Question

The period of $\left| \sin \ \left. 3x \right|\ \text{is} \right.$
A).$2\pi $
B).${2\pi }/{3}\;$
C).${\pi }/{3}\;$
D).$3\pi .$

Answer 1

Hint: In this question, we see the modulus of the trigonometric function. The modulus of sine and cotangent functions Period of function is π. Period of function is π. So, proceeding in this manner, keeping modulus function in mind, we will get our final result.

Complete step by step solution: A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians. The period is represented as “T”. A period is a distance among two repeating points on the graph function.
We are asked to find the period of the function $|\sin 3x|$
Period of a function means after a certain interval function repeat.
So, period of $\sin \,x=2\pi $
$\Rightarrow \ \text{Period}\ \text{of}\ \text{sin}\ x=2\pi $
So, period of $|\sin 3x|\,=\,\dfrac{2\text{ }\!\!\pi\!\!\text{ }}{2}\,=\,2$
$\Rightarrow \ \text{period}\ \text{of}\ \left| \sin \ \left. x \right| \right.=\pi $
$\therefore \ \text{period}\ \text{of}\ |\sin 3x|\,=\,\dfrac{\text{ }\!\!\pi\!\!\text{ }}{3}$

So, option $B$ is the correct answer to this question.

Note: In order of finding the period of sine function;
If we have a function f(x) = sin (xs), where s > 0, then the graph of the function makes complete cycles between 0 and 2π and each of the function have the period, p = 2π/s.
Here, we are given the periodic of the trigonometric function, which is as follows:-
Sine function: Period 2π
cosine function: Period 2π
tangent function: Period π
cosecant function: Period 2π
secant function: Period 2π
cotangent function: Period π
In this question, we notice the modulus of trigonometric functions and periods.
As such, we draw the graph of modulus function by taking the mirror image of the corresponding core graph in the x-axis. The modulus of sine and cotangent functions Period of function is π. Period of function is π. So,
Period of $|\sin 3x|\,=\,\dfrac{\text{ }\!\!\pi\!\!\text{ }}{1}$
     $\Rightarrow \ \text{Period}\ \text{of}\ |\sin 3x|\,=\,\dfrac{\text{ }\!\!\pi\!\!\text{ }}{3}$