# How do you find theta if costheta=2?

## 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?

Apr 9, 2018

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

$x = \frac{\pi}{2} + i \ln \left(i z + \sqrt{1 - {z}^{2}}\right)$

#### Explanation:

Given: $\cos \left(\theta\right) = 2$

$\theta = {\cos}^{-} 1 \left(2\right)$

$\theta = \frac{\pi}{2} + i \ln \left(2 i + \sqrt{3} i\right)$