find-solution-set-of-equation-sin-x-2-x-C-

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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