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.
How to convert from degrees to
radians: deg ree
radians
180


r
θ
Standard position
Ex 1: Convert 120° to radians
deg ree radians

180

120 radians

180

2 radians
π( )(
)π
3

2
 radians
3
Ex 2: Convert
deg ree radians

180


deg rees 4

180

deg rees  1
 
180
4 

4
to degrees
deg
rees
1
180(
) ( )180
180
4
deg rees  45
Arc Length and Area of a Sector
Arc Length:
s  r
1 2
Area of Sector: r 
2
(θ is in radians)
Arc Length
(s)
θ
r
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?
Surface Area = Area of the sector
1 2
1 2 
3
 r   3 ( ) 
2
2
3
2
Crust = Arc Length

s  r  3( )  
3
θ
r= 3

3
So…What does the a radian LOOK
like?
A circle has 360°, how many radians is that?
360 radians If 360° is 2π, then how many

radians is 180° do you
180

suppose?
radians
π( 2)(

)π
2  radians

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

3
4
Ex 5: Draw an angle with
the given measure in
standard position.
2

3
4
2

2

3
2
Ex 6: Draw an angle with the given
measure in standard position.
810°
This is obviously larger then 360°, so what will this
angle look like?
810°
810°-720°=90°

720°
360°
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°?
Are there negative
coterminal angles of 90°?
Terminal Side

90°/810°
2