Question

find all of the angles which satisfy the given equation. (use the parameter k as necessary to represent any
sin(\theta)=\frac{sqrt{3}}{2}
\theta=

Explanation:

Step1: Recall sine - value on unit - circle

We know that $\sin(\theta)=\frac{\sqrt{3}}{2}$ has solutions in the first and second quadrants. The reference angle $\theta_{r}$ for which $\sin(\theta_{r})=\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$ (or $60^{\circ}$).

Step2: Find solutions in radians

In the first quadrant, $\theta=\frac{\pi}{3}+2k\pi$, where $k\in\mathbb{Z}$ (the set of all integers). In the second quadrant, $\theta = \pi-\frac{\pi}{3}+2k\pi=\frac{2\pi}{3}+2k\pi$, where $k\in\mathbb{Z}$.

Answer:

$\theta=\frac{\pi}{3}+2k\pi,\frac{2\pi}{3}+2k\pi,k\in\mathbb{Z}$