28 lines
877 B
TeX
28 lines
877 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 \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
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\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
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\tan(\theta) = \frac{\sin(\theta)}{\cos{\theta}} = -\frac{i(-1 + e^{2i\theta})}{1 + e^{2i\theta}}
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\end{gather*}
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\begin{gather*}
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\text{let} \quad \tan(\theta) = x \\
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x(1 + e^{2i\theta}) = -i(-1 + e^{2i\theta}) \\
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x + xe^{2i\theta} = i - ie^{2i\theta} \\
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xe^{2i\theta} + ie^{2i\theta} = i - x \\
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e^{2i\theta}(i + x) = i - x \\
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e^{2i\theta} = \frac{i - x}{i + x} \\
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2i\theta = \ln(\frac{i - x}{i + x}) \\
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i\theta = \frac{1}{2}\ln(\frac{i - x}{i + x}) \\
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\theta = -\frac{i}{2}\ln(\frac{i - x}{i + x})
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\end{gather*}
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\begin{gather*}
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\therefore \arctan(\theta) = -\frac{i}{2}\ln(\frac{i - \theta}{i + \theta})
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\end{gather*}
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\end{document}
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