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sin x = 2.md

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