1. Why do we use angles? Here is the theory… Ancient Babylonians measured the path of the stars from night to night and noticed that they traveled in a circle over one year.

2. What does the night sky and a year have to do with angles? Why do degrees measure 360 and not 365 ¼? 360 also has the factors 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 60…. Ancient Babylonians used a number system based on 60.

3. 7.1 Measurement of Angles I will be able to measure central angles using radians and convert between degrees and radians Objective

4. Angles of the Coordinate System An angle (θ) is in standard position if its vertex is at the origin of a coordinate system 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.

5. Special Angles Reference Angles are acute angles formed by the terminal side of the given angle and the x-axis. Reference Angles aid in graphing. Coterminal angles are two angles that share the same terminal ray. They should also share the same reference angle when graphing.

6. Graph the following angles in standard position. Then find one positive and one negative coterminal angle for each. a. 20º angleb. –120º angle. Examples Find the reference angle for each. a. 120º angleb. 225º angle.

7. Examples- find two angles, one positive and one negative, that are coterminal with each given angle. Then, state the reference angle. 1.10° 2.100° 3.-5° 4.400° 1.370°,-350° Ref: 10° 2.460°,-260° Ref: 80° 3.355°,-365° Ref: 5° 4.40°,-320° Ref: 40°

8. So what is wrong with degrees then? We could divide a circle into anything… If we lived on Mars then maybe we would have 680 o instead of 360 o. (This is the number of days in Mars year.)

9. We have a second system to measure angles that has more to do with a circle than the days in a year. We call these radians.

10. Cut the pipe cleaner the length of the radius of your circle. Lay the radius pipe cleaner along your circle and mark the length. Repeat until you get all the way around the circle. A radian is an arc length of the circle that equals the length of the radius.

11. Lay your pipe cleaner along the remaining portion that didn’t make a whole radius. Bend the pipe cleaner to measure that length. Find what percent the remaining portion is to a radius. How many pipe cleaners did it take to cover the entire circumference of the circle?

12. Have we seen this number before? The exact number is 6.283185307… 2π

13. So, in a full circle, we have 360 ⁰ = 2π radians How many radians are in half a circle (180 ⁰)? How many radians are in 1 ⁰? How many degrees are in 1 radian?

14. Convert each to either radians or degrees

15. Homework Page 261 #3, 7, 18  GRAPH TOO! Fill out Unit Circle organizer with radians

16. Exit Slip