2.1 ${\sin ^{ - 1}}x$:

2.1 ${\sin ^{ - 1}}x$:
2.2 ${\cos ^{ - 1}}x$:
2.3 ${\tan ^{ - 1}}x$:
2.4 ${\text{cose}}{{\text{c}}^{ - 1}}x$:
2.5 ${\sec^{ - 1}}x$
2.6 ${\cot^{ - 1}}x$
2.7 Summary

3.1 Property 1
3.2 Property 2
3.3 Property 3
3.4 Property 4
3.5 Property 5
3.6 Property 6
3.7 Property 7
3.8 Property 8
3.9 Property 9
3.10 Property 10
3.11 Property 11
3.12 Property 12
3.13 Property 13

We denote the inverse of sine function by ${\sin ^{ - 1}}x$ (arc $\sin x$) function.

From our previous discussion, the inverse of a function is possible only when it is one-one and onto i.e., bijective. So, we have to choose a certain set of domain and range to consider ${\sin ^{ - 1}}x$ as a function.

a certain set of domain and range to consider ${\sin ^{ - 1}}x$ as a function.

As we know, the domain of sine function is $R$(all real numbers) and range is $[-1,1]$. If we restrict its domain to $\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]$ then it becomes one-one and onto with in range of $[-1,1]$. From the graph of sine function as shown in figure, we can also say that it is one-one and onto not only with in the domain of $\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]$ but also in $\left[ { - \frac{{3\pi }}{2}, - \frac{\pi }{2}} \right]$, $\left[ {\frac{\pi }{2},\frac{{3\pi }}{2}} \right]$,... etc and its range is $[-1,1]$.

We conclude that the inverse of sine function exists with in all the above mentioned set of domain and range of $[-1,1]$ out of which domain $\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]$ is the principle value set. We can write the inverse of sine function as $$y = {\sin ^{ - 1}}x$$ where domain becomes $[-1,1]$ and range becomes $\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]$.