Finding the value of Inverse Trigonometric functions beyond their Real Domain I wanted to ask how can we calculate the values of the inverse of trigonometric functions beyond their domain of definition, for example $\arcsin{2}$ beyond its domain of $-\frac{\pi}{2}<x<\frac{\pi}{2}$.
I tried to use the Euler form but did not get much. Thanks.

Let us start with the well known Euler form:$$e^{i\theta}=\cos\theta+i\sin\theta$$Now we solve for $cos\theta$.$$\cos\theta=e^{i\theta}-isin\theta$$Now take the $arccos$ of both sides.$$\theta=\arccos(e^{i\theta}-i\sin\theta)=\arccos[f(\theta)]$$Attempt to find the inverse of $f(\theta)$.$$f(\theta)=e^{i\theta}-i\sin\theta=x$$I do not have the skills to do this, if it is possible.  I will assume I have the answer to be $\theta=f^{-1}(x)$.  You would then substitute it up above.$$\theta=\arccos[f(\theta)]$$$$f^{-1}(x)=\arccos[f(f^{-1}(x))]=\arccos(x)$$
And that is your solution.
