# 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

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

## Question 1The 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.
## solutionLocate 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 2The 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.## solutionThere 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 3The 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.## solutionThere 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 4The 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.## solutionThe 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 5The 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.
## solutionThe 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 6Find the period of each of the following functions1) y = sin(x)cos(x) - 3 2) y = 2 + 5 cos ^{2}(x)
3) y = cos(x) + sin(x) ## solution1) 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
## Question 7Suppose 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?## solutionIf 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 FunctionsProperties of The Six Trigonometric Functions

Interactive tutorials on Period of trigonometric functions.

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