MP: Additionstheoreme (für Sinus/Kosinus) - Herleitung (Forum Matroids Matheplanet)

$<math> \def\anno{0} % 1 "yes", 0 "no" \ifnum\anno=0 \tikzset{ every label/.style={text=black!1} } \fi \begin{tikzpicture}[%scale=0.7, font=\footnotesize, %nodes={align=left, inner sep=1pt}, Anno/.style={align=left, inner sep=2pt, midway,}, Fill1/.style={fill=magenta!33}, Fill2/.style={fill=blue!22}, declare function={ Sin(\x)=sin(\x); }, ] % Gegebene Gr��en \pgfmathsetmacro{\u}{5.6} % unit length \pgfmathsetmacro{\u}{7.6} % unit length \pgfmathsetmacro{\v}{0.666} % unit length \pgfmathsetmacro{\Alpha}{33} % \pgfmathsetmacro{\Beta}{39} % \pgfmathsetmacro{\SinA}{\u*sin(\Alpha)} \pgfmathsetmacro{\SinB}{\u*sin(\Beta)} \pgfmathsetmacro{\SinAB}{\u*sin(\Alpha+\Beta)} \pgfmathsetmacro{\CosA}{\u*cos(\Alpha)} \pgfmathsetmacro{\CosB}{\u*cos(\Beta)} \pgfmathsetmacro{\CosAB}{\u*cos(\Alpha+\Beta)} \pgfmathsetmacro{\CosACosB}{\u*cos(\Alpha)*cos(\Beta)} \pgfmathsetmacro{\CosASinB}{\u*cos(\Alpha)*sin(\Beta)} \pgfmathsetmacro{\SinACosB}{\u*sin(\Alpha)*cos(\Beta)} \pgfmathsetmacro{\SinASinB}{\u*sin(\Alpha)*sin(\Beta)} % Dreieckskonstruktion \coordinate[label=below:$A$] (A) at (0,0); \coordinate[label=$B$] (B) at (\Alpha+\Beta:\u); \coordinate[label=$C$] (C) at (0,\SinAB); % Triangles \draw[local bounding box=dreieck] (A) -- (B) -- (C) --cycle;% \ifnum\anno=1 \draw[densely dashed] (A) -- +(\Alpha:\CosB+\v); \fi \draw[Fill1] (A) -- +(\Alpha:\CosB) coordinate[label=right:$L_1$] (L1) node[Anno, sloped, above]{ \ifnum\anno=1 $y_2=$ \fi $\cos(\beta)$} -- (B) node[Anno, sloped, below]{ \ifnum\anno=1 $x_2=$ \fi $\sin(\beta)$} -- cycle; \ifnum\anno=1 \draw[densely dashed] (A) -- (\CosACosB+\v,0); \fi \draw[Fill2] (A) -- (\CosACosB,0) coordinate[label=below:$L_2$](L2) node[Anno, below]{ $\cos(\alpha)\cos(\beta)$ \ifnum\anno=1 \\ $=x_3$ \fi} -- (L1) --cycle; \path[] (L1) -- (L2) node[Anno, sloped, above]{ \ifnum\anno=1 $y_3=$ \\ \fi $\sin(\alpha)\cos(\beta)$}; \ifnum\anno=1 \draw[densely dashed] (L2) -- +(0,\SinACosB+\CosASinB+\v); \draw[densely dashed] (C) -- +(\CosAB+\SinASinB+\v,0); \fi \draw[] (L1) -- +(0,\CosASinB) coordinate[label=45:$S$](S) -- (B) node[Anno, above]{ \ifnum\anno=1 $x_4=$ \\ \fi $\sin(\alpha)\sin(\beta)$} --cycle; \path[] (S) -- (L1) node[Anno, sloped, above]{ \ifnum\anno=1 $y_4=$ \\ \fi $\cos(\alpha)\sin(\beta)$}; % Annotations \path[] ($(A)!0.5!(B)$) -- (C) node[midway]{ \ifnum\anno=1 $\Delta_1$ \fi}; \draw[thick] (A) -- (B) node[midway, sloped, above, font=\bfseries]{1}; \path[] (B) -- (C) node[Anno, above]{ \ifnum\anno=1 $x_1=$\\ \fi $\cos(\alpha+\beta)$}; \path[] (A) -- (C) node[Anno, sloped, above]{ \ifnum\anno=1 $y_1=$ \fi $\sin(\alpha+\beta)$}; \path[] ($(A)!0.5!(B)$) -- (L1) node[near start]{ \ifnum\anno=1 $\Delta_2$ \fi}; \path[] ($(A)!0.5!(L1)$) -- (L2) node[near start]{ \ifnum\anno=1 $\Delta_3$ \fi}; \path[] ($(B)!0.5!(L1)$) -- (S) node[near start]{ \ifnum\anno=1 $\Delta_4$ \fi}; % Angles \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=B--C--A}; \draw pic [draw, angle radius=4mm, angle eccentricity=2.1, "$\alpha+\beta$", ] {angle =C--B--A}; \draw pic [draw, angle radius=7mm, %angle eccentricity=2.1, "$\beta$", ] {angle =L1--A--B}; \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=B--L1--A}; \draw pic [draw, angle radius=6mm, %angle eccentricity=2.1, "$\alpha$", ] {angle =L2--A--L1}; \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=L1--L2--A}; \draw pic [draw, angle radius=4mm, angle eccentricity=1.9, "$90^\circ$-$\alpha$\;", ] {angle =A--L1--L2}; \draw pic [draw, angle radius=6mm, %angle eccentricity=2.1, "$\alpha$", ] {angle =S--L1--B}; \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=L1--S--B}; % Points \ifnum\anno=1 \foreach \P in {A,B,C,L1,L2,S} \draw[fill=black] (\P) circle (1.5pt); \fi % AnnotationS - Task \ifnum\anno=1%%%%%%%%%%%%% \begin{scope}[shift={($(C)+(-\u cm-7mm,-7mm)$)}, ] \draw[|-|] (0,0) -- (\u,0) node[midway, above]{1}; \end{scope} \begin{scope}[shift={($(C)+(-\u cm-7mm,-17mm)$)}, ] \pgfmathsetmacro{\Winkel}{\Alpha} \pgfmathsetmacro{\WinkelXShift}{\Winkel > 90 ? -cos(\Winkel) : 0} % \draw[] (\Winkel:1) coordinate(P) -- (0,0) coordinate(Q) -- (1,0) coordinate(R); \draw pic [draw, angle radius=5mm, %angle eccentricity=1.3, % pic text={$\Winkel$}, pic text options={}, "$\alpha$", ] {angle =R--Q--P}; \end{scope} \begin{scope}[shift={($(C)+(-\u cm-7mm+15mm,-17mm)$)}, ] \pgfmathsetmacro{\Winkel}{\Beta} \pgfmathsetmacro{\WinkelXShift}{\Winkel > 90 ? -cos(\Winkel) : 0} % \draw[] (\Winkel:1) coordinate(P) -- (0,0) coordinate(Q) -- (1,0) coordinate(R); \draw pic [draw, angle radius=5mm, %angle eccentricity=1.3, % pic text={$\Winkel$}, pic text options={}, "$\beta$", ] {angle =R--Q--P}; \end{scope} \fi%%%%%%%%%%%%%%%%%% \end{tikzpicture}</math>$

