60 lines
638 B
Markdown
60 lines
638 B
Markdown
#Math #Trig
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$$
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\sin x = 2
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$$
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$$
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\frac {e^{ix} - e^{-ix}} {2i} = 2
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$$
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$$
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e^{ix} - e^{-ix} = 4i \\
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$$
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$$
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e^{ix} - (e^{ix})^{-1} = 4i
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$$
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Let $u = e^{ix}$:
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$$
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u - u^{-1} = 4i
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$$
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$$
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u^2 - 1 = 4iu
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$$
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$$
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u^2 - 4iu - 1 = 0
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$$
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$$
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u^2 - 4iu - 4 = -3
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$$
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$$
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(u - 2i)^2 = -3
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$$
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$$
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u - 2i = \pm \sqrt {-3}
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$$
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$$
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u = 2i \pm \sqrt {-3}
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$$
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$$
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u = i(2 \pm \sqrt 3)
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$$
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Substitute back into $u$, for $n \in \mathbb{Z}$:
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$$
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e^{ix} = i(2 \pm \sqrt 3) \\
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$$
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$$
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ix = \ln (i(2 \pm \sqrt 3)) \\
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$$
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$$
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ix = \ln i + 2\pi n+ \ln(2 \pm \sqrt 3) \\
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$$
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$$
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ix = \frac {i\pi} 2 + 2\pi n + \ln(2 \pm \sqrt 3)
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$$
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$$
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x = \frac \pi 2 - i\ln(2 \pm \sqrt 3) + 2\pi n
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$$
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