28 lines
844 B
TeX
28 lines
844 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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\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{2i}{e^{i\theta} - e^{-i\theta}}
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\end{gather*}
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\begin{gather*}
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\text{let} \quad \csc(\theta) = x \\
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\frac{2i}{x} = e^{i\theta} - e^{-i\theta} \\
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\frac{2i}{x}e^{i\theta} = (e^{i\theta})^2 - 1 \\
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(e^{i\theta})^2 + (-\frac{2i}{x})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{-(\frac{2i}{x}) \pm \sqrt{(\frac{2i}{x})^2 - 4(-1)}}{2} = x^{-1}i \pm \sqrt{1 - x^2} \\
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i\theta = \ln(x^{-1}i \pm \sqrt{1 - x^2}) \\
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\theta = -i\ln(x^{-1}i \pm \sqrt{1 - x^2})
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\end{gather*}
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\begin{gather*}
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\therefore \text{arccsc}(\theta) = -i\ln(i\theta^{-1} \pm \sqrt{1 -\theta^2})
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\end{gather*}
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\end{document}
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