How do you find $theta$ if $costheta=2$ ?

Question

I know that the cosine function taken over the reals only has values between $-1$ and $1$ . However, I think there is some complex value of $theta$ for any complex output of a sine or cosine function. How would you find such a value?

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Answer 1 · 2018-04-09T17:23:47

The reference Inverse Trigonometric Functions gives us the complex form of the inverse cosine function:

$x = pi/2+iln(iz+sqrt(1-z^2))$

Given: $cos(theta) = 2$

$theta = cos^-1(2)$

$theta = pi/2 + iln(2i+sqrt3i)$