810°-720°=90°

 2 

Did you notice that 810° looked a lot like 90°? We call these angles Coterminal angles, because their terminal sides (ending sides) coincide.

What are some other coterminal angles of 90°?

Terminal Side 90°/810°

Are there negative coterminal angles of 90°?

2 