Answer :

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We are asked to determine x from the function sin x = sqrt 3/2. In this case, the right hand side number belongs to the 30-60-90 triangle. Using a calculator, in degrees mode, we input in the calculator, arc sin sqrt 3/2 equal to 60 degrees or  120 degrees since it can be in quadrants 1 or 2.
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Answer:

[tex] x=\frac{\pi}{3} and x=\frac{2\pi}{3}[/tex]

Step-by-step explanation:

We are given that [tex] sin x=\frac{\sqrt3}{2}[/tex]

We have to find all solutions of the given equation

We know that [tex] sin \frac{\pi}{3} =sin60^{\circ}=\frac{\sqrt3}{2}[/tex]

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 .

[tex] sin x =sin \frac{\pi}{3}[/tex] ...(I equation )and [tex] sin x =sin(\pi-\frac{\pi}{3})[/tex]...(II equation)

In II quadrant [tex]\theta[/tex] change into[tex] (\pi-\theta )[/tex]

Cancel  sin on both side of equation I

Then, we get

[tex] x=\frac{\pi}{3}[/tex]

[tex]sin x =sin (\frac{3\pi-\pi}{3})[/tex]

[tex] sin x =sin \frac{2\pi}{3}[/tex]...(II equation )

Cancel sin on both side of equation II

Then we get

[tex] x=\frac{2\pi}{3}[/tex]

Hence, the solutions of equation are

[tex] x=\frac{\pi}{3} and x=\frac{2\pi}{3}[/tex]

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