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katex fix

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This commit is contained in:
Cayden Yap
2026-02-04 11:08:58 -08:00
parent e73b7959cc
commit c8c59df0c7
1 changed files with 28 additions and 10 deletions
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38
sin x = 2.md
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@@ -19,22 +19,40 @@ $$
u - u^{-1} = 4i u - u^{-1} = 4i
$$ $$
$$ $$
u^2 - 1 = 4iu \\ $$$$ u^2 - 1 = 4iu \\ $$
$$
u^2 - 4iu - 1 = 0 u^2 - 4iu - 1 = 0
$$$$ $$
u^2 - 4iu - 4 = -3 $$$$ $$
(u - 2i)^2 = -3 \\ $$$$ u^2 - 4iu - 4 = -3
u - 2i = \pm \sqrt {-3} $$$$ $$
u = 2i \pm \sqrt {-3} \\ $$$$ $$
(u - 2i)^2 = -3 \\
$$
$$
u - 2i = \pm \sqrt {-3}
$$
$$
u = 2i \pm \sqrt {-3} \\
$$
$$
u = i(2 \pm \sqrt 3) u = i(2 \pm \sqrt 3)
$$ $$
Substitute back into $u$, for $n \in \mathbb{Z}$: Substitute back into $u$, for $n \in \mathbb{Z}$:
$$ $$
e^{ix} = i(2 \pm \sqrt 3) \\ $$$$ e^{ix} = i(2 \pm \sqrt 3) \\
ix = \ln (i(2 \pm \sqrt 3)) \\ $$$$ $$
ix = \ln i + 2\pi n+ \ln(2 \pm \sqrt 3) \\ $$$$ $$
ix = \frac {i\pi} 2 + 2\pi n + \ln(2 \pm \sqrt 3) $$$$ ix = \ln (i(2 \pm \sqrt 3)) \\
$$
$$
ix = \ln i + 2\pi n+ \ln(2 \pm \sqrt 3) \\
$$
$$
ix = \frac {i\pi} 2 + 2\pi n + \ln(2 \pm \sqrt 3)
$$
$$
x = \frac \pi 2 - i\ln(2 \pm \sqrt 3) + 2\pi n x = \frac \pi 2 - i\ln(2 \pm \sqrt 3) + 2\pi n
$$ $$