29 lines
984 B
TeX
29 lines
984 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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\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{1 + e^{2i\theta}}{i(-1 + e^{2i\theta})}
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\end{gather*}
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\begin{gather*}
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\text{let} \quad \cot(\theta) = x \\
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x(i + ie^{2i\theta}) = -1(1 + e^{2i\theta}) \\
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ix + ixe^{2i\theta} = -e^{2i\theta} - 1 \\
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ixe^{2i\theta} + e^{2i\theta} = -1 - ix \\
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(ix + 1)e^{2i\theta} = -1 - ix \\
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e^{2i\theta} = -\frac{1 - ix}{1 + ix} \\
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2i\theta = \ln(\frac{x + i}{x - i}) \\
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i\theta = \frac{1}{2}\ln(\frac{x + i}{x - i}) \\
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\theta = -\frac{i}{2}\ln(\frac{x + i}{x - i})
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\end{gather*}
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\begin{gather*}
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\therefore \text{arccot}(\theta) = -\frac{i}{2}\ln(\frac{\theta + i}{\theta - i})
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\end{gather*}
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\end{document}
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