\Section[
    \eng{Geometry}
    \ger{Geometrie}
    ]{geo}

\Subsection[
    \eng{Trigonometry}
    \ger{Trigonometrie}
]{trig}

    \begin{formula}{exponential_function}
        \desc{Exponential function}{}{}
        \desc[german]{Exponentialfunktion}{}{}
        \eq{\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}}
    \end{formula}

    \begin{formula}{sine}
        \desc{Sine}{}{}
        \desc[german]{Sinus}{}{}
        \eq{\sin(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n+1)}}{(2n+1)!} \\
                    &= \frac{e^{ix}-e^{-ix}}{2i}}
    \end{formula}

    \begin{formula}{cosine}
        \desc{Cosine}{}{}
        \desc[german]{Kosinus}{}{}
        \eq{\cos(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n)}}{(2n)!} \\
                    &= \frac{e^{ix}+e^{-ix}}{2}}
    \end{formula}


    \begin{formula}{hyperbolic_sine}
        \desc{Hyperbolic sine}{}{}
        \desc[german]{Sinus hyperbolicus}{}{}
        \eq{\sinh(x) &= -i\sin{ix} \\ &= \frac{e^{x}-e^{-x}}{2}}
    \end{formula}

    \begin{formula}{hyperbolic_cosine}
        \desc{Hyperbolic cosine}{}{}
        \desc[german]{Kosinus hyperbolicus}{}{}
        \eq{\cosh(x) &= \cos{ix} \\ &= \frac{e^{x}+e^{-x}}{2}}
    \end{formula}

\Subsection[
    \eng{Various theorems}
    \ger{Verschiedene Theoreme}
]{theorems}
    \begin{formula}{sum}
        \desc{Hypthenuse in the unit circle}{}{}
        \desc[german]{Hypothenuse im Einheitskreis}{}{}
        \eq{1 &= \sin^2 x + \cos^2 x}
    \end{formula}

    \begin{formula}{addition_theorems}
        \desc{Addition theorems}{}{}
        \desc[german]{Additionstheoreme}{}{}
        \eq{
            \sin(x\pm y) &= \sin x \cos y \pm \cos x \sin y \\
            \cos(x\pm y) &= \cos x \cos y \mp \sin x \sin y \\
            \tan(x\pm y) &= \frac{\sin(x \pm y)}{\cos(x \pm y)} = \frac{\tan x\pm \tan y}{1\mp \tan x \tan y}
        }
    \end{formula}

    \begin{formula}{double_angle}
        \desc{Double angle}{}{}
        \desc[german]{Doppelwinkelfunktionen}{}{}
        \eq{
            \sin 2x &= 2\sin x \cos x \\
            \cos 2x &= \cos^2 x - \sin^2 x = 1 - 2\sin^2 x \\
            \tan 2x &= \frac{2\tan x}{1 - \tan^2x}
        }
    \end{formula}

    \begin{formula}{other}
        \desc{Other}{}{$\tan\theta = b$}
        \desc[german]{Sonstige}{}{$\tan\theta = b$}
        \eq{\cos x + b\sin x = \sqrt{1 + b^2}\cos(x-\theta)}
    \end{formula}


    \Subsubsection[
        \eng{Table of values}
        \ger{Wertetabelle}
    ]{value_table}
        \begingroup
            \setlength{\tabcolsep}{0.9em}  % horizontal
            \renewcommand{\arraystretch}{2}  % vertical
            \begin{table}[h]
                \centering
                % \caption{caption}
                \label{tab:sin_cos_table}
                \begin{tabular}{c|c|c|c|c|c|c|c|c}
                    \GT{angle_deg} & 0°  & 30°                  & 45°                  & 60°                   & 90°             & 120°                 & 180°          & 270° \\ \hline
                    \GT{angle_rad} & $0$ & $\frac{\pi}{6}$      & $\frac{\pi}{4}$      & $\frac{\sqrt{\pi}}{3}$ & $\frac{\pi}{2}$ & $\frac{2\pi}{3}$    & $\pi$         & $\frac{3\pi}{2}$ \\ \hline
                    $\sin(x)$      & $0$ & $\frac{1}{2}       $ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$  & $1     $        & $\frac{\sqrt{3}}{2}$ & $ 0$ & $-1    $ \\ 
                    $\cos(x)$      & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2}       $  & $0     $        & $\frac{-1}{2}      $ & $-1$ & $ 0    $ \\ 
                    $\tan(x)$      & $0$ & $\frac{1}{\sqrt{3}}$ & $\frac{1}{\sqrt{2}}$ & $\frac{1}{2}       $  & $\infty$        & $-\sqrt{3}         $ & $ 0$ & $\infty$ \\
                \end{tabular}
            \end{table}
        \endgroup
