1. Periodic Function Review
Vocabulary:
Periodic Function
Cycle
Period
Midline
Amplitude = 1 2 (max. – min.)
Degrees vs. Radians
⋅ 𝜋 180°
Degrees = Radians
⋅ 180° 𝜋
Radians = Degrees

2. 𝒚= 𝐬𝐢𝐧 𝒙 𝒚= 𝐜𝐨𝐬 𝒙 𝒚= 𝐭𝐚𝐧 𝒙 𝑦= 𝒂 sin 𝒃(𝑥−𝒉) +𝒌 D: All Reals R: −𝟏≤𝒚≤𝟏
= 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
D: All Reals
R: −𝟏≤𝒚≤𝟏
Period: 𝟐𝝅
D: All Reals
R: −𝟏≤𝒚≤𝟏
Period: 𝟐𝝅
D: All Reals but odd values of 𝝅 𝟐
R: All reals
Period: 𝝅
𝑦= 𝒂 sin 𝒃(𝑥−𝒉) +𝒌
𝑎 = Amplitude
𝑏= frequency
ℎ= Phase shift
𝑘= Vertical shift
Reciprocal Functions
𝐜𝐬𝐜 𝜽 = 𝟏 𝐬𝐢𝐧 𝜽
𝐬𝐞𝒄 𝜽 = 𝟏 𝐜𝐨𝐬 𝜽
𝐜𝐨𝐭 𝜽 = 𝟏 tan 𝜽 = 𝐜𝐨𝐬 𝜽 𝐬𝐢𝐧 𝜽

3. Periodic Functions Review
Unit Circle
Equation Parts
Equations from Graphs
Graphing:
sin and cos
tan, sec, csc 10 20 30 40 50

4. Find the exact value of:
sin 23𝜋 6
Answer

5. sin 23𝜋 6
=− 1 2

6. Find the exact value of:
Answer

7. c𝑜𝑠 − 17𝜋 4 =

8. Find the exact value of:
𝑡𝑎𝑛 𝜋 6
Answer

9. 𝑡𝑎𝑛 𝜋 6 = sin 𝜋 6 cos 𝜋 6 = =
= 1 2 ⋅ 2 3

10. Find the exact value of:
sec 5𝜋 6
Answer

11. sec 5𝜋 6 = 1 cos 5𝜋 6
= −
= 1 − 3 2

12. Find the exact value of:
csc 5𝜋 4
Answer

13. csc 5𝜋 4 = 1 sin 5𝜋 4
= 1 − 2 2
=− 2

14. 𝑦=2 sin 3𝑥 +1 Identify the following Amplitude: Phase shift: Period:
Vertical Shift:
Range:
Answer

15. 𝑦=2 sin 3𝑥 +1 Identify the following Amplitude: Phase shift: 2 Period:
Vertical Shift:
Range: 2
None or 0
2𝜋 3
Up 1
−1≤𝑦≤3

16. 𝑦=−3 sin 2 𝑥− 𝜋 2 −5 Identify the following Amplitude: Phase shift:
Period:
Vertical Shift:
Range:
Answer

17. 𝑦=−3 sin 2 𝑥− 𝜋 2 −5 Identify the following Amplitude: Phase shift: 3
Period:
Vertical Shift:
Range: 3
Right 𝜋 2 𝜋
Down 5
−8≤𝑦≤−2

18. 𝑦=2𝜋 cos 𝜋 𝑥+ 𝜋 6 −4 Identify the following Amplitude: Phase shift:
Period:
Vertical Shift:
Range:
Answer

19. 𝑦=2𝜋 cos 𝜋 𝑥+ 𝜋 6 −4 Identify the following Amplitude: Phase shift: 2𝜋
Period:
Vertical Shift:
Range: 2𝜋
Left 𝜋 6 2
Down 4
−4−2𝜋≤𝑦≤−4+2𝜋

20. 𝑦= tan 3𝑥 −2 Identify the following Amplitude: Phase shift: Period:
Vertical Shift:
Range:
Answer

21. 𝑦= tan 3𝑥 −2 Identify the following Amplitude: Phase shift: None
Period:
Vertical Shift:
Range:
None
𝜋 3
Down 2
All Real Numbers

22. 𝑦= 1 2 tan 𝑥+ 𝜋 4 +3 Identify the following Amplitude: Phase shift:
Period:
Vertical Shift:
Range:
Answer

23. 𝑦= 1 2 tan 𝑥+ 𝜋 4 +3 Identify the following Amplitude: Phase shift:
Period:
Vertical Shift:
Range:
Vertical Stretch of 1 2
Left 𝜋 4 𝜋
Up 3
All Reals

24. Write a sine function for the graph.
Answer

25. 𝑦=2 sin 3𝑥 −4

26. Write a sine function for the graph.
Answer

27. 𝑦=2 sin 𝑥+ 𝜋

28. Write a cosine function for the graph.
Answer

29. 𝑦=3 cos 𝑥− 𝜋 3 −1

30. Write a cosine function for the graph.
Answer

31. 𝑦=2 cos 𝑥− 3𝜋

32. Write a sine function for the graph.
Answer

33. 𝑦=−2 sin 2 𝑥+ 𝜋

34. Graph the following from 0 to 4𝜋: 𝑦= sin 𝑥
Answer

35. 𝑦= sin 𝑥

36. Graph the following from 0 to 4𝜋: 𝑦= cos 𝑥
Answer

37. 𝑦= cos 𝑥

38. Graph the following from 0 to 4𝜋: 𝑦= 2sin 0.5𝑥
Answer

39. 𝑦= 2sin 0.5𝑥

40. Graph the following from 0 to 4𝜋: 𝑦= sin 𝑥− 𝜋 3 +1
Answer

41. 𝑦= sin 𝑥− 𝜋

42. Graph the following from 0 to 4𝜋: 𝑦= cos 𝑥+ 𝜋 4 −2
Answer

43. 𝑦= cos 𝑥+ 𝜋 4 −2

44. Graph the following from 0 to 4𝜋: 𝑦= tan 𝑥
Answer

45. 𝑦= tan 𝑥

46. Graph the following from 0 to 4𝜋: 𝑦= 2tan 0.5𝑥
Answer

47. 𝑦= 2tan 0.5𝑥

48. Graph the following from 0 to 4𝜋: 𝑦= sec 𝑥
Answer

49. 𝑦= sec 𝑥

50. Graph the following from 0 to 4𝜋: 𝑦= csc 𝑥
Answer

51. 𝑦= csc 𝑥

52. Graph the following from 0 to 4𝜋: 𝑦=3 sin 2 𝑥+ 𝜋 3 −1
Answer

53. 𝑦=3 sin 2 𝑥+ 𝜋 3 −1