how to find the period of a trig function

Find Period of Trigonometric Functions

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

In the problems beneath, we will use the formula for the menstruum P of trigonometric functions of the grade 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 commencement exist used to sympathize this concept.

Question 1

The graph below is that of a trigonometric function of the class y = a sin(b 10), with b > 0. Find its period and the parameter b.

solution

Locate ii zeros that delimit a whole cycle or an integer number of cycles. In this example, we can run across that from the nada at x = 0 to the zero at ten = 1, there are two cycles. Hence the flow P is equal to:
P = (ane - 0) / 2 = 1 / ii
Nosotros now calculate b by equating the value of the period establish 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 role of the form y = a sin(b ten), with b >0, is shown below. Find its period and the parameter b.

solution

There is one bicycle from the naught at x = -π/4 to the cipher at 10 = π/iv. Hence the period P is equal to:
P = π/4 - (-π/four) = π/2
Nosotros at present equate the value of the flow found using the graph to the higher up formula and solve for b.
π/two = 2π / b
b = 4

Question iii

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

solution

In that location are two zeros that delimit half a cycle. We first find these zeros.
Zero on the left: (-π / 4 - π / 8 ) / two = - 3π / 16 (assuming it is in the middle of x = -π / iv and -π / viii)
Zippo on the right: (0 + π / eight ) / 2 = π / xvi (assuming it is in the middle of x = 0 and π / viii)
Hence half a flow is equal to:
(π / 16 - (- 3π / 16)) = π / 4
and a period P is equal to:
P = two � π / 4 = π / 2
We now equate the value of the catamenia establish using the graph to the higher up formula and solve for b.
π/2 = 2π / b
b = iv

Question iv

The graph beneath is that of a trigonometric office of the form y = a sin(b x + c) + d and points A and B are maximum and minimum points respectively. Observe the menstruation 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 / iii
The period P of the function is given past
P = ii� two π / 3 = 4 π / three
b is plant by solving
2 π / b = iv π / iii
b = three / 2

Question 5

The graph of a trigonometric function of the grade 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 bicycle betwixt points A and B. Hence catamenia P is given by
P = 0.i - (-0.3) = 0.4
b is found past solving
ii π / b = 0.4
b = 5π

Question 6

Find the menstruum of each of the following functions
1) y = sin(ten)cos(ten) - iii
2) y = 2 + 5 cos ^two (x)
three) y = cos(ten) + sin(x)

solution

ane) Employ the identity sin(2x) = 2 sin(10)cos(x) to rewrite the given part as follows:
y = (1 / 2) sin(2x) - three
Use the formula P = 2π / b to find the period equally
P = 2π / 2 = π
2) Utilize the identity cos²(ten) = (1 / 2)(cos(2x) + one)to rewrite the given function as follows:
y = 2 + 5 cos ² (x) = 2 + 5((1 / 2)(cos(2x) + ane)) = (5 / 2) cos(2 10) + nine / 2
Utilise the formula P = 2π / b to notice the menstruation as
P = 2π / 2 = π
iii) Rewrite the given function every bit follows:
y = cos(10) + sin(x) = (2 / √2)(√2 / 2 cos(x) + √2 / 2 sin(x))
Utilise the identity:
sin(π / 4 + x) = sin(π / iv) cos(ten) + cos(π / 4) sin(x) = √2 / ii cos(x) + √two / 2 sin(x)
to rewrite the given function as:
y = cos(10) + sin(x) = (2 / √2) sin(x + π / iv)
Apply the formula P = 2π / b to find the period equally
P = 2π / 1 = 2 π

Question 7

Suppose f(10) is periodic office with period p. What is the catamenia of the role h(x) = f(1000 x), where 1000 is a positive abiding?

solution

If p is the menstruum of function f, and then
f(10 + p) = f(x) for all ten in the domain of f.
Let x = grand X , where k is a constant.
f(k X + p) = f(chiliad Ten)
Rewrite the above every bit
f(k(X + p / chiliad)) = f (k X)
Let h(10) = f(k ten). The above may be written every bit
h(10 + p / grand) = h(10)
Which indicates that h(10) = f(1000 x) is periodic and has a menses equal to p / thousand.

More References and links

Periods of Trigonometric Functions
Properties of The Vi Trigonometric Functions
Interactive tutorials on Catamenia of trigonometric functions.
High School Maths (Grades ten, 11 and 12) - Free Questions and Problems With Answers
Center School Maths (Grades six, 7, 8, 9) - Free Questions and Bug With Answers
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