complex analysis - $arccos$ of an imaginary number

How can I solve a $\arccos$ of an imaginary number? like:

$$\cos x = 0.9i$$

Because I can't make the $\arccos$ of a imaginary number

Answer

Well, you can start by noting that $$\cos x=\frac12\left(e^{ix}+e^{-ix}\right).$$ Thus, multiplying by $2e^{ix}$ yields $$e^{2ix}+1=1.8ie^{ix},$$ and the substitution $u=e^{ix}$ yields $$u^2+1=1.8iu.$$ Solve this quadratic for $u,$ then go from there. Bear in mind that $e^{2\pi i}=1,$ so there will be not just two solutions, but infinitely many.