13.2 – Angles and the Unit Circle

1. 13.2 – Angles and the Unit Circle

2. Angles and the Unit Circle For each measure, draw an angle with its vertex at the origin of the coordinate plane. Use the positive x-axis as one ray of the angle. 1. 90° 2. 45° 3. 30° 4. 150° 5. 135° 6. 120°

3. Angles and the Unit Circle For each measure, draw an angle with its vertex at the origin of the coordinate plane. Use the positive x-axis as one ray of the angle. 1. 90° 2. 45° 3. 30° 4. 150° 5. 135° 6. 120° Solutions 1. 2. 3. 4. 5. 6.

4. The Unit Circle • The Unit Circle • Radius is always one unit • Center is always at the origin • Points on the unit circle relate to the periodic function Let’s pick a point on the unit circle. The positive angle always goes counter-clockwise from the x-axis. 1 30 The x-coordinate of this has a value of the cosine of the angle. The y-coordinate has a value of the sine of the angle. -1 1 -1 In order to determine the sine and cosine we need a right triangle.

5. The Unit Circle The angle can also be negative. If the angle is negative, it is drawn clockwise from the x axis. 1 -1 1 - 45 -1

6. Angles and the Unit Circle Find the measure of the angle. The angle measures 60° more than a right angle of 90°. Since 90 + 60 = 150, the measure of the angle is 150°.

7. Angles and the Unit Circle Sketch each angle in standard position. a. 48° b. 310° c. –170°

8. Let’s Try Some • Draw each angle of the unit circle. • 45o • -280 o • -560 o

9. The Unit Circle (0, 1) |-------1-------| (-1,0) (0 , 0) (1,0) (0, -1) Definition: A circle centered at the origin with a radius of exactly one unit.

10. What are the angle measurements of each of the four angles we just found? π/2 90° 0° 0 2π 180° 360° π 270° 3π/2

11. The Unit Circle Let’s look at an example The x-coordinate of this has a value of the cosine of the angle. The y-coordinate has a value of the sine of the angle. 1 In order to determine the sine and cosine we need a right triangle. 30 -1 1 -1

12. The Unit Circle 1 • Create a right triangle, using the following rules: • The radius of the circle is the hypotenuse. • One leg of the triangle MUST be on the x axis. • The second leg is parallel to the y axis. 30 -1 1 Remember the ratios of a 30-60-90 triangle- 2 60 -1 1 30

13. The Unit Circle 2 60 1 1 30 P X- coordinate 30 -1 1 Y- coordinate -1

14. From the figure, the x-coordinate of point A is – , so cos 135° = – , or about –0.71. 2 2 2 2 opposite leg = adjacent leg 2 2 =    Substitute. 0.71   Simplify. The coordinates of the point at which the terminal side of a 135° angle intersects are about (–0.71, 0.71), so cos 13 –0.71 and sin 135° 0.71. Angles and the Unit Circle Find the cosine and sine of 135°. Use a 45°-45°-90° triangle to find sin 135°.

15. Step 1:  Sketch an angle of –150° in standard position. Sketch a unit circle. Step 2:  Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x-axis. (The other leg will be parallel to the y-axis.) x-coordinate = cos (–150°) y-coordinate = sin (–150°) Angles and the Unit Circle Find the exact values of cos (–150°) and sin (–150°).

16. 1 2 shorter leg = The shorter leg is half the hypotenuse. 1 2 longer leg = 3 = The longer leg is 3 times the shorter leg. Since the point lies in Quadrant III, both coordinates are negative. The longer leg lies along the x-axis, so cos (–150°) = – , and sin (–150°) = – . 1 2 3 2 3 2 Angles and the Unit Circle (continued) The triangle contains angles of 30°, 60°, and 90°. Step 3: Find the length of each side of the triangle. hypotenuse = 1 The hypotenuse is a radius of the unit circle.

17. Let’s Try Some • Draw each Unit Circle. Then find the cosine and sine of each angle. • 45o • 120o

18. 45° Reference Angles - Coordinates ( , ) 3π/4 ( , ) 135° 45° π/4 7π/4 5π/4 225° 315° ( , ) ( , ) Remember that the unit circle is overlayed on a coordinate plane (that’s how we got the original coordinates for the 90°, 180°, etc.) Use the side lengths we labeled on the QI triangle to determine coordinates.

19. 30-60-90 Green Triangle Holding the triangle with the single fold down and double fold to the left, label each side on the triangle. Unfold the triangle (so it looks like a butterfly) and glue it to the white circle with the triangle you just labeled in quadrant I, on top of the blue butterfly.

20. 60° Reference Angles - Coordinates 2π/3 π/3 60° 120° ( , ) ( , ) 5π/3 4π/3 ( , ) ( , ) 240° 300° Use the side lengths we labeled on the QI triangle to determine coordinates.

21. 30-60-90 Yellow Triangle Holding the triangle with the single fold down and double fold to the left, label each side on the triangle. Unfold the triangle (so it looks like a butterfly) and glue it to the white circle with the triangle you just labeled in quadrant I, on top of the green butterfly.

22. 30° Reference Angles π/6 150° 30° 5π/6 11π/6 330° 210° 7π/6 We know that the quadrant one angle formed by the triangle is 30°. That means each other triangle is showing a reference angle of 30°. What about in radians? Label the remaining three angles.

23. 30° Reference Angles - Coordinates ( , ) ( , ) 150° 30° π/6 5π/6 7π/6 11π/6 330° 210° ( , ) ( , ) Use the side lengths we labeled on the QI triangle to determine coordinates.

24. Final Product

25. The Unit Circle