Question

How to find the solution of the general solution of \[\sin 2x=\dfrac{\sqrt{3}}{2}\] ?

Answer 1

Hint: The given equation is a trigonometric equation and the general solution of this type of equation is given by \[\sin x=\sin y,x=n\pi +{{(-1)}^{n}}y\]. Here \[n\] belongs to the set of integers. So, in order to find the general solution of the above equation, we will use the above equation. We will convert the given trigonometric equation in \[\sin x=\sin y\] form and then compare and write the general solution.

Complete step by step answer:
The above equation belongs to the concept of trigonometric equations and their solution. In order to find the solution of the trigonometric equation let us get a brief intro of what they are. Trigonometric equations are the equations having a trigonometric function and if it holds true for any value of angle then it is called trigonometric identity.

Now in the question we have \[\sin 2x=\dfrac{\sqrt{3}}{2}\] , in order to find the solution of the equation we need to convert it in \[\sin x=\sin y\] form.
Therefore,
\[\sin 2x=\dfrac{\sqrt{3}}{2}\]
As we know that \[\sin \left( \frac{\pi }{3} \right)=\dfrac{\sqrt{3}}{2}\]. So, we will replace it in the above equation.
\[\Rightarrow \sin 2x=\sin \left( \dfrac{\pi }{2} \right)\]
This implies that \[y=\dfrac{\pi }{3}\]
Therefore, the general solution is
\[\begin{align}
  & \Rightarrow 2x=n\pi +{{(-1)}^{n}}\dfrac{\pi }{3} \\
 & \Rightarrow x=\frac{n\pi }{2}+{{(-1)}^{n}}\dfrac{\pi }{6} \\
\end{align}\]
Hence, we can conclude that the general solution of \[\sin 2x=\dfrac{\sqrt{3}}{2}\] is \[x=\dfrac{n\pi }{2}+{{(-1)}^{n}}\dfrac{\pi }{6}\] where \[n\] belongs to the set of integers.

Note:
 To solve the trigonometric equations we need to have the knowledge of trigonometric identities, formulas, and the graph of trigonometric functions thoroughly. While writing the general equation notice that it gives the value of x and not of 2x thus you have to divide by two in order to get the general solution for the equation. Also, you should get through with a few trigonometric values.