In order to find a domain that works for Tangent to have an inverse function, it is easiest to consider the Tangent Graph:

Now, I happen to feel this works out quite neatly, but you can see that between the two asymptotes, Tangent is monotonically increasing and is thus, fully invertible! How nice is that? This leads us naturally to a domain restriction of \(\left(-\frac{\pi}{{2}},\frac{\pi}{{2}}\right)\). However, different from Sine and Cosine, the range of Tangent on this restricted domain is actually all real numbers. As a result, the Inverse Tangent has the following properties:

• \(\theta=\tan^{-1}\left(y\right)\):
• Domain: All Real Numbers
• Range: \(\left(-\frac{\pi}{{2}},\frac{\pi}{{2}}\right)\)

A very practical and useful result of the domain being all real numbers, is that for most practical purposes, the Inverse Tangent function is by far the "best" choice to use if you need to calculate an angle. "Best" in this case means that it will work in almost any situation due to its domain being unrestricted, whereas Inverse Sine and Cosine suffer from very limited domains.