resoudre-dans-c-l-equation-sinx-2-erly-rolvinst-lt-erly-rolvinst-gt-

Question Number 196523 by ERLY last updated on 27/Aug/23

r e s o u d r e d a n s c l e q u a t i o n s i n x = 2 ★ e r l y r o l v i n s t ★ < e r l y r o l v i n s t >

Answered by Frix last updated on 26/Aug/23

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

e^{i x} - \frac{1}{e^{i x}} = 4 i

e^{2 i x} - {4 ie}^{i x} - 1 = 0

t = e^{i x} \Leftrightarrow x = 2 n π - i \ln t

t^{2} - 4 i t - 1 = 0

t = (2 \pm \sqrt{3}) i

\dots

x = \frac{4 n + 1}{2} π \pm i \ln (2 + \sqrt{3})

Answered by Skabetix last updated on 26/Aug/23

O n p o s e z = x + i y a v e c x e t y \in R

s i n (x + i y) = s i n (x) c o s h (y) + i c o s (x) s i n h (y)

O n o b t i e n t l e s y s t e m e d e q u a t i o n s u i v a n t

s i n (x) c o s h (y) = 2 (1)

c o s (x) s i n h (y) = 0 (2)

E n r e s o l v a n t l e s y s t e m e o n a

x = (\frac{p i}{2} + k \times p i) (k p a i r)

y = \pm l n (2 + \sqrt{} 3)

i l e s t p o s s i b l e d e t r o u v e r t o u t e s l e s s o l u t i o n s

e n r e s o l v a n t l e q u a t i o n

e^{i t} - e^{- i t} = 4 i

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