101 lines
3.5 KiB
TeX
101 lines
3.5 KiB
TeX
\Part[
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\eng{Geometry}
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\ger{Geometrie}
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]{geo}
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\Section[
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\eng{Trigonometry}
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\ger{Trigonometrie}
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]{trig}
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\begin{formula}{exponential_function}
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\desc{Exponential function}{}{}
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\desc[german]{Exponentialfunktion}{}{}
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\eq{\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}}
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\end{formula}
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\begin{formula}{sine}
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\desc{Sine}{}{}
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\desc[german]{Sinus}{}{}
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\eq{\sin(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n+1)}}{(2n+1)!} \\
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&= \frac{e^{ix}-e^{-ix}}{2i}}
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\end{formula}
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\begin{formula}{cosine}
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\desc{Cosine}{}{}
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\desc[german]{Kosinus}{}{}
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\eq{\cos(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n)}}{(2n)!} \\
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&= \frac{e^{ix}+e^{-ix}}{2}}
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\end{formula}
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\begin{formula}{hyperbolic_sine}
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\desc{Hyperbolic sine}{}{}
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\desc[german]{Sinus hyperbolicus}{}{}
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\eq{\sinh(x) &= -i\sin{ix} \\ &= \frac{e^{x}-e^{-x}}{2}}
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\end{formula}
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\begin{formula}{hyperbolic_cosine}
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\desc{Hyperbolic cosine}{}{}
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\desc[german]{Kosinus hyperbolicus}{}{}
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\eq{\cosh(x) &= \cos{ix} \\ &= \frac{e^{x}+e^{-x}}{2}}
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\end{formula}
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\Subsection[
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\eng{Various theorems}
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\ger{Verschiedene Theoreme}
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]{theorems}
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\begin{formula}{sum}
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\desc{}{}{}
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\desc[german]{}{}{}
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\eq{1 &= \sin^2 x + \cos^2 x}
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\end{formula}
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\begin{formula}{addition_theorems}
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\desc{Addition theorems}{}{}
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\desc[german]{Additionstheoreme}{}{}
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\eq{
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\sin(x\pm y) &= \sin x \cos y \pm \cos x \sin y \\
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\cos(x\pm y) &= \cos x \cos y \mp \sin x \sin y \\
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\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}
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}
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\end{formula}
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\begin{formula}{double_angle}
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\desc{Double angle}{}{}
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\desc[german]{Doppelwinkelfunktionen}{}{}
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\eq{
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\sin 2x &= 2\sin x \cos x \\
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\cos 2x &= \cos^2 x - \sin^2 x = 1 - 2\sin^2 x \\
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\tan 2x &= \frac{2\tan x}{1 - \tan^2x}
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}
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\end{formula}
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\begin{formula}{name}
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\desc{}{}{$\tan\theta = b$}
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\desc[german]{}{}{$\tan\theta = b$}
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\eq{\cos x + b\sin x = \sqrt{1 + b^2}\cos(x-\theta)}
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\end{formula}
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\Subsection[
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\eng{Table of values}
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\ger{Wertetabelle}
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]{value_table}
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\begingroup
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\setlength{\tabcolsep}{0.9em} % horizontal
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\renewcommand{\arraystretch}{2} % vertical
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\begin{table}[h]
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\centering
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% \caption{caption}
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\label{tab:sin_cos_table}
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\begin{tabular}{c|c|c|c|c|c|c|c|c}
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\GT{angle_deg} & 0° & 30° & 45° & 60° & 90° & 120° & 180° & 270° \\ \hline
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\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
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$\sin(x)$ & $0$ & $\frac{1}{2} $ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & $1 $ & $\frac{\sqrt{3}}{2}$ & $ 0$ & $-1 $ \\
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$\cos(x)$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2} $ & $0 $ & $\frac{-1}{2} $ & $-1$ & $ 0 $ \\
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$\tan(x)$ & $0$ & $\frac{1}{\sqrt{3}}$ & $\frac{1}{\sqrt{2}}$ & $\frac{1}{2} $ & $\infty$ & $-\sqrt{3} $ & $ 0$ & $\infty$ \\
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\end{tabular}
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\end{table}
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\endgroup
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