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