Question

what is the value of $\sin(\frac{2\pi}{3})$?
$ - \frac{1}{2}$
$\frac{\sqrt{3}}{2}$
$ - \frac{\sqrt{3}}{2}$
$\frac{1}{2}$

Explanation:

Step1: Convert angle to degree

We know that \(\pi = 180^{\circ}\), so \(\frac{2\pi}{3}=\frac{2\times180^{\circ}}{3}=120^{\circ}\)

Step2: Use sine formula

\(\sin(120^{\circ})=\sin(180^{\circ} - 60^{\circ})\)
Since \(\sin(A - B)=\sin A\cos B-\cos A\sin B\), here \(A = 180^{\circ}\), \(B=60^{\circ}\), \(\sin(180^{\circ} - 60^{\circ})=\sin180^{\circ}\cos60^{\circ}-\cos180^{\circ}\sin60^{\circ}\)
We know that \(\sin180^{\circ}=0\), \(\cos180^{\circ}=- 1\), \(\cos60^{\circ}=\frac{1}{2}\), \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\)
\(\sin(180^{\circ} - 60^{\circ})=0\times\frac{1}{2}-(-1)\times\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}\)

Another way:
We know that for the unit - circle, the coordinates of the point corresponding to the angle \(\theta=\frac{2\pi}{3}\) are \((-\frac{1}{2},\frac{\sqrt{3}}{2})\) and \(\sin\theta=y\) coordinate of the point on the unit - circle.

Answer:

\(\frac{\sqrt{3}}{2}\) (the second option)