13-3 The Unit Circle

13.3.1 Convert angle measures between degrees and radians.

13.3.2Find the values of trigonometric functions on the unit circle.

13.3.3 Learn strategies to quickly recall the angles (in degrees and radians) as well as the trig. values for sin, cos, and tan of the unit circle.

Learning Goals

You have measured angles in degrees; today you will

learn to measure angles on another scale called

_____________.

Example 1: Converting Between Degrees and Radians

Convert each measure from degrees to radians or from radians to degrees.

A. – 60° B.

C. 80° D.

Warm-Up

Label all angles that are

multiples of 30o, 45o, 60o,

and 90o around the origin.

13-3 The Unit CircleTHE UNIT CIRCLEWhat are the following exact values?

sin(45)= sin(30)= sin(60)=

cos(45)= cos(30)= cos(60)=

tan(45)= tan(30)= tan(60)=

13-3 The Unit Circle

Example 2: Using the Unit Circle to Evaluate Trig Functions

Use the unit circle to find the exact value of each trigonometric function.

cos 225°A. tan B.

sin 315° tan 180°C.

E.

D.

QII

Sin θ

Cos θ

Tan θ

QI

Sin θ

Cos θ

Tan θ

QIII

Sin θ

Cos θ

Tan θ

QIV

Sin θ

Cos θ

Tan θ

Example 3: Using Reference Angles to Evaluate Trig Functions

A. Use a reference angle to find the exact value of the sine, cosine, and tangent of 330°.

13-3 The Unit Circle

Use a reference angle to find the exact value of the sine, cosine, and tangent of the angles listed below.

A. 270° B. 11π 6

C. -30°

13-3 The Unit Circle

Example 4: Automobile Application

A. A tire of a car makes 653 complete rotations in 1 min. The diameter of the tire is 0.65 m. To the nearest meter, how far does the car travel in 1 s?

Step 1 Find the radius of the tire.

Step 2 Find the angle θ through which the tire rotates in 1 second.

Step 3 Find the length of the arc intercepted by radians.