Man entliest: $
\boxed{\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)}$
und

$
\cos(\alpha+\beta) +\sin(\alpha)\sin(\beta) = \cos(\alpha)\cos(\beta)
~\Leftrightarrow~
\boxed{\cos(\alpha+\beta) = \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)}.$

Konstruktion.

[]

Nach der Abbildung gilt $
\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta) +\cos(\alpha)\sin(\beta)$ bzw. ${
\cos(\alpha+\beta) = \cos(\alpha)\sin(\beta)-\sin(\alpha)\sin(\beta)
}$ für Winkel $\alpha$, $\beta$ und $\alpha+\beta$, die positiv und kleiner als $90^\circ$ sind.

Satz: Die Additionstheoreme $
\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta) +\cos(\alpha)\sin(\beta)
$ und $
\cos(\alpha+\beta) = \cos(\alpha)\sin(\beta)-\sin(\alpha)\sin(\beta)
$ gelten für beliebige Winkel $
{ \alpha,~ \beta ~\in~ \mathbb{R} }.$

Insbesondere ist $
\boxed{ \sin(\alpha-\beta)=\sin(\alpha)\cos(\beta) -\cos(\alpha)\sin(\beta) }$

und $\boxed{
\cos(\alpha-\beta)=\cos(\alpha)\cos(\beta) +\sin(\alpha)\sin(\beta) }.$

Beweis.

[]

