Question

Find the period of the function $\sin \left( {3x} \right)$.

Answer 1

We know that for a function represented as $y = A\sin \left( {Bx + C} \right)\;{\text{and}}\;y = A\cos \left( {Bx + C} \right)$the period is given by $\dfrac{{2\pi }}{B}$ radians. So the function $\sin \left( {3x} \right)$ has to be converted into the above form. Also by converting it we can then find the period directly.

Complete step by step solution:
Given, $\sin \left( {3x} \right)............................\left( i \right)$
We know that a periodic function is a function which repeats its values on regular intervals or periods. Also a function $f$is said to be periodic with a period $n$ if it follows the following condition:
$f\left( {a + n} \right) = f\left( a \right)\;\;\forall \;\;n > 0$
Also for a function which can be represented as $y = A\sin \left( {Bx + C} \right)\;{\text{and}}\;y = A\cos \left( {Bx + C} \right)$ the period is given by $\dfrac{{2\pi }}{B}$ radians.
Now our given function is $\sin \left( {3x} \right)$ which has to represent in the form$y = A\sin \left( {Bx + C} \right)$.

Now we know that in the Cartesian system the trigonometric function sine is positive in Quadrant I and Quadrant IV. So we can rewrite $\sin \left( {3x} \right)$ as below:
$\sin \left( {3x} \right) = \sin \left( {3x + 2\pi } \right).........................\left( {ii} \right)$
Now we have the general equation $y = A\sin \left( {Bx + C} \right)$.
So on comparing it with (ii) we can write that:
$B = 3\;\;{\text{and}}\;\;C = 2\pi$

Also from the general equation the period of the function is given by: $\dfrac{{2\pi }}{B}$ radians
Also here we have found that the value of$B\;{\text{is}}\;3$. Therefore the period of the function $\sin \left( {3x} \right)$ would be $\dfrac{{2\pi }}{3}$ radians.It simply implies that the arc would be rotating 3 times after which the function $\sin \left( {3x} \right)$ would come back to its initial value again.

Note: Here $y = A\sin \left( {Bx + C} \right)\;{\text{and}}\;y = A\cos \left( {Bx + C} \right)$ are the basic representations of sine and cosine functions respectively. Also we know that frequency and period are inverse to each other or in order to find the frequency of a function we just have to find the reciprocal of the period of the function.