1. 6.3 Angles & Radian Measure
Objectives:
Use a rotating Ray to extend the definition of angle measure to negative angles and angles greater than 180°.
Define Radian Measure and convert angle measures between degrees and radians.

2. Angles of Rotation
Positive angles are rotated counter-clockwise & negative angles clockwise.
Standard position has the initial side on the x-axis & the vertex on the origin.

3. Radians & the Unit Circle
Radians are used to measure angles using arc length.
r = 1
0° = 360° = 2π
180° = π
Circumference:

4. Example #1 Convert from Radians to Degrees

5. Example #2 Convert from Degrees to Radians
150°
-330°
540°

6. Example #3 Find the angle measures from each graph.
360° - 60°= 300°
-360° + 90° + 115° = -155°
5(180°) = 900°

7. Example #4 Draw the following angles in standard position
Example #4 Draw the following angles in standard position. State the quadrant in which the terminal side is located.
-110°
530°

8. Example #4 Draw the following angles in standard position
Example #4 Draw the following angles in standard position. State the quadrant in which the terminal side is located.
3400°

9. Example #4 (continued…) Draw the following angles in standard position
Example #4 (continued…) Draw the following angles in standard position. State the quadrant in which the terminal side is located.

10. Arc Length of a Circle For radians: For degrees:
Depending on whether an angle is given in radians or degrees the formulas for arc length vary slightly, although the concept remains the same.
For radians:
For degrees:
The key to learning this is not to memorize either formula, but to build on what you already know. The length of an arc is a fraction of the distance around the entire circle (circumference).
Multiply that fraction by the circumference of the circle and you get the arc length.

11. Sector Area of a Circle For radians: For degrees:
Depending on whether an angle is given in radians or degrees the formulas for sector area also vary.
For radians:
For degrees:
And just like arc length, the formulas for sector area are based on the same concept:

12. Example #5 Find the Arc Length & Sector Area of the following:

13. Example #5 Find the Arc Length & Sector Area of the following:B.

14. Example #6 Arc Length
The second hand on a clock is 5 inches long. How far does the tip of the hand move in 45 seconds? 12 6 3 9 1 2 4 5 7 8 10 11
5’’