Transcript Angles and radians

Radians and Angles
Welcome to Trigonometry!!
Starring
Angles
The Coterminal Angles
Rad Radian
Supp & Comp Angles
Degree
The Converter

And introducing…
THE UNIT CIRCLE
1
You & I are
gonna be great
friends!
0.5
-1
1
-0.5
-1
2
Angle- formed by rotating a ray about its endpoint
(vertex)
Terminal Side Ending position
Initial Side Starting position
Initial side on positive x-axis
and the vertex is on the origin
Standard Position
Angle describes the amount and direction of rotation
120°
–210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
Angles can be measured in two kinds of units:
Degrees and Radians
Degrees are 360 equal sized sections of a circle
Radians, or “Rads” are… based on a circle &
easier seen than said, but first a reminder about
circles.
Angles can be measured in two kinds of units:
Degrees and Radians
Degrees are 360 equal sized sections of a circle
Radians, or “Rads” are… based on a circle &
easier seen than said, but first a reminder about
circles.
“Circles” are like pizza
The key to any pizza
is the crust
At Uncle Luggi’s…

The Crust is D
C = D 3+
C = 2r
A Portion of the Circumference: a Radian
This is:
5cm
1
1 radian
2
2
2
5cm
What it means is at the
place where the radius
equals the portion of the
circumference is:
1 radian
1 Radian = measure of central angle, , that intercepts the arc that
has the same length as the radius of the circle
IF the Radius is “One Unit” …
A semi-circle’s circumference can be found by algebra
By Algebra:
1
unit
1
2
2
2
1 unit
1
2
1
2
C = 2r
2
2
C = r
C = r
IF the Radius is “One Unit” …
It’s called the unit circle, and it means that
1
2
 Radians
180° Degrees
2
360° Degrees
2
1 unit
2 Radians
OR, as uncle luggi, would say you get 2 Pi’s in one
circle – Just slice’m to get rad pizza
Slice the pizza in 90° intervals

2
2 = 90°
4 = 360°
2
=
180°
2
3
2 = 270°
OR, in 45° intervals
2
3
4
4 = 135°
4 = 90°
 = 45°
4
4 = 180°
5
4 = 225°
7
4 = 315°
So, here’s a few common angles
in both degrees and radians
30º
45º
60º
90º
180º
360º
To convert from degrees
To convert from radians
radians, multiply by
degrees, multiply by
Convert to radians:
135 

180
3

4
 80 

180
4

9

180
180

To convert from degrees
To convert from radians
Convert to degrees:
 8 180

  480
3

5 180


6 
150
radians, multiply by
degrees, multiply by

180
180

So, you think you
got it now?
Angles describes the amount and direction of rotation,
but since they are periodic, many more angles can
terminate at the same location. When they do, we call
them “CO-TERMINAL”
120°
–240°
70°
–290°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
Coterminal Angles: Two angles with the same initial
and terminal sides
Find a positive coterminal angle to 20º
Find a negative coterminal angle to 20º
15
Find 2 coterminal angles to
4
20  360  380
20  360  340
23
15 8
15


 2 
4
4
4
4
15
15 8
 2 

4
4
4
7 8




4
4
4
Now, you try…
 2
Find two coterminal angles (+ & -) to
3
What did you find?
4  8
,
3
3
These are just two possible answers.
Remember…there are more! 
Complementary Angles: Two angles whose sum is 90

6

2


 3    2  
6
6 6
6
3
Supplementary Angles: Two angles whose sum is 180
2
3
2
3 2 




3
3
3
3
1 degree = 60 minutes
1 minute = 60 seconds
1° = 60 
1  = 60 
3600
So … 1 degree = _________seconds
Express 365010as decimal degrees
50
36 
60
10
3000
10

 36 

3600
3600 3600
3010
 36 
3600
 36.8361
Express 50.525 in degrees, minutes, seconds
50º + .525(60) 
50º + 36.5
50º + 36 + .5(60) 
50 degrees, 36 minutes, 30 seconds