We have seen that in order to utilize the inverse Sine function, we were required to limit the domain of the original Sine function to just \(\left[-\frac{\pi}{{2}},\frac{\pi}{{2}}\right]\). In the Unit Circle, this had the effect that for any Horizontal Line (aka, \(y=\sin(\theta)\), there was only one intersection with the Unit Circle.

Try this out: What requirements would you need to enforce on the domain of Cosine so that the resulting Cosine Function is invertible? You should look at the graph of Cosine as well as the Unit Circle before you make up your mind.

Basic Cosine Graph Here Basic Unit Circle Here

What did you come up with? In this case, the domain of Cosine which permits an inverse and is near 0 happens to be: \([0,\pi]\). We can see this working visually with both the Cosine Graph and the Unit Circle (where vertical lines must only intersect once):

Cosine Inverse Domain Limited And the Unit Circle:Half Unit Circle (Cosine inverse)

Taken altogether, the features of the inverse Cosine function are then:

• \(\theta=\cos^{-1}(y)\):
• Domain: \([-1,1]\)
• Range: \([0,\pi]\)