Evaluate the following expressions. Your answer must be an exact angle in radians and in the interval $[-\frac{\pi }{2},\frac{\pi }{2}]$. Example: Enter pi/6 for $\frac{\pi }{6}$.
(a) ${\tan }^{-1}(-1)=$
(b) ${\tan }^{-1}(1)=$
(c) ${\tan }^{-1}(-\frac{\sqrt{3}}{3})=$

Step 1 ：The question is asking for the inverse tangent (also known as arctangent) of several values. The inverse tangent function, denoted as ${\tan }^{-1}(x)$ or $arctan(x)$, is the inverse of the tangent function. It returns the angle whose tangent is x. The range of the inverse tangent function is $-\frac{\pi }{2}$ to $\frac{\pi }{2}$.

Step 2 ：Let's calculate the inverse tangent of the given values.

Step 3 ：For (a) ${\tan }^{-1}(-1)$, the result is -0.7853981633974483 in radians.

Step 4 ：For (b) ${\tan }^{-1}(1)$, the result is 0.7853981633974483 in radians.

Step 5 ：For (c) ${\tan }^{-1}(-\frac{\sqrt{3}}{3})$, the result is -0.5235987755982988 in radians.

Step 6 ：The results are in radians, but they are not in the form that the question asked for. We need to convert these decimal results into fractions of $\pi$.

Step 7 ：For (a) ${\tan }^{-1}(-1)$, the result is -0.250000000000000 in terms of $\pi$.

Step 8 ：For (b) ${\tan }^{-1}(1)$, the result is 0.250000000000000 in terms of $\pi$.

Step 9 ：For (c) ${\tan }^{-1}(-\frac{\sqrt{3}}{3})$, the result is -0.166666666666667 in terms of $\pi$.

Step 10 ：Finally, we can write the results in the form that the question asked for.

Step 11 ：$\boxed{{\displaystyle {\tan }^{-1}(-1)=-\frac{\pi }{4}}}$

Step 12 ：$\boxed{{\displaystyle {\tan }^{-1}(1)=\frac{\pi }{4}}}$

Step 13 ：$\boxed{{\displaystyle {\tan }^{-1}(-\frac{\sqrt{3}}{3})=-\frac{\pi }{6}}}$