Question

1. $\tan(-\frac{2\pi}{3})$

a) $-\sqrt{3}$ b) $-\frac{\sqrt{3}}{3}$ c) $\frac{\sqrt{3}}{3}$ d) $\sqrt{3}$

1. $\tan(630^{\circ})$

Explanation:

Step1: Use the property of tangent function \(\tan(-\alpha)=-\tan\alpha\)

For \(\tan(-\frac{2\pi}{3})\), we have \(\tan(-\frac{2\pi}{3})=-\tan\frac{2\pi}{3}\)

Step2: Rewrite the angle \(\frac{2\pi}{3}\)

\(\frac{2\pi}{3}=\pi - \frac{\pi}{3}\), and \(\tan(\pi-\theta)=-\tan\theta\). So \(\tan\frac{2\pi}{3}=\tan(\pi - \frac{\pi}{3})=-\tan\frac{\pi}{3}\)

Step3: Calculate \(\tan\frac{\pi}{3}\)

We know that \(\tan\frac{\pi}{3}=\sqrt{3}\)

Step4: Substitute back

\(\tan(-\frac{2\pi}{3})=-(-\tan\frac{\pi}{3})=\tan\frac{\pi}{3}=\sqrt{3}\)

Answer:

D. \(\sqrt{3}\)