Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
$\tan^{-1}\sqrt3$

Verified step by step guidance

1. Understand that ${\tan }^{-1}(x)$ represents the angle whose tangent is $x$. Therefore, ${\tan }^{-1}(\sqrt{3})$ is the angle whose tangent is $\sqrt{3}$.
2. Recall the special angles and their tangent values. The tangent of $\frac{\pi }{3}$ (or 60 degrees) is $\sqrt{3}$.
3. Since ${\tan }^{-1}(\sqrt{3})$ is asking for the angle whose tangent is $\sqrt{3}$, and we know that $\tan (\frac{\pi }{3})=\sqrt{3}$, the angle is $\frac{\pi }{3}$.
4. Verify that $\frac{\pi }{3}$ is within the range of the inverse tangent function, which is $(-\frac{\pi }{2},\frac{\pi }{2})$. Since $\frac{\pi }{3}$ is within this range, it is a valid solution.
5. Conclude that ${\tan }^{-1}(\sqrt{3})=\frac{\pi }{3}$.