angle measures in degrees & radians trigonometry 1.0 students understand the notation of angle...

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!