angles and the unit circle. an angle is in standard position when: 1) the vertex is at the origin....

Section 13.2

Angles and the Unit Circle

An angle is in standard position when:

1) The vertex is at the origin.

2) One leg is on the positive x – axis.

(This is the initial side.)

3) The second ray moves in the direction of the angle

(This is the terminal side.)

Standard Position Angle (of 60°)

If the movement

from the initial side

to the terminal side

of the angle is

counterclockwise,

then the angle

measures positive.

Reading Angles

+135°

If the movement

from the initial side

to the terminal side

of the angle is

clockwise, then the

angle measures

negative.

Reading Angles

– 225°

Measuring Angles

–315° 240° –110°

1) 2) 3)

Two angles in standard

position that have the same

terminal side are coterminal

angles.

To find a coterminal angle

between 0 ° and 360 ° either

add or subtract 360 ° until you

get the number that you want.

Find the measure of an angle between 0 ° and 360 ° coterminal with each given angle:

4) 575°215 °

5) –356°4 °

6) –210°150 °

7) –180°180 °

Coterminal Angles

The Unit Circle:

1) Is centered at the origin,

2) Has a radius of 1,

3) Has points that relate to

periodic functions.

Normally, the angle measurement

is referred to as θ (theta).

The Unit Circle

1

1

For all values using SOH, CAH, TOA the H value is always 1.

We can use the Pythagorean Theorem to find the rest.

cos θ is the x coordinate. sin θ is the y coordinate. Let’s find sin (60°) and

cos (60°).

On a 30-60-90 triangle the short side is ½ the hypotenuse.

So, cos (60°) = ½. a2 + b2 = c2

(½)2 + b2 = 12

¼ + b2 = 1

b2 = ¾

b = √(¾) = √(3)/2 So, sin (60°) = √(3)/2

Finding Values on the Unit Circle

1

½

√3 2

Continue to find the values on the Unit Circle Find cos 0° and sin 0°

Find cos 30° and sin 30°

Find cos 45° and sin 45°

Find cos 90° and sin 90°

Finding Values on the Unit Circle

1

1

(1, 0)

(0, 1)(½, √3/2)

(√2/2, √2/2)

(√3/2, ½)

These patterns repeat for the right x and y values.

The values can be either positive or negative based on the x and y axes.

Use this information to fill in the worksheets with exact values

All Four Quadrants

Finding Values

Locate the Unit Circle diagram from before.

8) 9)

sin (–60°) = –√(3)/2cos (–60°) = ½

sin (–60°) = –½cos (–60°) = √(3)/2

10) –390°

11) –30°

During this lesson we completed page 708 # 1 – 27 odd.

For more practice, complete the even problems

Extra Practice