the-honk/maths/trigometric functions/sin.tex

\documentclass{../style}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}
\euler

\begin{gather*}
	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}
\end{gather*}

\begin{gather*}
	\text{let} \quad \sin(\theta) = x \\
	2ix = e^{i\theta} - e^{-i\theta} \\
	2ixe^{i\theta} = (e^{i\theta})^2 - 1 \\ 
	(e^{i\theta})^2 + (-2ix)e^{i\theta} - 1 = 0
\end{gather*}

\begin{gather*}
	e^{i\theta} = \frac{-(-2ix) \pm \sqrt{(-2ix)^2 - 4(-1)}}{2} = ix \pm \sqrt{1 - x^2} \\
	i\theta = \ln(ix \pm \sqrt{1 - x^2}) \\
	\theta = -i\ln(ix \pm \sqrt{1 - x^2})
\end{gather*}

\begin{gather*}
	\therefore \arcsin(\theta) = -i\ln(i\theta \pm \sqrt{1 -\theta^2})
\end{gather*}
\end{document}