1. Section 3.4
Periodic Functions

2. Objectives:
1. To identify periodic functions and
their periods.
2. To state the cofunction, period,
and odd-even identities and use
them to evaluate trigonometric
functions.

3. The curve described by the rotation of a point on a tire is one type of periodic curve called a cycloid.

4. The horizontal distance before it repeats is called the period, and each repeating part is a cycle.

5. Definition
Periodic function A function is periodic if, for some given constant c, ƒ(x + c) = ƒ(x)  x. The smallest such positive value of c is called the period of the function.

6. EXAMPLE 1 Give the graph, period, domain, and range of the function
EXAMPLE 1 Give the graph, period, domain, and range of the function. Then express it in a simpler form.
f(x) =
∙∙∙
0 if -4  x  -1
1 if -1  x  0
0 if 0  x  3
1 if 3  x  4
0 if 4  x  7
1 if 7  x  8    

7. EXAMPLE 1 Give the graph, period, domain, and range of the function
EXAMPLE 1 Give the graph, period, domain, and range of the function. Then express it in a simpler form.
D = {real numbers}
R = {0, 1}
period = 4

8. EXAMPLE 1 Give the graph, period, domain, and range of the function
EXAMPLE 1 Give the graph, period, domain, and range of the function. Then express it in a simpler form.
Let n be the period number. Since the period is 4, at the end of one period 4n = 4. Likewise, 4n – 1 = 3 and 4n – 4 = 0.
period = 4

9. EXAMPLE 1 Give the graph, period, domain, and range of the function
EXAMPLE 1 Give the graph, period, domain, and range of the function. Then express it in a simpler form.
f(x) =
0 if 4n – 4  x  4n – 1
1 if 4n – 1  x  4n   

10. Practice: Give the graph, period, domain, and range of the function
Practice: Give the graph, period, domain, and range of the function. Then express it in a simpler form.
f(x) =
∙∙∙
2 if -1  x  0
3 if 0  x  1
2 if 1  x  2
3 if 2  x  3    

11. Practice: Give the graph, period, domain, and range of the function
Practice: Give the graph, period, domain, and range of the function. Then express it in a simpler form. y x
D = {real numbers}
R = {2, 3}
period = 2

12. To express the function in simpler terms trace one complete period starting from the y-axis.

13. Let n be the number of periods
Let n be the number of periods. Since the period is 2, after one period 2n = 2. Then 2n – 1 = 1 and 2n – 2 = 0. y x

14. n is the number of periods.    3 if 2n – 2  x  2n - 1
f(x) = , where
n is the number of periods.   
3 if 2n – 2  x  2n - 1
2 if 2n – 1  x  2n y x

15. Give the graph, period, domain, and range of the function
Give the graph, period, domain, and range of the function. Then express it in a simpler form.
f(x) =    
∙∙∙
0 if 0  x  1
1 if 1  x  3
2 if 3  x  6
0 if 6  x  7
1 if 7  x  9
2 if 9  x  12

16. y x 3 6 9 12
period = 6
D = {real numbers}
R = {0, 1, 2}

17. Let n be the number of periods
Let n be the number of periods. Since the period is 6, after one period 6n = 6. Then 6n – 3 = 3, 6n – 5 = 1, and 6n – 6 = 0. y x 3 6 9 12

18.    0 if 6n – 6  x  6n - 5 1 if 6n – 5  x  6n – 3
f(x) = , where
n is the number of periods.   
0 if 6n – 6  x  6n - 5
1 if 6n – 5  x  6n – 3
2 if 6n – 3  x  6n y x 3 6 9 12

19. EXAMPLE 2 Write the period relationship for cosine as an identity.
cos (x + 2k) = cos x  x, where k  {integers}

20. EXAMPLE 3 Find cos 29/3. cos 29/3 = cos (5/3 + 8) = cos 5/3

21. Find cos 41/6.
1. 1/2
2. 3/2
3. 2/2
4. 3/3
5. None of these

22. Find cos 41/6.
cos 41/6 = cos (5/6 + 6) = cos 5/6
cos 5/6 = cos /6 = 3/2

23. Since the acute angles of a right triangle are complementary, mA + mB = /2 radians or 90°. Therefore mB = /2 - mA. You can use this substitution to prove the cofunction relationship.

