Find Period of Trigonometric Functions

Grade 12 trigonometry problems and questions on how to find the period of trigonometric functions given its graph or formula, are presented along with detailed solutions.

In the problems below, we will use the formula for the period P of trigonometric functions of the form y = a sin(bx + c) + d or y = a cos(bx + c) + d and which is given by

P = 2π / | b | and becomes P = 2π / b for b > 0.

Interactive tutorials on Period of trigonometric functions may first be used to understand this concept.

Question 1

The graph below is that of a trigonometric function of the form y = a sin(b x), with b > 0. Find its period and the parameter b.

solution

Locate two zeros that delimit a whole cycle or an integer number of cycles. In this example, we can see that from the zero at x = 0 to the zero at x = 1, there are two cycles. Hence the period P is equal to:
P = (1 - 0) / 2 = 1 / 2
We now calculate b by equating the value of the period found using the graph to the above formula and solve for b.
1 / 2 = 2π / b
b = 4 π

Question 2

The graph of a trigonometric function of the form y = a sin(b x), with b >0, is shown below. Find its period and the parameter b.

solution

There is one cycle from the zero at x = -π/4 to the zero at x = π/4. Hence the period P is equal to:
P = π/4 - (-π/4) = π/2
We now equate the value of the period found using the graph to the above formula and solve for b.
π/2 = 2π / b
b = 4

Question 3

The graph below is that of a trigonometric function of the form y = a cos(b x + c) with b > 0. Find the period of this function and the value of b.

solution

There are two zeros that delimit half a cycle. We first find these zeros.
Zero on the left: (-π / 4 - π / 8 ) / 2 = - 3π / 16 (assuming it is in the middle of x = -π / 4 and -π / 8)
Zero on the right: (0 + π / 8 ) / 2 = π / 16 (assuming it is in the middle of x = 0 and π / 8)
Hence half a period is equal to:
(π / 16 - (- 3π / 16)) = π / 4
and a period P is equal to:
P = 2 × π / 4 = π / 2
We now equate the value of the period found using the graph to the above formula and solve for b.
π/2 = 2π / b
b = 4

Question 4

The graph below is that of a trigonometric function of the form y = a sin(b x + c) + d and points A and B are maximum and minimum points respectively. Find the period of this function and the value of b, assuming b > 0.

solution

The distance along the x axis between points A and B is equal to half a period and is given by
7π / 6 - 3π / 6 = 2 π / 3
The period P of the function is given by
P = 2× 2 π / 3 = 4 π / 3
b is found by solving
2 π / b = 4 π / 3
b = 3 / 2

Question 5

The graph of a trigonometric function of the form y = a cos(b x + c) + d is shown below where points A and B are minimum points with x coordinates - 0.3 and 0.1 respectively. Find the value of b.

solution

The is one whole cycle between points A and B. Hence period P is given by
P = 0.1 - (-0.3) = 0.4
b is found by solving
2 π / b = 0.4
b = 5π

Question 6

Find the period of each of the following functions
1) y = sin(x)cos(x) - 3
2) y = 2 + 5 cos ² (x)
3) y = cos(x) + sin(x)

solution

1) Use the identity sin(2x) = 2 sin(x)cos(x) to rewrite the given function as follows:
y = (1 / 2) sin(2x) - 3
Use the formula P = 2π / b to find the period as
P = 2π / 2 = π
2) Use the identity cos²(x) = (1 / 2)(cos(2x) + 1)to rewrite the given function as follows:
y = 2 + 5 cos ² (x) = 2 + 5((1 / 2)(cos(2x) + 1)) = (5 / 2) cos(2 x) + 9 / 2
Use the formula P = 2π / b to find the period as
P = 2π / 2 = π
3) Rewrite the given function as follows:
y = cos(x) + sin(x) = (2 / √2)(√2 / 2 cos(x) + √2 / 2 sin(x))
Use the identity:
sin(π / 4 + x) = sin(π / 4) cos(x) + cos(π / 4) sin(x) = √2 / 2 cos(x) + √2 / 2 sin(x)
to rewrite the given function as:
y = cos(x) + sin(x) = (2 / √2) sin(x + π / 4)
Use the formula P = 2π / b to find the period as
P = 2π / 1 = 2 π

Question 7

Suppose f(x) is periodic function with period p. What is the period of the function h(x) = f(k x), where k is a positive constant?

solution

If p is the period of function f, then
f(x + p) = f(x) for all x in the domain of f.
Let x = k X , where k is a constant.
f(k X + p) = f(k X)
Rewrite the above as
f(k(X + p / k)) = f (k X)
Let h(x) = f(k x). The above may be written as
h(X + p / k) = h(X)
Which indicates that h(x) = f(k x) is periodic and has a period equal to p / k.

More References and links

Periods of Trigonometric Functions
Properties of The Six Trigonometric Functions
Interactive tutorials on Period of trigonometric functions.
High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers
Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
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