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solve-sin-x-2-x-C-




Question Number 22856 by FilupS last updated on 23/Oct/17
solve: sin(x)=2,   x∈C
$$\mathrm{solve}:\:\mathrm{sin}\left({x}\right)=\mathrm{2},\:\:\:{x}\in\mathbb{C} \\ $$
Commented by Tinkutara last updated on 23/Oct/17
Write as e^(iθ) =cos θ+isin θ and  e^(−iθ) =cos θ−isin θ  Subtract to get sin θ in terms of e^(iθ)   and e^(−iθ) . Assume one of them =x, so  other will be (1/x). You will get e^(iθ)  and  then take log.
$${Write}\:{as}\:{e}^{{i}\theta} =\mathrm{cos}\:\theta+{i}\mathrm{sin}\:\theta\:{and} \\ $$$${e}^{−{i}\theta} =\mathrm{cos}\:\theta−{i}\mathrm{sin}\:\theta \\ $$$${Subtract}\:{to}\:{get}\:\mathrm{sin}\:\theta\:{in}\:{terms}\:{of}\:{e}^{{i}\theta} \\ $$$${and}\:{e}^{−{i}\theta} .\:{Assume}\:{one}\:{of}\:{them}\:={x},\:{so} \\ $$$${other}\:{will}\:{be}\:\frac{\mathrm{1}}{{x}}.\:{You}\:{will}\:{get}\:{e}^{{i}\theta} \:{and} \\ $$$${then}\:{take}\:{log}. \\ $$

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