The document contains worked examples of finding the reference angle given a rotation angle in different quadrants. It demonstrates converting the angle to an equivalent angle in the first quadrant by adding or subtracting multiples of 2π or π to change the quadrant, then taking the reference angle as the angle between the new terminal

1. Chapter 5
Trigonometric Functions of Real Numbers
5.1 The Unit Circle
Day 2
Philippians 4:13 I can do all things through
Christ who strengthens me.

2. Reference Angles

3. Reference Angles
Let θ be a first-wrap angle of rotation. θ ' (the
reference angle) is the shortest rotation along the unit
circle between the terminal point and the x-axis.

4. Reference Angles
Let θ be a first-wrap angle of rotation. θ ' (the
reference angle) is the shortest rotation along the unit
circle between the terminal point and the x-axis.
Q I rotation
θ'=θ

5. Reference Angles
Let θ be a first-wrap angle of rotation. θ ' (the
reference angle) is the shortest rotation along the unit
circle between the terminal point and the x-axis.
Q I rotation Q II rotation
θ'=θ θ ' = π −θ

6. Reference Angles
Let θ be a first-wrap angle of rotation. θ ' (the
reference angle) is the shortest rotation along the unit
circle between the terminal point and the x-axis.
Q III rotation
θ'=θ −π

7. Reference Angles
Let θ be a first-wrap angle of rotation. θ ' (the
reference angle) is the shortest rotation along the unit
circle between the terminal point and the x-axis.
Q III rotation Q IV rotation
θ'=θ −π θ ' = 2π − θ

8. Find the reference angle

9. Find the reference angle
17π
1. θ =
6

10. Find the reference angle
17π
1. θ =
6
which quadrant is it in?

11. Find the reference angle
17π
1. θ =
6
which quadrant is it in?

12. Find the reference angle
17π
1. θ =
6
which quadrant is it in?
Now, convert to 1st wrap

13. Find the reference angle
17π
1. θ = 17π 12π
6 −
6 6
which quadrant is it in?
Now, convert to 1st wrap

14. Find the reference angle
17π
1. θ = 17π 12π
6 −
6 6
which quadrant is it in?
5π
∴ QII
6
Now, convert to 1st wrap

15. Find the reference angle
17π
1. θ = 17π 12π
6 −
6 6
which quadrant is it in?
5π
∴ QII
6
6π 5π π
θ'= − =
6 6 6
Now, convert to 1st wrap

16. Find the reference angle
17π
1. θ = 17π 12π
6 −
6 6
which quadrant is it in?
5π
∴ QII
6
6π 5π π
θ'= − =
6 6 6
Now, convert to 1st wrap
what is wrong with
this diagram?

17. Find the reference angle
11π
2. θ =
8

18. Find the reference angle
11π
2. θ =
8

19. Find the reference angle
11π
2. θ =
8
Q III

20. Find the reference angle
11π
2. θ =
8
11π 8π 3π
θ'= − =
8 8 8
Q III

21. Find the reference angle
11π
2. θ =
8
11π 8π 3π
θ'= − =
8 8 8
Q III

22. Find the reference angle
9π
3. θ =
5

23. Find the reference angle
9π
3. θ =
5

24. Find the reference angle
9π
3. θ =
5
Q IV

25. Find the reference angle
9π
3. θ =
5
10π 9π π
θ'= − =
5 5 5
Q IV

26. Find the reference angle
9π
3. θ =
5
10π 9π π
θ'= − =
5 5 5
Q IV

27. Find the reference angle
−7π
4. θ =
6

28. Find the reference angle
−7π
4. θ =
6
Get the 1st wrap angle

29. Find the reference angle
−7π
4. θ =
6
Get the 1st wrap angle
12π −7π
+
6 6

30. Find the reference angle
−7π
4. θ =
6
Get the 1st wrap angle
12π −7π
+
6 6
5π
6

31. Find the reference angle
−7π
4. θ =
6
Get the 1st wrap angle
12π −7π
+
6 6
5π
6
Q II

32. Find the reference angle
−7π
4. θ =
6
Get the 1st wrap angle
6π 5π π
12π −7π θ'= − =
+ 6 6 6
6 6
5π
6
Q II

33. Find the reference angle
−7π
4. θ =
6
Get the 1st wrap angle
6π 5π π
12π −7π θ'= − =
+ 6 6 6
6 6
5π
6
Q II

34. 5. Find the reference angle and the terminal point if
−41π
θ=
4

35. 5. Find the reference angle and the terminal point if
−41π
θ=
4
Convert to 1st wrap angle

36. 5. Find the reference angle and the terminal point if
−41π
θ=
4
Convert to 1st wrap angle
48π −41π
+
4 4

37. 5. Find the reference angle and the terminal point if
−41π
θ=
4
Convert to 1st wrap angle
48π −41π
+
4 4
7π
4

38. 5. Find the reference angle and the terminal point if
−41π
θ=
4
Convert to 1st wrap angle
48π −41π
+
4 4
7π
4
Q IV

39. 5. Find the reference angle and the terminal point if
−41π
θ=
4
Convert to 1st wrap angle
8π 7π π
48π −41π θ'= − =
+ 4 4 4
4 4
7π
4
Q IV

40. 5. Find the reference angle and the terminal point if
−41π
θ=
4
Convert to 1st wrap angle
8π 7π π
48π −41π θ'= − =
+ 4 4 4
4 4
Terminal Point:
7π
4
Q IV

41. 5. Find the reference angle and the terminal point if
−41π
θ=
4
Convert to 1st wrap angle
8π 7π π
48π −41π θ'= − =
+ 4 4 4
4 4
Terminal Point:
7π
4 ⎛ 2 − 2 ⎞
⎜ 2 , 2 ⎟
Q IV ⎝ ⎠

42. HW #2
Nothing so conclusively proves a man’s ability to lead
others as what he does from day to day to lead himself.
Thomas J. Watson