1. 5.1 Angles and Degree Measure

2. Angle- formed by rotating a ray about its endpoint (vertex) Initial Side Starting position Terminal SideEnding position Standard Position Initial side on positive x-axis and the vertex is on the origin

3. Angle describes the amount and direction of rotation 120°–210° Positive Angle- rotates counter-clockwise (CCW) Negative Angle- rotates clockwise (CW)

4. An angle is formed by joining the endpoints of two half-lines called rays. The side you measure from is called the initial side. Initial Side The side you measure to is called the terminal side. Terminal Side This is a counterclockwise rotation. This is a clockwise rotation. Angles measured counterclockwise are given a positive sign and angles measured clockwise are given a negative sign. Positive Angle Negative Angle

5. Standard Position: An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis

6. Angle Measurement The two most common units of measurement for angles are degrees and radians. There are two variations used in the “degree system” – here we describe only the one called degreedecimal. In the degree decimal system… 1 complete revolution = 360 degrees.

7. Degree Measure Over 2500 years ago, the Babylonians used a number system based on 60 The number system we use today is based on 10 However we still use the Babylonian idea to measure certain things such as time and angles. That is why there are 60 minutes in an hour and 60 seconds in a minute.

8. Degree Measurement and the Babylonians In a full circle there are 360 degrees Each degree is split up into 60 parts, each part being 1/60 of a degree. These parts are called minutes Each minute is split up into 60 parts, each part being 1/60 of a minute. These parts are called seconds

9. Converting …say 121.135 degrees The whole units of degrees will remain the same (i.e. in 121.135° longitude, start with 121°). Multiply the decimal by 60 (i.e..135 * 60 = 8.1). The whole number becomes the minutes (8'). Take the remaining decimal and multiply by 60. (i.e..1 * 60 = 6). The resulting number becomes the seconds (6"). Seconds can remain as a decimal. Take your three sets of numbers and put them together, using the symbols for degrees (°), minutes (‘), and seconds (") (i.e. 121°8'6" longitude)

10. Convert 112.42 0 to DMS Convert the fractional part Convert the fractional part of the minutes into seconds

12. Let’s look at the special angles called the quadrantal angles. The quadrantal angles are those angles that lie on the axis of the Cartesian coordinate system:,,, and.

13. Ex: Give the measure of the angle represented by 3.3 rotations counterclockwise.

14. Coterminal Angles Angles with the same initial side and same terminal side, but have different rotations, are called coterminal angles. 50° and 410° are coterminal angles. Their measures differ by a multiple of 360.

15. Q: Can we ever rotate the initial side counterclockwise more than one revolution? Answer – YES! EXIT BACKNEXT

16. Note: Complete Revolutions Rotating the initial side counter- clockwise 1 rev., 2 revs., 3revs.,... generates the angles which measure 360 , 720 , 1080 ,... EXIT BACKNEXT

17. Picture EXIT BACKNEXT

18. ANGLES 360, 720, & 1080 ARE ALL COTERMINAL ANGLES!

19. What if we start at 30  and now rotate our terminal side counter- clockwise 1 rev., 2 revs., or 3 revs. EXIT BACKNEXT

20. Suppose we are given an angle measure of 775 degrees. What would the angle measure be of its terminal side?

21. If A is an angle in standard positon, its reference angle Ar is the acute angle formed by the x axis and the terminal side of angle A

22. Reference angle rule Based on the location of the terminal side…

23. Ex: Find the measure of the reference angle for -135 degrees.