solve-in-x-C-sin-x-3-2-

Question Number 153568 by liberty last updated on 08/Sep/21

s o l v e i n x \in C

\sin x = \frac{3}{2}

Answered by Ar Brandon last updated on 08/Sep/21

\sin x = \frac{3}{2} \Rightarrow e^{i x} - e^{- i x} = 3 i

\Rightarrow e^{2 i x} - 3 {i e}^{i x} - 1 = 0 \Rightarrow e^{i x} = \frac{3 i \pm i \sqrt{5}}{2}

\Rightarrow i x = \ln (i) + \ln (3 \pm \sqrt{5}) - \ln 2

\Rightarrow x = i [\ln 2 - \ln (3 \pm \sqrt{5}) - \frac{π}{2} i]

= \frac{π}{2} + i \ln (\frac{2}{3 \pm \sqrt{5}}) = \frac{π}{2} - i \ln (\frac{3 \pm \sqrt{5}}{2})