Question

How do you find the period of $\sin 4x$?

Answer 1

Hint: Normalize the function and find out the coefficient and substitute in formula to get the required. From the given equation, we need to compare it to the normalized sine function equation of \[y{\text{ }} = {\text{ }}A{\text{ }}sin\left( {B\left( {x{\text{ }} + {\text{ }}C} \right)} \right){\text{ }} + {\text{ }}D\] and also a simpler form \[y = A \cdot sin(Bx)\]. If it’s in the required form, no need for simplification. Then later, we identify the value of B which is a coefficient of x, which will be used in the period formula for the given sine function, for which the formula is $Period = \dfrac{{2\pi }}{{\left| B \right|}}$.

Complete step-by-step solution:
The given question whose period we have to find is $\sin 4x$.
The given equation is of the form\[y = A \cdot sin(Bx)\].
Since the formula for period is given by $Period = \dfrac{{2\pi }}{{\left| B \right|}}$. We need to find the value of B, which is the coefficient of x.
On comparing the given equation and the simple form, we can get the value of B which is; 4.
As, we have found B, we can substitute it in the formula and get the period.
$\Rightarrow Period = \dfrac{{2\pi }}{{\left| 4 \right|}}$
Since it cannot further be simplified, this is the final answer that is the period of $\sin 4x$ is
$\Rightarrow Period = \dfrac{{2\pi }}{4}$.

Hence the answer is $Period = \dfrac{{2\pi }}{4}$

Note: It is necessary to have the equation be normalized first, as it makes it easy for identifying the coefficients of A, B, C and so on. And also, we should be careful on not missing out the coefficients which are in fraction forms as well as the nature of sign of the coefficient. We can forget to consider denominators as a part of the coefficient value. We should always remember that period and amplitude is always positive.