1. Angles

2. I. Definitions

3. Ray : A part of a line with a single endpoint

4. Angle : Two rays with a common endpoint
Terminal Side
The space between the two rays is the measure of the angle.
Vertex
Initial Side

5. QUADRANTS II I
III IV

6. An angle with its vertex at the origin and its initial side on the positive x-axis is in standard form.

7. Positive Angles: Angles measured from the initial side counterclockwise to its terminal side.

8. Negative Angles: Angles measured from the initial side clockwise to its terminal side.

9. II. Measuring Angles

10. Acute Angle : Angles between 0 and 90o

11. Right Angle : Angles measuring 90o

12. Obtuse Angle : Angles between 90o and 180o

13. Straight Angle : Angles that measure 180o

14. The size of an angle is determined by the space between the two sides (rays).
These are measured in two different units:
Degrees
Radians

15. Degrees
Radians
360o 2p

16. Degrees
Radians
180o p

17. So we can concluded
180o = p radians

18. III. Converting between Radians and Degrees

19. 30 o ∙ 𝜋 180 o
30 o
= 𝜋 6
radians 6

20. 2
120 o ∙ 𝜋 180 o
= 2𝜋 3
120 o
radians 3
120o

21. 5
225 o ∙ 𝜋 180 o
= 5𝜋 4
225 o
radians 4
225o

22. 11
330 o ∙ 𝜋 180 o
= 11𝜋 6
330 o
radians 6
330o

23. 60
∙ 180 o 𝜋
𝜋 3
=60 o
𝝅 𝟑

24. 30
5𝜋 6
∙ 180 o 𝜋
=150 o 1 5
𝟓𝝅 𝟔

25. 60
4𝜋 3
∙ 180 o 𝜋
=240 o
𝟒𝝅 𝟑

26. 45
7𝜋 4
∙ 180 o 𝜋
=315 o
𝟕𝝅 𝟒

27. IV. Reference & Co-terminal Angles

28. Reference angle: the acute angle measured from the terminal side to the nearest x-axis.

29. Find the reference angle.
80 ∘

30. −295 ∘
65 ∘

31. 260 ∘
80 ∘

32. −690 ∘
30 ∘

33. 2 60
2𝜋 3
∙ 180 o 𝜋
=120 o 1 1
60 ∘
2𝜋 3
120 o

34. IV. Arc Lengths

35. q
𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
𝑠=𝑟∙𝜃

36. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒
𝑎𝑟𝑐𝑙𝑒𝑛𝑔𝑡ℎ.
4𝜋 3 𝑟𝑎𝑑
12 𝑓𝑡
𝑠=𝑟∙𝜃
𝑠=12 𝑓𝑡∙ 4𝜋 3 4 1
𝑠=16𝜋 𝑓𝑡

37. 𝑠=𝑟∙𝜃 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑐𝑙𝑒𝑛𝑔𝑡ℎ. ∙ 𝜋 𝑟𝑎𝑑 180 ∘ 𝑠=4𝑚∙ 45 ° 𝑠=4𝑚∙ 𝜋 4
∙ 𝜋 𝑟𝑎𝑑 180 ∘
𝑠=𝑟∙𝜃
𝑠=4𝑚∙ 45 ° 4
𝑠=4𝑚∙ 𝜋 4
𝑠=𝜋 𝑚𝑒𝑡𝑒𝑟𝑠