Answer:
\(x=\frac{\pi}{3} and x=\frac{2\pi}{3}\)
Step-by-step explanation:
We are given that \(sin x=\frac{\sqrt3}{2}\)
We have to find all solutions of the given equation
We know that \(sin \frac{\pi}{3} =sin60^{\circ}=\frac{\sqrt3}{2}\)
sin x is positive then the value of sin x will lie in I quadrant and II quadrant.The value of sin x is negative in III and IV quadrant .
We are given that sin x is positive then the solution will lie in I and II quadrant only.Therefore, the solution of sin x will not lie in III and IV quadrant .
\(sin x =sin \frac{\pi}{3}\) ...(I equation )and \(sin x =sin(\pi-\frac{\pi}{3})\)...(II equation)
In II quadrant \(\theta\) change into\((\pi-\theta )\)
Cancel sin on both side of equation I
Then, we get
\(x=\frac{\pi}{3}\)
\(sin x =sin (\frac{3\pi-\pi}{3})\)
\(sin x =sin \frac{2\pi}{3}\)...(II equation )
Cancel sin on both side of equation II
Then we get
\(x=\frac{2\pi}{3}\)
Hence, the solutions of equation are
\(x=\frac{\pi}{3} and x=\frac{2\pi}{3}\)
