Question

Find all solutions to the equation. sin x = sqrt(3)/2

Answer 1

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

$x=\frac{\pi}{3} and x=\frac{2\pi}{3}$

Answer 2

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.