TUC-1 Measurements of Angles

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

2. 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. Terminal Ray Initial Ray The Coordinate Plane Positive Rotation – counterclockwise Negative Rotation - clockwise Precalculus

3. 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. r r r The Radian In the circle shown, the length of the intercepted arc equals the radius of the circle. Hence, the angle theta measures 1 radian. Precalculus

4. Wait! What is a radian? • Visual animation of what a radian represents. • This visual was created by LucasVB, this is a link to his blog post about radians Precalculus

5. 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!! Precalculus

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

7. Converting from Degrees to Radians • To convert from degrees to radians, multiply by • Example 1 Convert 320 to radians. • Example 2 Convert -153 to radians. Precalculus

8. Converting from Radians to Degrees • To convert from degrees to radians, multiply by • Example 1 Convert to degrees. Precalculus

9. Converting from Radians to Degrees • Example 2 Convert to degrees. • Example 3 Convert 1.256 radians to degrees. Precalculus

10. 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 Precalculus

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

12. Coterminal Angles • Example 2 Find two coterminal angles (one positive and one negative) for The general expression would be: Precalculus

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

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

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

16. Supplementary Angles • The supplement of /8 is 7/8. • The supplement of 2.891R is 0.251R. Precalculus