2. TUC-1 Measurements of Angles “Things I’ve Got to Remember from the Last Two Years”

3. MPH 9/11Precalculus The Coordinate Plane In the coordinate plane, angles in standard position are created by rotating about the origin (the vertex of the angle). The initial ray is the positive x –axis. In the coordinate plane, angles in standard position are created by rotating about the origin (the vertex of the angle). The initial ray is the positive x –axis. Initial Ray Terminal Ray Positive Rotation – counterclockwise Negative Rotation - clockwise

4. MPH 9/11Precalculus The Radian Angles can also be measured in radians. A central angle measures one radian when the measure of the intercepted arc equals the radius of the circle. In the circle shown, the length of the intercepted arc equals the radius of the circle. Hence, the angle theta measures 1 radian. r r r

5. Wait! What is a radian? Visual animation of what a radian represents. Visual animation of what a radian represents. This visual was created by LucasVB, this is a link to his blog post about radians This visual was created by LucasVB, this is a link to his blog post about radianslink to his blog post link to his blog post CRO 8/14Precalculus

6. MPH 9/11Precalculus Radians If one investigated one revolution of a circle, the arc length would equal the circumference of the circle. The measure of the central angle would be 2 radians. Since 1 revolution of a circle equals 360, 2 radians = 360!!

7. MPH 9/11Precalculus Radians This implies that 1 radian  57.2958 . This implies that 1 radian  57.2958 . The coordinate plane now has the following labels. The coordinate plane now has the following labels. 0 , 0 90 ,  /2 180 ,  360 , 2  270 , 3  /2

8. MPH 9/11Precalculus Converting from Degrees to Radians To convert from degrees to radians, multiply by To convert from degrees to radians, multiply by Example 1 Convert 320  to radians. Example 1 Convert 320  to radians. Example 2 Convert -153  to radians. Example 2 Convert -153  to radians.

9. MPH 9/11Precalculus Converting from Radians to Degrees To convert from degrees to radians, multiply by To convert from degrees to radians, multiply by Example 1 Convert to degrees. Example 1 Convert to degrees.

10. MPH 9/11Precalculus Converting from Radians to Degrees Example 2 Convert to degrees. Example 2 Convert to degrees. Example 3 Convert 1.256 radians to degrees. Example 3 Convert 1.256 radians to degrees.

11. MPH 9/11Precalculus Coterminal Angles Angles that have the same initial and terminal ray are called coterminal angles. Graph 30 and 390 to observe this. Coterminal angles may be found by adding or subtracting increments of 360 or 2

12. MPH 9/11Precalculus Coterminal Angles Example 1 Example 1 Find two coterminal angles (one positive and one negative) for 425 . 425  - 360  = 65  65  - 360  = -295  65  - 360  = -295  The general expression would be: 425  + 360  n where n  integer (I)

13. MPH 9/11Precalculus Coterminal Angles Example 2 Example 2 Find two coterminal angles (one positive and one negative) for The general expression would be:

14. MPH 9/11Precalculus Coterminal Angles Example 3 Example 3 Find two coterminal angles (one positive and one negative) for -3.187 R. -3.187 – 2π = -9.470 R -3.187 + 2π = 3.096 R -3.187 + 2π = 3.096 R The general expression would be: -3.187 + 2πn where n  I

15. MPH 9/11Precalculus Complementary Angles Two angles whose measures sum to 90  or are called complementary angles. Two angles whose measures sum to 90  or  /2 are called complementary angles. The complement of 37  is 53 . The complement of 37  is 53 . The complement of is 3. The complement of  /8 is 3  /8. The complement of 1.274 R is 0.297 R. The complement of 1.274 R is 0.297 R.

16. MPH 9/11Precalculus Supplementary Angles Two angles whose measures sum to 180  or are called supplementary angles. Two angles whose measures sum to 180  or  are called supplementary angles. The supplement of 85  is 95 . The supplement of 85  is 95 . The supplement of 217  does not exist. Why? The supplement of 217  does not exist. Why?

17. MPH 9/11Precalculus Supplementary Angles The supplement of is 7. The supplement of  /8 is 7  /8. The supplement of The supplement of 2.891 R is 0.251 R.