find solution set of equation sin x = 2 , x ∈ C .


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Question Number 114409 by bemath last updated on 19/Sep/20

$f i n d s o l u t i o n s e t o f e q u a t i o n$ $\sin x = 2, x \in C .$
	Answered by bobhans last updated on 19/Sep/20

	$\sin x = \frac{e^{i x} - e^{- i x}}{2 i} = 2$ $e^{i x} - e^{- i x} = 4 i, l e t e^{i x} = z$ $z - z^{- 1} = 4 i; z^{2} - 4 i z - 1 = 0$ $z = \frac{4 i \pm \sqrt{- 16 + 4}}{2} = (2 \pm \sqrt{3}) i$ $t h e n e^{i x} = (2 \pm \sqrt{3}) i = (2 \pm \sqrt{3}) e^{i . \frac{π}{2}}$ $i x = \ln (2 \pm \sqrt{3}) + i (\frac{π}{2} + 2 n π); n \in Z$ $x = (4 n + 1) \frac{π}{2} \pm i \ln (2 + \sqrt{3}); n \in Z$
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