Warm Up Find the measure of the reference angle for each given angle. 1. 120° 2. 225° Convert the following from degrees to radians or radians to degrees. 3. 150° 4. 2π radians

A unit circle is a circle with a radius of 1 unit. For every point P(x, y) on the unit circle, the value of r is 1. (0,1) (-1,0) (1,0) (0,-1)

Why the unit circle? We use the unit circle to help us evaluate any trigonometric function. Ex. sin 60° = Ex. cos Π/2 =

Explore the Unit Circle http://www.geogebratube.org/student/m26809 Take a few moments to go to this website and explore the unit circle. Write down any interesting things you notice about the sine, cosine, and tangent values as you change the angle.

Creating/ completing the unit circle We will keep in mind 2 things The unit circle is symmetric in many ways (reference angles) The quadrant helps us to find out which is positive and which is negative.

The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position.

So the coordinates of P can be written as (cosθ, sinθ). tanθ = sinθ cosθ

Example 2A: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. cos 225°

Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. tan

Check It Out! Example 1a Use the unit circle to find the exact value of each trigonometric function. sin 315°

Check It Out! Example 1b Use the unit circle to find the exact value of each trigonometric function. tan 180°

Check It Out! Example 1c Use the unit circle to find the exact value of each trigonometric function.

You can use reference angles and Quadrant I of the unit circle to determine the values of trigonometric functions around the unit circle.

The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position.

Example 3: Using Reference Angles to Evaluate Trigonometric functions Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of 330°.

Check It Out! Example 3a Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of 270°. 270°

Check It Out! Example 3b Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of each angle.

Check It Out! Example 3c Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of each angle. –30° –30°

Check It Out! Example 3c Use a coterminal angle and the unit circle to find the exact value of the sine, cosine, and tangent of each angle. 390°

Check It Out! Example 3c Use a coterminal angle and or the unit circle to find the exact value of the sine, cosine, and tangent of each angle. 5π/2

Arc Length

If you know the measure of a central angle of a circle, you can determine the length s of the arc intercepted by the angle.

Example 4: Automobile Application 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?

Example 4 Continued

Example 4 Continued Step 3 Find the length of the arc intercepted by radians.

Check It Out! Example 4 An minute hand on Big Ben’s Clock Tower in London is 14 ft long. To the nearest tenth of a foot, how far does the tip of the minute hand travel in 1 minute? Step 1 Find the radius of the clock. r =14

Check It Out! Example 4 Continued

Check It Out! Example 4 Continued Step 3 Find the length of the arc intercepted by radians.

Lesson Quiz: Part I Convert each measure from degrees to radians or from radians to degrees. 1. 100° 2. 3. Use the unit circle to find the exact value of . 4. Use a reference angle to find the exact value of the sine, cosine, and tangent of

Lesson Quiz: Part II 5. A carpenter is designing a curved piece of molding for the ceiling of a museum. The curve will be an arc of a circle with a radius of 3 m. The central angle will measure 120°. To the nearest tenth of a meter, what will be the length of the molding?