Question

What is the period and frequency for $\sin \left( 2\pi \dfrac{t}{5} \right)$?

Answer 1

This type of question depends on the period and frequency of a sine function. Frequency is the number of complete cycles of waves passing a point in unit time. Period is the time taken by a complete cycle of the wave to pass a point. Hence frequency is the reciprocal of period. Hence, we can write it as, $Frequency=\dfrac{1}{Period}$. If T is the period of periodic function $f\left( x \right)$ then $f\left( x \right)=f\left( x+T \right)$. Also we know that the $\sin t$ is a periodic function with period $2\pi$ relative to $t$.

Complete step by step solution:
Now, we have to find the period and frequency for $\sin \left( 2\pi \dfrac{t}{5} \right)$.
We know that $\sin t$ is a periodic function with period $2\pi$ relative to $t$.
Hence, we can say that,
$\Rightarrow \sin \left( 2\pi \dfrac{t}{5} \right)$ has a period of $2\pi$ relative to $2\pi \dfrac{t}{5}$.
$\Rightarrow \sin \left( 2\pi \dfrac{t}{5} \right)$ has a period of $\dfrac{2\pi }{\left( \dfrac{2\pi }{5} \right)}$ relative to $\dfrac{2\pi \dfrac{t}{5}}{\left( \dfrac{2\pi }{5} \right)}$.
$\Rightarrow \sin \left( 2\pi \dfrac{t}{5} \right)$ has a period of $2\pi \times \dfrac{5}{2\pi }=5$ relative to $2\pi \dfrac{t}{5}\times \dfrac{5}{2\pi }=t$
$\Rightarrow \sin \left( 2\pi \dfrac{t}{5} \right)$ has a period of $5$ relative to $t$.
Now, as we know that, Frequency is the number of complete cycles of waves passing a point in unit time. Period is the time taken by a complete cycle of the wave to pass a point. Hence frequency is the reciprocal of period.
$\Rightarrow Frequency=\dfrac{1}{Period}$
$\Rightarrow Frequency=\dfrac{1}{5}$
Thus, we can write,
Period of $\sin \left( 2\pi \dfrac{t}{5} \right)=5$ and
Frequency of $\sin \left( 2\pi \dfrac{t}{5} \right)=\dfrac{1}{5}$.

Note: In this type of question students may make mistakes in calculation of period. As we have to find the period relative to $t$ and given function of sin is $\sin \left( 2\pi \dfrac{t}{5} \right)$ , student have to divide $2\pi$ by $\dfrac{2\pi }{5}$ to obtain the value of period.