Question

find all solutions of the equation.

\\( \sin x = \frac { \sqrt { 3 } } { 2 } \\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\\( \bigcirc \\) a. \\( x = \\)

(simplify your answer. type your answer(s) as an expression, using \\( n \\) as the variable, in the form \\( a + b n \\) where \\( 0 \leq a < 2 \pi \\). type any angle measures in radians, using \\( \pi \\) as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

\\( \bigcirc \\) b. there is no solution.

Explanation:

Step1: Recall the unit - circle values

We know that \(\sin x=\frac{\sqrt{3}}{2}\) when \(x = \frac{\pi}{3}+ 2n\pi\) (in the first - quadrant) and \(x=\pi-\frac{\pi}{3}=\frac{2\pi}{3}+2n\pi\) (in the second - quadrant) for \(n\in\mathbb{Z}\), because the sine function has a period of \(2\pi\).

Answer:

A. \(x=\frac{\pi}{3}+2n\pi,\frac{2\pi}{3}+2n\pi\)