Question

find the exact value.

tan\frac{2\pi}{3}

type + or -

?\sqrt{}

Explanation:

Step1: Determine the reference angle

The angle \(\frac{2\pi}{3}\) is in the second quadrant. The reference angle \(\theta'\) for an angle \(\theta=\frac{2\pi}{3}\) is \(\pi - \frac{2\pi}{3}=\frac{\pi}{3}\).

Step2: Use the tangent identity

We know that \(\tan\theta=-\tan\theta'\) for \(\theta\) in the second quadrant (since \(\tan x=\frac{\sin x}{\cos x}\), \(\sin x> 0\) and \(\cos x<0\) in the second quadrant). And \(\tan\frac{\pi}{3}=\sqrt{3}\).

Answer:

\(-\sqrt{3}\)