”Solve

To solve for $\theta$ in the equation $\sin(\theta) = \frac{\sqrt{3}}{2}$ on the unit circle, we need to determine the angles corresponding to this sine value. The reference angle for $\sin(\theta) = \frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$. Since sine is positive in the first and second quadrants:

1. In the first quadrant, $\theta = \frac{\pi}{3}$

2. In the second quadrant, $\theta = \pi – \frac{\pi}{3} = \frac{2\pi}{3}$

Thus, the solutions are:

$\theta = \frac{\pi}{3}, \frac{2\pi}{3}$

Given $sin( heta) = frac{sqrt{3}}{2}$, recognize that $sin( heta)$ yields this value at specific angles on the unit circle. The principal angle is $frac{pi}{3}$.

We know $sin( heta)$ is positive in two quadrants:

1. In the first quadrant, the angle is directly $frac{pi}{3}$.

2. In the second quadrant, we use the formula $pi – alpha$ where $alpha = frac{pi}{3}$.

So, $ heta = pi – frac{pi}{3} = frac{2pi}{3}$.

Therefore, the solutions are:

$ heta = frac{pi}{3}, frac{2pi}{3}$

Given $sin( heta) = frac{sqrt{3}}{2}$, find $ heta$ in [0, 2$pi$). The solutions are where:

1. $ heta = frac{pi}{3}$

2. $ heta = frac{2pi}{3}$

Thus, the answers are:

$ heta = frac{pi}{3}, frac{2pi}{3}$

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