1. PARTS OF ANGLES AND DMS Fall, 2015 Dr. Shildneck

2. Angles and Their Measures v ertex Recall that an angle is defined as two non‐colinear rays that share a common endpoint, called the v ertex. i nitial side An angle in (circle) trigonometry has a fixed (starting) side, called the i nitial side ‐ which points in the direction of the positive x‐axis. v ertex The v ertex is the origin. t erminal side The second side is rotated (like the hands of a clock) about the vertex. This side is called the t erminal side. S tandard Position An angle with this placement is said to be in S tandard Position. 2

3. y terminal side initial side x 3

4. 4 measure The measure of the angle describes the amount and direction of the rotation necessary to move from the initial side to the terminal side. positive angle A positive angle indicates that the terminal side has been rotated counter‐clockwise. negative angle A negative angle indicates that the terminal side has been rotated clockwise.

5. y x 5 Positive Angle Negative Angle

6. 6 The most common angular unit of measure is called the "degree," which is equivalent to 1/360 of a full rotation around a circle. Key (basic) Angle Measures (counterclockwise) 0 o ‐ No rotation 90 o ‐ ¼ of a rotation 180 o ‐ ½ of a rotation 270 o ‐ ¾ of a rotation 360 o ‐ 1 full rotation (back to the initial side) Quadrantal Angles These angles are called the "Quadrantal Angles" since they end on the boundaries between the quadrants of the coordinate plane.

7. Angles based on rotation can be any size and either sign (positive or negative). If you were to continue to rotate the terminal side past 360 o, you would simply continue to add (or subtract if negative) to the angle. **Note that Additional rotations are indicated by a rotation mark. 7 120 o -240 o

8. Degrees – Not exact Obviously the terminal side can end up not on a "degree" mark. Thus, you can have measures that use parts of a degree. There are two ways that this might be indicated. 1. Decimal form: This form simply indicates the "fractional" part of the degree. Example:35.75 o ‐ means 35 degrees and 3/4 more of a degree 8

9. 9 2. Degrees-Minutes-Seconds (DMS) A more formal "part form" of a degree uses minutes and seconds. The conversion ratio between decimal form and DMS is like that of a clock. 1 degree is made up of 60 minutes ( 1 o = 60') 1 minute is made up of 60 seconds (1' = 60")

10. 10 Converting by Hand Convert 65.755 o into DMS form Convert 20 o 21' into Decimal form

11. 10 Converting by Calculator Convert 65.75382 o into DMS form Convert 20 o 15' 54" into Decimal form

12. 11 ASSIGNMENT 2 WS ‐ Measures of Angles