Answer:
![x=\frac{\pi}{3} and x=\frac{2\pi}{3}](https://tex.z-dn.net/?f=%20x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%20and%20x%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D)
Step-by-step explanation:
We are given that ![sin x=\frac{\sqrt3}{2}](https://tex.z-dn.net/?f=%20sin%20x%3D%5Cfrac%7B%5Csqrt3%7D%7B2%7D)
We have to find all solutions of the given equation
We know that ![sin \frac{\pi}{3} =sin60^{\circ}=\frac{\sqrt3}{2}](https://tex.z-dn.net/?f=%20sin%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%3Dsin60%5E%7B%5Ccirc%7D%3D%5Cfrac%7B%5Csqrt3%7D%7B2%7D)
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 .
...(I equation )and
...(II equation)
In II quadrant
change into![(\pi-\theta )](https://tex.z-dn.net/?f=%20%28%5Cpi-%5Ctheta%20%29)
Cancel sin on both side of equation I
Then, we get
![x=\frac{\pi}{3}](https://tex.z-dn.net/?f=%20x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D)
![sin x =sin (\frac{3\pi-\pi}{3})](https://tex.z-dn.net/?f=sin%20x%20%3Dsin%20%28%5Cfrac%7B3%5Cpi-%5Cpi%7D%7B3%7D%29)
...(II equation )
Cancel sin on both side of equation II
Then we get
![x=\frac{2\pi}{3}](https://tex.z-dn.net/?f=%20x%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D)
Hence, the solutions of equation are
![x=\frac{\pi}{3} and x=\frac{2\pi}{3}](https://tex.z-dn.net/?f=%20x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%20and%20x%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D)