1. Holt Geometry 8-Ext Trigonometry and the Unit Circle 8-Ext Trigonometry and the Unit Circle Holt Geometry Lesson Presentation Lesson Presentation

2. Holt Geometry 8-Ext Trigonometry and the Unit Circle Define trigonometric ratios for angle measures greater than or equal to 90°. Objective

3. Holt Geometry 8-Ext Trigonometry and the Unit Circle reference angle unit circle Vocabulary

4. Holt Geometry 8-Ext Trigonometry and the Unit Circle Rotations are used to extend the concept of trigonometric ratios to angle measures greater than or equal to 90°. Consider a ray with its endpoint at the origin, pointing in the direction of the positive x- axis. Rotate the ray counterclockwise around the origin. The acute angle formed by the ray and the nearest part of the positive or negative x-axis is called the reference angle. The rotated ray is called the terminal side of that angle.

5. Holt Geometry 8-Ext Trigonometry and the Unit Circle

6. Holt Geometry 8-Ext Trigonometry and the Unit Circle Example 1: Finding Reference Angles Sketch each angle on the coordinate plane. Find the measure of its reference angle. A.252º B.297º Reference angle: 252° – 180° = 72° Reference angle: 360° – 297° = 63°

7. Holt Geometry 8-Ext Trigonometry and the Unit Circle Check It Out! Example 1 Sketch each angle on the coordinate plane. Find the measure of its reference angle. a. 309°b. 410° Reference angle: 360° – 309° = 51° Reference angle: 410° – 360° = 50°

8. Holt Geometry 8-Ext Trigonometry and the Unit Circle The unit circle is a circle with a radius of 1 unit, centered at the origin. It can be used to find the trigonometric ratios of an angle.

9. Holt Geometry 8-Ext Trigonometry and the Unit Circle Consider the acute angle θ. Let P(x, y) be the point where the terminal side of θ intersects the unit circle. Draw a vertical line from P to the x-axis. Since and, the coordinates of P can be written as (cos θ, sin θ). Thus if you know the coordinates of a point on the unit circle, you can find the trigonometric ratios for the associated angle.

10. Holt Geometry 8-Ext Trigonometry and the Unit Circle Example 2A: Finding Trigonometric Ratios Find the trigonometric ratio. sin 240° Sketch the angle on the coordinate plane. The reference angle is 60º.

11. Holt Geometry 8-Ext Trigonometry and the Unit Circle Example 2A Continued Let P(x, y) be the point where the terminal side of the angle intersects the unit circle. Since P is in Quadrant III, its x-coordinate is negative, and its y-coordinate is negative. So the coordinates of P are. The sine of 240º is the y-coordinate of P, so

12. Holt Geometry 8-Ext Trigonometry and the Unit Circle Example 2B: Finding Trigonometric Ratios B. tan225° Sketch the angle on the coordinate plane. The reference angle is 45º. Find the trigonometric ratio.

13. Holt Geometry 8-Ext Trigonometry and the Unit Circle Example 2B: Finding Trigonometric Ratios Let P(x, y) be the point where the terminal side of the angle intersects the unit circle. Since P is in Quadrant III, its x-coordinate is negative, and its y-coordinate is negative. So the coordinates of P are. Remember that. So.

14. Holt Geometry 8-Ext Trigonometry and the Unit Circle Check It Out! Example 2a Find the trigonometric ratio. The reference angle is 60º. cos 240°

15. Holt Geometry 8-Ext Trigonometry and the Unit Circle Check It Out! Example 2a Continued Let P(x, y) be the point where the terminal side of the angle intersects the unit circle. Since P is in Quadrant III, its x-coordinate is negative, and its y-coordinate is negative. So the coordinates of P are. The cosine of 240º is the x-coordinate of P, so

16. Holt Geometry 8-Ext Trigonometry and the Unit Circle Check It Out! Example 2b sin 135° The reference angle is 45º. Find the trigonometric ratio.

17. Holt Geometry 8-Ext Trigonometry and the Unit Circle Check It Out! Example 2b Continued Let P(x, y) be the point where the terminal side of the angle intersects the unit circle. Since P is in Quadrant II, its x-coordinate is negative, and its y-coordinate is positive. So the coordinates of P are. The sine of 135º is the y-coordinate of P, so