Transcript TrigonometryRadianDegreeMeasures

Angle Measures
in Degrees & Radians
B

O
A
Trigonometry 1.0
Students understand the notation of angle and how to measure it, in both
degrees and radians. They can convert between degrees and radians.
Angles in Standard Position
B•

O
•
A
The vertex is always at the origin.
The initial side (OA) - (the side
where the  begins) – is always
the positive x-axis.
The terminal side (OB) - is the ray that forms the 
An angle is in standard position when:
1. The initial side is the positive x-axis
2. The vertex is at the origin.
Measuring Angles
terminal
side
y
90
150
0
initial side 360x
180
vertex
270
Degrees?
90˚
II
I
180˚
Quadrants?
0˚ or 360˚
III
IV
270˚
135˚
Draw a 135˚ .
Positive s are drawn
counterclockwise.
Start on the positive x-axis.
The terminal side ends up in quadrant __.
II
Negative angles are drawn clockwise.
(Start on the positive x-axis.)
-270˚
- 60˚
What Quadrant? ___
IV
-180˚
0˚or -360˚
-90˚
II
- 210˚ What Quadrant? __
-60˚
-210˚
Radian Measure
The distance around a circle is 360°.
y
The distance around a circle is also 2πr.
So, 2πr = 360°.
r
In trigonometry, we deal with
a “unit circle” where the
radius is 1.
Therefore:
2π = 360°
or
π = 180°
That’s radian measure!
x
y

2

2
3
3
3 
90
60
120
4
4
45
135

5
30 6
6 150
 180
0 0
Unit Circle
7 210
330 11
6
6
315
5 225
7
240
300
4 4
270
5 4
3
3
3
2
x

To change radians to degrees,
multiply by 180 .


30°
 ___
6
5
 300
___ °
3
5 180
•
3

 180
•
6 
You try it:
4
80°
 ___
9 
9
 810°
___
2
To change degrees to radians,
multiply by  .
180


20˚ = __
9
60˚ = ___
3

20 •
180

60 •
180
You try it:
4
80˚ = ___
9


45˚ = __
4
Coterminal Angles in Radians
Coterminal Angles have the same initial side
the same vertex
the same terminal side
but different measures
Angle has measure of π/4 (45°)
Angle has measure of -7π/4 (-315°)
Angle has measure of 9π/4 (405°)
To find coterminal angles in radians, add or subtract 2π.
Find two coterminal angles, one
positive and one negative.
Positive
Negative
8π/3
-4π/3
9π/7
-19π/7
7π/4
-π/4
2π/3
± 6π/3
- 5π/7
± 14π/7
15π/4
- 8π/4
Find two coterminal angles,
one positive and one negative for 140°.
140°
y
To find coterminal angles in degrees:
Add 360° or Subtract 360°
140° + 360° = 500°
140° - 360° = -220°
Find two coterminal angles,
one positive and one negative.
320°
Positive
Negative
680 °
-40 °
115 °
-605 °
160 °
-200 °
± 360°
- 245°
± 360°
880°
- 720°
- 360°
Complementary &
Supplementary Angles

Complementary angles add to 90° or
Supplementary angles add to 180° or
2

If possible, find the complement and
supplement of the angle.
Complement
70°
20 °
90°- 70°

6
4
5

3
3


6
6
none
5
8

10
10
Supplement
110 °
180°- 70°
5
6
 

6
54
 
5
Arc Length
s = rθ
arc length = radius · angle (in radians)
s
r
θ
Determine the arc length
of a circle of radius 6 cm
intercepted by an angle
of π/2.
s = (π/2)·6
s = 3π cm
If the central angle is given in degrees,
change it to radians in the problem!
Find the arc length to the nearest tenth of a
centimeter of a circle of radius 7 cm that is
intercepted by a central angle of 85°.
s = 7(85)(π/180)
s = 10.4 cm
Homework
Page
Memorize the unit circle!