27 lines
680 B
TeX
27 lines
680 B
TeX
\documentclass{../style}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\begin{document}
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\euler
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\begin{gather*}
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\therefore \cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}
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\end{gather*}
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\begin{gather*}
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\text{let} \quad \cos(\theta) = x \\
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2x = e^{i\theta} + e^{-i\theta} \\
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2xe^{i\theta} = (e^{i\theta})^2 + 1 \\
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(e^{i\theta})^2 + (-2x)e^{i\theta} + 1 = 0
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\end{gather*}
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\begin{gather*}
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e^{i\theta} = \frac{-(-2x) \pm \sqrt{(-2x)^2 - 4}}{2} = x \pm \sqrt{x^2 - 1} \\
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i\theta = \ln(x \pm \sqrt{x^2 - 1}) \\
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\theta = -i\ln(x \pm \sqrt{x^2 - 1})
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\end{gather*}
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\begin{gather*}
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\therefore \arccos(\theta) = -i\ln(\theta \pm \sqrt{\theta^2 - 1})
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\end{gather*}
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\end{document}
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