When working with inverse trigonometric functions, it's essential to understand the specific intervals for which these functions are defined. For the inverse tangent function, the input values can range from negative infinity to positive infinity, but the output angles are restricted to the interval between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This restriction ensures that the function remains one-to-one, allowing us to find unique angle values.

To visualize the inverse tangent function, we can reflect the tangent function's graph over the line \(y = x\). However, since the tangent function is not one-to-one over its entire range, we only consider the segment between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) for this reflection.

For example, to evaluate \(\tan^{-1}(\sqrt{3})\), we seek an angle \(\theta\) such that \(\tan(\theta) = \sqrt{3}\). Within the specified interval, we find that \(\theta = \frac{\pi}{3}\) satisfies this condition, as the tangent of \(\frac{\pi}{3}\) equals \(\sqrt{3}\).

Similarly, to find \(\tan^{-1}(-1)\), we look for an angle \(\theta\) where \(\tan(\theta) = -1\). In the interval from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), this occurs at \(\theta = -\frac{\pi}{4}\).

When evaluating inverse trigonometric functions using a calculator, the process is straightforward. You typically press the 'second' button followed by the corresponding trigonometric function (sine, cosine, or tangent) and then input the value you wish to evaluate. It's important to ensure that your calculator is set to radian mode unless specified otherwise.

By understanding these concepts and practicing various examples, you can confidently evaluate inverse trigonometric functions and apply them in different mathematical contexts.