2. Angles and Their Measure Section 4.1

3. Objectives I can label the unit circle for radian angles I can draw and angle showing correct rotation in Standard Format I can calculate Reference Angles I can determine what quadrant an angle is in

4. Angles An angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. A è B C Terminal Side Initial Side Vertex

5. Section 4.1: Figure 4.2, Angle in Standard Position

6. Section 4.1: Figure 4.3, Positive and Negative Angles

7. Angles of the Rectangular Coordinate System An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis. x y Terminal Side Initial Side Vertex   is positive Positive angles rotate counterclockwise. x y Terminal Side Initial Side Vertex   is negative Negative angles rotate clockwise.

8. Measuring Angles Using Degrees The figures below show angles classified by their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.  Acute angle 0º <  < 90º 90º Right angle 1/4 rotation  Obtuse angle 90º <  < 180º 180º Straight angle 1/2 rotation

9. Definition of a Radian One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.

10. Radian Measure Consider an arc of length s on a circle or radius r. The measure of the central angle that intercepts the arc is  = s/r radians.  O r s r

11. Section 4.1: Figure 4.6, Illustration of Six Radian Lengths

12. Section 4.1: Figure 4.7, Common Radian Angles

13. Definition of a Reference Angle Let  be a non-acute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle  ´ prime formed by the terminal side or  and the x-axis.

14. Example a b   a b P(a,b) Find the reference angle , for the following angle:  =315º Solution:  =360 º - 315 º = 45 º

15. Example Find the reference angles for:

16. Homework WS 8-1 Start learning Unit circle