

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

The graph of a trigonomteric 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

The graph below is that of a trigonomteric 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

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

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

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 cos^{2}(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

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.