24. EXAMPLE 4 Prove the cofunction identity cos (/2 - ) = sin 
Consider the following right triangle. a b c 
/2 – 

25. EXAMPLE 4 Prove the cofunction identity cos (/2 - ) = sin 
cos (/2 - ) = a/c; sin  = a/c
cos (/2 - ) = sin  a b c 
/2 – 

26. sin ( + 2k) = sin  sin (/2 – ) = cos 
Period Identities Cofunction Identities
k  {integers}
sin ( + 2k) = sin  sin (/2 – ) = cos 
cos ( + 2k) = cos  cos (/2 – ) = sin 
sec ( + 2k) = sec  sec (/2 – ) = csc 
csc ( + 2k) = csc  csc (/2 – ) = sec 
tan ( + k) = tan  tan (/2 – ) = cot 
cot ( + k) = cot  cot (/2 – ) = tan 

27. EXAMPLE 5 List the special angles up to 2 and give the cosine of each.

28. EXAMPLE 5 List the special angles up to 2 and give the cosine of each.QI
 cos 
0 1
/6 3/2
/4 2/2
/3 1/2
/2 0

29. EXAMPLE 5 List the special angles up to 2 and give the cosine of each.
QII
 cos 
2/3 -1/2
3/4 - 2/2
5/6 - 3/2
 -1

30. EXAMPLE 5 List the special angles up to 2 and give the cosine of each.
QIII
 cos 
7/6 - 3/2
5/4 - 2/2
4/3 -1/2
3/2 0

31. EXAMPLE 5 List the special angles up to 2 and give the cosine of each.
QIV
 cos 
5/3 1/2
7/4 2/2
11/6 3/2
2 1

32. Odd-even identities sin (-) = -y/r -sin  = -(y/r) sin (-) = -sin x r y -y
cos (-) = x/r
cos  = x/r
cos (-) = cos 

33. Homework: pp

34. ►B. Exercises
Evaluate. Give exact values when possible.
9. tan
32 3

35. ►B. Exercises 17. tan (/2 – ) = cot 
Prove these cofunction relations using a right triangle diagram.
17. tan (/2 – ) = cot  a b c 
/2 –  A C B

36. ►B. Exercises Consider ∙∙∙  0 if 0  x  2  1 if 2  x  4
g(x) =
∙∙∙
0 if 0  x  2
1 if 2  x  4
2 if 4  x  5
0 if 5  x  7
1 if 7  x  9
2 if 9  x  10    

37. ►B. Exercises 18. Graph f(x). Is it periodic? ∙∙∙  0 if 0  x  2 
g(x) =
∙∙∙
0 if 0  x  2
1 if 2  x  4
2 if 4  x  5
0 if 5  x  7
1 if 7  x  9
2 if 9  x  10    

38. ►B. Exercises
18. Graph f(x). Is it periodic?

39. ►B. Exercises 19. Give the domain, range, and period. period = 4
D = {real numbers}
R = {0, 1, 2}

40. ►B. Exercises 20. Find g(17), g( 899.4), and g(729.58).
g(17) = g(3p + 2) = g(2) = 0
g( ) = g(29.99) = g(4p )
= g(4.99) = 2
g(729.58) = g(145p ) = g(4.58) = 2

41. ►B. Exercises 21. Simplify the function rule for g(x).
Let n  {integers}. After 2 periods, 5n = 10.
g(x) =
0 if 5n – 5  x  5n – 3
1 if 5n – 3  x  5n – 1
2 if 5n - 1  x  5n   

42. ►B. Exercises Write a period relation for
23. the function of exercise 18.
g(x + 5) = g(x)

43. ■ Cumulative Review Use synthetic division to answer exercises 28-30.
28. Divide 4x³ – 3x² – 15x – 19 by x – 3.

44. ■ Cumulative Review Use synthetic division to answer exercises 28-30.
29. Find f(5) if f(x)= 4x³ – 3x² – 15x – 19.

45. ■ Cumulative Review Use synthetic division to answer exercises 28-30.
30. Factor and find the zeros of
f(x) = 3x4 + 16x3 + 24x2 – 16.

46. ■ Cumulative Review
31. Graph f(x) as given in exercise 30.

47. ■ Cumulative Review
32. Solve for x in the triangle.

48. 2 1 6 sin = p 2 4 sin = p 2 3 sin = p 2 3 6 cos = p 2 4 cos = p 2 1 3
Exact values you should know. 2 1 6
sin = p 2 4
sin = p 2 3
sin = p 2 3 6
cos = p 2 4
cos = p 2 1 3
cos = p 3 6
tan = p 1 4
tan = p 3
tan = p