Question

tan^-1 $\left(\tan \frac{2\mathrm{\pi }}{3}\right)$is equal to __________________.

Answer 1

$$\begin{gathered} {\tan }^{-1}\left(\tan \frac{2\mathrm{\pi }}{3}\right) \\ ={\tan }^{-1}\left[\tan \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right)\right] \\ ={\tan }^{-1}\left(-\tan \frac{\mathrm{\pi }}{3}\right) \end{gathered}$$
$$\begin{gathered} ={\tan }^{-1}\left[\tan \left(-\frac{\mathrm{\pi }}{3}\right)\right]\left[\tan \left(-\theta \right)=-\tan \theta \right] \\ =-\frac{\mathrm{\pi }}{3}\left[{\tan }^{-1}\left(\tan x\right)=x,\mathrm{if}x\in \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)\right] \end{gathered}$$

tan⁻¹$\left(\tan \frac{2\mathrm{\pi }}{3}\right)$ is equal to $\underline{-\frac{\mathrm{\pi }}{3}}$.