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Section 3.4
Periodic Functions
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.
The curve described by the rotation of
a point on a tire is one type of
periodic curve called a cycloid.
1
2
3
4
5
6
7
8
The horizontal distance before it
repeats is called the period, and each
repeating part is a cycle.
1
2
3
4
5
6
7
8
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.
EXAMPLE 1 Give the graph, period,
domain, and range of the function. Then
express it in a simpler form.
 ∙∙∙
 0
if -4  x  -1
 1
if -1  x  0
 0
if
0

x

3
f(x) = 
if 3  x  4
 1
if 4  x  7
 0
 1
if 7  x  8
 ∙∙∙
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
EXAMPLE 1 Give the graph, period,
domain, and range of the function. Then
express it in a simpler form.
period = 4
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.
EXAMPLE 1 Give the graph, period,
domain, and range of the function. Then
express it in a simpler form.
0
f(x) = 
1
if
if
4n – 4  x  4n – 1
4n – 1  x  4n
Practice: Give the graph, period,
domain, and range of the function. Then
express it in a simpler form.
 ∙∙∙
 2
if -1  x  0
 3
if
0

x

1
f(x) = 
if 1  x  2
 2
 3
if 2  x  3
 ∙∙∙
Practice: Give the graph, period,
domain, and range of the function. Then
express it in a simpler form.
y
period = 2
x
D = {real
numbers}
R = {2, 3}
To express the function in simpler terms
trace one complete period starting from
the y-axis.
y
x
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
 3 if 2n – 2  x  2n - 1
f(x) = 
, where
 2 if 2n – 1  x  2n
n is the number of periods.
y
x
Give the graph, period, domain, and range
of the function. Then express it in a
simpler form.
 ∙∙∙

 0 if 0  x  1
 1 if 1  x  3
 2 if 3  x  6
f(x) = 
 0 if 6  x  7
 1 if 7  x  9

 2 if 9  x  12
 ∙∙∙
y
3
6
9
period = 6
D = {real numbers}
R = {0, 1, 2}
x
12
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
3
6
9
x
12
 0 if 6n – 6  x  6n - 5
f(x) =  1 if 6n – 5  x  6n – 3 , where
 2 if 6n – 3  x  6n
n is the number of periods.
y
3
6
9
x
12
EXAMPLE 2 Write the period
relationship for cosine as an identity.
cos (x + 2k) = cos x  x, where k 
{integers}
EXAMPLE 3 Find cos 29/3.
cos 29/3 = cos (5/3 + 8) = cos 5/3
cos 5/3 = cos /3 = 1/2
Find cos 41/6.
1. 1/2
2. 3/2
3. 2/2
4. 3/3
5. None of these
Find cos 41/6.
cos 41/6 = cos (5/6 + 6) = cos 5/6
cos 5/6 = cos /6 = 3/2
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.
EXAMPLE 4 Prove the cofunction
identity cos (/2 - ) = sin 
Consider the following right triangle.
c

b
/
2
–
a
EXAMPLE 4 Prove the cofunction
identity cos (/2 - ) = sin 
cos (/2 - ) = a/c; sin  = a/c
cos (/2 - ) = sin 
c

b
/
2
–
a
Period Identities
k  {integers}
Cofunction Identities
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 
EXAMPLE 5 List the special angles
up to 2 and give the cosine of each.
EXAMPLE 5 List the special angles
up to 2 and give the cosine of each.
QI

0
/
6
/
4
/
3
/
2
cos 
1
3/
2
2/
2
1/
2
0
EXAMPLE 5 List the special angles
up to 2 and give the cosine of each.
QII

2/
3
cos 
-1/2
3/
4
5/
6
- 2/2
- 3/2

-1
EXAMPLE 5 List the special angles
up to 2 and give the cosine of each.
QIII

7/
6
5/
4
4/
3
3/
2
cos 
- 3/2
- 2/2
-1/2
0
EXAMPLE 5 List the special angles
up to 2 and give the cosine of each.
QIV

5/
3
cos 
1/
2
7/
4
11/
6
2/
2
3/
2
2
1
Odd-even identities
-y
sin (-) = /r
y
-sin  = -( /r)
sin (-) = -sin 
x
cos (-) = /r
x
cos  = /r
cos (-) = cos 
r

- x
r
y
-y
Homework:
pp. 132-133
►B. Exercises
Evaluate. Give exact values when
possible.
32
9. tan
3
►B. Exercises
Prove these cofunction relations using a
right triangle diagram.
17. tan (/2 – ) = cot 
c
B
/
2
–
a

A
b
C
►B. Exercises
Consider




g(x) = 




∙∙∙
0
1
2
0
1
2
∙∙∙
if
if
if
if
if
if
0x2
2x4
4x5
5x7
7x9
9  x  10
►B. Exercises
18. Graph f(x). Is it periodic?
 ∙∙∙
 0
if 0  x  2
 1
if 2  x  4
 2
if
4

x

5
g(x) = 
if 5  x  7
 0
if 7  x  9
 1
 2
if 9  x  10
 ∙∙∙
►B. Exercises
18. Graph f(x). Is it periodic?
►B. Exercises
19. Give the domain, range, and period.
period = 4
D = {real numbers}
R = {0, 1, 2}
►B. Exercises
20. Find g(17), g( 899.4), and g(729.58).
g(17) = g(3p + 2) = g(2) = 0
g( 899.4) = g(29.99) = g(4p + 4.99)
= g(4.99) = 2
g(729.58) = g(145p + 4.58) = g(4.58) = 2
►B. Exercises
21. Simplify the function rule for g(x).
Let n  {integers}. After 2 periods, 5n = 10.
if 5n – 5  x  5n – 3
 0
if 5n – 3  x  5n – 1
g(x) =  1
 2
if 5n - 1  x  5n
►B. Exercises
Write a period relation for
23. the function of exercise 18.
g(x + 5) = g(x)
■ Cumulative Review
Use synthetic division to answer
exercises 28-30.
28. Divide 4x³ – 3x² – 15x – 19 by x – 3.
■ Cumulative Review
Use synthetic division to answer
exercises 28-30.
29. Find f(5) if f(x)= 4x³ – 3x² – 15x – 19.
■ Cumulative Review
Use synthetic division to answer
exercises 28-30.
30. Factor and find the zeros of
f(x) = 3x4 + 16x3 + 24x2 – 16.
■ Cumulative Review
31. Graph f(x) as given in exercise 30.
■ Cumulative Review
32. Solve for x in the triangle.
Exact values you should know.
 1
sin =
6 2


2
3
sin =
sin =
4
2
3
2


 1
3
2
cos =
cos =
cos =
6
2
3 2
4
2


3
tan =
tan = 1
6
3
4

tan = 3
3