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How do you find the period of $y=\sin \left( -\dfrac{1}{3}x \right)?$

seo-qna
Last updated date: 20th Jun 2024
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Answer
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Hint: Here from the given equation we have to calculate period.
For that we have to use the formula of period i.e.
Period $=2\dfrac{\pi }{b}$
Where,
$b=$ Period which is to be calculated but sometimes it is given to calculate actual period.

Complete step by step solution:
By using this formula we can calculate the period. Before starting to solve the equation, remember to transfer the negative sign to the starting of the equation, otherwise the answer will get negative.
The standard formula is also used, i.e.
Where, $a=$ amplitude
$b=$ period
$c=$ phase shift
$d=$ mid line
Given that, the equation is as follows:
$y=\sin \left( -\dfrac{1}{3}x \right)$
We have to find the period from the above equation. The standard equation for calculating the amplitude. Frequency and period is.
$y=a\sin \left( bx+c \right)+d$
Where, Period $=2\dfrac{\pi }{b}$
From above we have to calculate the period.
The given equation is,
$y=\sin \left( -\dfrac{1}{3}x \right)$
Draw the negative sign $\left( - \right)$ to the starting of the equation.
i.e. $y=-\sin \left( \dfrac{1}{3}x \right)$
Now, the formula for calculating the period is
Period $=2\dfrac{\pi }{b}$
As, $b=\dfrac{1}{3}$ (Given in the numerical)
Period $=2\dfrac{\pi }{\dfrac{1}{3}}$
$=2\dfrac{\pi \times 3}{1}$
$=2\pi 3$
Period $=6\pi $

Therefore from above calculation we can say that the period of given equation is $6\pi .$

Additional Information:
The frequency is nothing but some quantity or any other thing which repeats for one per second that means it has a period of one second. The frequency is denoted by $F$ and its unit is Hertz which is written as Hz. In simple term, one cycle per second is known as frequency and this can also called as $1Hz$ as $F=\dfrac{1}{5}=1Hz$
The frequency and period are related to each other.
As the period is the inverse of frequency. It means that period is nothing but time duration for one cycle in the repeating event. Period is denoted by $T.$
i.e. Period $=\dfrac{1}{Frequency}$
$T=\dfrac{1}{F}$

Note: In this numerical we have to calculate the period from the given equation. The time duration between two ware is known as period.
The period is calculated from the formula as
Period $=2\dfrac{\pi }{b}$
Where, $b$ is also the period which we have to calculate by dividing it by $2\pi $ in the case when period and phase shift are the same.