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x y initial side terminal side vertex Measuring Angles
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Angle Measures in Degrees & Radians
Trigonometry 1.0Students understand the notation of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians.
O A
B
- (the side where the begins) – is always the positive x-axis.
O A
B•
• The initial side (OA)
The vertex is always at the origin.
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.
Angles in Standard Position
x
y
0
90
180
270
360initial side
terminal side
vertex
150
Measuring Angles
Degrees?
180˚
The terminal side ends up in quadrant __.
Positive s are drawn counterclockwise.
Draw a 135˚ .
90˚
270˚
0˚
Start on the positive x-axis.
Quadrants?III
III IV
or 360˚
135˚
II
Negative angles are drawn clockwise. (Start on the positive x-axis.)
- 60˚What Quadrant? ___IV
- 210˚ What Quadrant? __II
0˚
-90˚
-180˚
-270˚
or -360˚
-60˚
-210˚
Radian Measure
The distance around a circle is 360°.
x
y
r
The distance around a circle is also 2πr.
So, 2πr = 360°.
In trigonometry, we deal with a “unit circle” where the radius is 1.
Therefore: 2π = 360° or π = 180°
That’s radian measure!
Unit Circle
x
y
030
456090120
150
135
210
225240
270300
315
330
6
43
0
22
3
34
56
180
76
54
43
32
53
74
116
180
6
•
30° 5 ___3
53
• 180
300 °
4 ___9
80°9 ___2
810°
___6
To change radians to degrees, multiply by .180
You try it:
To change degrees to radians,multiply by .
180
60˚ = ___
60 • 180
3
20 • 180
20˚ = __9
80˚ = ___49
4
45˚ = __
You try it:
Coterminal Angles in Radians
Angle has measure of 9π/4 (405°)
Angle has measure of -7π/4 (-315°)
Angle has measure of π/4 (45°)
To find coterminal angles in radians, add or subtract 2π.
Coterminal Angles have the same initial side
the same vertex
the same terminal side
but different measures
Find two coterminal angles, one positive and one negative.
2π/3
- 5π/7
15π/4
Positive Negative
8π/3
9π/7
7π/4
-4π/3
-19π/7
-π/4
± 6π/3
± 14π/7
- 8π/4
Find two coterminal angles, one positive and one negative for 140°.
To find coterminal angles in degrees: Add 360° or Subtract 360°
140°
140° + 360° = 500°
140° - 360° = -220°
y
Find two coterminal angles, one positive and one negative.
320°
- 245°
880°
Positive Negative
680 ° -40 °
115 °
160 °
-605 °
-200 °
± 360°
± 360°
- 720° - 360°
Complementary & Supplementary Angles
Complementary angles add to 90° or 2
Supplementary angles add to 180° or
If possible, find the complement and supplement of the angle.
70°
Complement Supplement
20 ° 110 °
6
45
90°- 70° 180°- 70°
36 6
3
6
56
none5 810 10
45
5
Arc Lengths = rθ
arc length = radius · angle (in radians)
s
rθ
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!