Question

How do you solve $\sin 3x=0$ ?

Answer 1

Hint : We first find the principal value of x for which $\sin 3x=0$ . In that domain, equal value of the same ratio gives equal angles. We find the angle value for x. at the end we also find the general solution for the equation $\sin 3x=0$ .

Complete step-by-step answer :
It’s given that $\sin 3x=0$ . The value in question is 0. We need to find x for which $\sin 3x=0$ .
We know that in the principal domain or the periodic value of
$-\dfrac{\pi }{2}\le x\le \dfrac{\pi }{2}$ for $\sin x$ , if we get $\sin a=\sin b$ where
$-\dfrac{\pi }{2}\le a,b\le \dfrac{\pi }{2}$ then $a=b$ . We have the value of
$\sin \left( 0 \right)$ as 0. $-\dfrac{\pi }{2}<0<\dfrac{\pi }{2}$ .
Therefore,
$\sin \left( 3x \right)=0=\sin \left( 0 \right)$ which gives $3x=0$ .
For ,
$\sin 3x=0$ , the value of $3x$ is $3x=0$ . Solving the equation, we get $x=0$ .
We also can show the solutions (primary and general) of the equation $\sin 3x=0$ through the graph. We take $y=\sin 3x=0$ . We got two equations $y=\sin \left( 3x \right)$ and
$y=0$ . We place them on the graph and find the solutions as their intersecting points.
We can see the primary solution in the interval $-\dfrac{\pi }{2}\le x\le \dfrac{\pi }{2}$ is the point A as $x=0$ .
All the other intersecting points of the curve and the line are general solutions.

We can see the primary solution in the interval $-\dfrac{\pi }{2}\le x\le \dfrac{\pi }{2}$ is the point A as $x=0$.
So, the correct answer is “ $x=0$ OR $x=\dfrac{n\pi }{3}$”.

Note : Although for elementary knowledge the principal domain is enough to solve the problem. But if mentioned to find the general solution then the domain changes to $-\infty \le x\le \infty$ . In that case we have to use the formula $x=n\pi +{{\left( -1 \right)}^{n}}a$ for $\sin \left( x \right)=\sin a$ where $-\dfrac{\pi }{2}\le a\le \dfrac{\pi }{2}$ . For our given problem $\sin 3x=0$ , the general solution will be $3x=n\pi +{{\left( -1 \right)}^{n}}\times 0=n\pi$ . Here $n\in \mathbb{Z}$ .
The simplified general solution for the equation $\sin 3x=0$ will be $x=\dfrac{n\pi }{3}$ .