Objectives Students will learn how to use special right triangles to find the radian and degrees.

Vocabulary standard position initial side terminal side angle of rotation coterminal angle reference angle

A unit circle is a circle with a radius of 1 unit 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. Therefore, for an angle θ in the standard position:

So the coordinates of P can be written as (x, y)  (cosθ, sinθ). The diagram shows the equivalent degree and radian measure of special angles, as well as the corresponding x- and y-coordinates of points on the unit circle.

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° The angle passes through the point on the unit circle. cos 225° = x Use cos θ = x.

Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. tan The angle passes through the point on the unit circle. Use tan θ = .

Because r is a distance, its value is always positive, regardless of the sign of x and y. Helpful Hint

Special Right Triangle Special Right Triangle The 45-45-90 Special Right Triangle The 30-60-90 Special Right Triangle

Trigonometric Functions and Reference Angles

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 to find the exact value of the sine, cosine, and tangent of 330°. Step 1 Find the measure of the reference angle. The reference angle measures 360-330 = 30°

Example 3 Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x.

Example 3 Continued Step 3 Adjust the signs, if needed. In Quadrant IV, sin θ is negative. In Quadrant IV, cos θ is positive. In Quadrant IV, tan θ is negative.

Evaluate the trigonometric function without using a calculator Evaluate the trigonometric function without using a calculator. Angles are given in degrees. cos 135° Find the reference angle for Quadrant II. Use the sides of a 45°-45°−90° triangle, then apply the sign of cosine in Quadrant II.

Evaluate the trigonometric function without using a calculator Evaluate the trigonometric function without using a calculator. Angles are given in degrees. cos 150° Find the reference angle for Quadrant II. Use the sides of a 30°−60°−90° triangle, then apply the sign of cosine in Quadrant II.

Evaluate the trigonometric function without using a calculator Evaluate the trigonometric function without using a calculator. Angles are given in degrees. tan(−60°) Find the reference angle for Quadrant IV. Use the sides of a 30°−60°−90° triangle, then apply the sign of cosine in Quadrant IV.