Find Period of Trigonometric Function Given its Graphs

Grade 12 trigonometry problems and questions on how to find the period of a trigonometric function 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.

 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 cos2(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 cos2(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.

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