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

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) \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\), 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) \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\), 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 ：\(\boxed{\tan^{-1}(-1) = -\frac{\pi}{4}}\)

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

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