Verwendet man die Additionstheoreme für Sinus und Kosinus und dividiert alle Größen im obigen Schaubild durch $\cos(\alpha)\cos(\beta)$ erhält man
$<math> \def\anno{0} % 1 "yes", 0 "no" \ifnum\anno=0 \tikzset{ every label/.style={text=black!1} } \fi \begin{tikzpicture}[%scale=0.7, font=\footnotesize, %nodes={align=left, inner sep=1pt}, Anno/.style={align=left, inner sep=2pt, midway,}, Fill1/.style={fill=magenta!33}, Fill2/.style={fill=blue!22}, ] % Gegebene Gr��en \pgfmathsetmacro{\u}{7.6} % unit length \pgfmathsetmacro{\v}{0.666} % \pgfmathsetmacro{\Alpha}{33} % \pgfmathsetmacro{\Beta}{39} % \pgfmathsetmacro{\SinA}{\u*sin(\Alpha)} \pgfmathsetmacro{\SinB}{\u*sin(\Beta)} \pgfmathsetmacro{\SinAB}{\u*sin(\Alpha+\Beta)} \pgfmathsetmacro{\CosA}{\u*cos(\Alpha)} \pgfmathsetmacro{\CosB}{\u*cos(\Beta)} \pgfmathsetmacro{\CosAB}{\u*cos(\Alpha+\Beta)} \pgfmathsetmacro{\CosACosB}{\u*cos(\Alpha)*cos(\Beta)} \pgfmathsetmacro{\CosASinB}{\u*cos(\Alpha)*sin(\Beta)} \pgfmathsetmacro{\SinACosB}{\u*sin(\Alpha)*cos(\Beta)} \pgfmathsetmacro{\SinASinB}{\u*sin(\Alpha)*sin(\Beta)} % Dreieckskonstruktion \coordinate[label=below:$A$] (A) at (0,0); \coordinate[label=$B$] (B) at (\Alpha+\Beta:\u); \coordinate[label=$C$] (C) at (0,\SinAB); % Triangles \draw[local bounding box=dreieck] (A) -- (B) -- (C) --cycle;% \ifnum\anno=1 \draw[densely dashed] (A) -- +(\Alpha:\CosB+\v); \fi \draw[Fill1] (A) -- +(\Alpha:\CosB) coordinate[label=right:$L_1$] (L1) node[Anno, sloped, above]{ \ifnum\anno=1 $y_2=$ \fi $1/\cos(\alpha)$} -- (B) node[Anno, sloped, below]{ \ifnum\anno=1 $x_2=$ \fi $\tan(\beta)/\cos(\alpha)$} -- cycle; \ifnum\anno=1 \draw[densely dashed] (A) -- (\CosACosB+\v,0); \fi \draw[Fill2] (A) -- (\CosACosB,0) coordinate[label=below:$L_2$](L2) node[Anno, below]{ %$\cos(\alpha)\cos(\beta)$ \ifnum\anno=1 \\ $=x_3$ \fi} -- (L1) --cycle; \path[] (L1) -- (L2) node[Anno, sloped, above]{ \ifnum\anno=1 $y_3=$ \\ \fi $\tan(\alpha)$}; \ifnum\anno=1 \draw[densely dashed] (L2) -- +(0,\SinACosB+\CosASinB+\v); \draw[densely dashed] (C) -- +(\CosAB+\SinASinB+\v,0); \fi \draw[] (L1) -- +(0,\CosASinB) coordinate[label=45:$S$](S) -- (B) node[Anno, above]{ \ifnum\anno=1 $x_4=$ \\ \fi $\tan(\alpha)\tan(\beta)$} --cycle; \path[] (S) -- (L1) node[Anno, sloped, above]{ \ifnum\anno=1 $y_4=$ \\ \fi $\tan(\beta)$}; % Annotations \path[] ($(A)!0.5!(B)$) -- (C) node[midway]{ \ifnum\anno=1 $\Delta_1$ \fi}; \draw[] (A) -- (B) node[midway, sloped, above]{ $1/\bigl( \cos(\alpha)\cos(\beta) \bigr)$ }; \draw[thick] (A) -- (L2) node[midway, sloped, above, font=\bfseries]{1}; \path[] (B) -- (C) node[Anno, above]{ \ifnum\anno=1 $x_1=$\\ \fi $1-\tan(\alpha)\tan(\beta)$}; \path[] (A) -- (C) node[Anno, sloped, above]{ \ifnum\anno=1 $y_1=$ \fi $\tan(\alpha)+\tan(\beta)$}; \path[] ($(A)!0.5!(B)$) -- (L1) node[near start]{ \ifnum\anno=1 $\Delta_2$ \fi}; \path[] ($(A)!0.5!(L1)$) -- (L2) node[near start]{ \ifnum\anno=1 $\Delta_3$ \fi}; \path[] ($(B)!0.5!(L1)$) -- (S) node[near start]{ \ifnum\anno=1 $\Delta_4$ \fi}; % Angles \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=B--C--A}; \draw pic [draw, angle radius=4mm, angle eccentricity=2.1, "$\alpha+\beta$", ] {angle =C--B--A}; \draw pic [draw, angle radius=7mm, %angle eccentricity=2.1, "$\beta$", ] {angle =L1--A--B}; \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=B--L1--A}; \draw pic [draw, angle radius=6mm, %angle eccentricity=2.1, "$\alpha$", ] {angle =L2--A--L1}; \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=L1--L2--A}; \draw pic [draw, angle radius=4mm, angle eccentricity=1.9, "$90^\circ$-$\alpha$\;", ] {angle =A--L1--L2}; \draw pic [draw, angle radius=6mm, %angle eccentricity=2.1, "$\alpha$", ] {angle =S--L1--B}; \draw pic [draw, angle radius=2.5mm, %angle eccentricity=1.3, ] {right angle=L1--S--B}; % Points \ifnum\anno=1 \foreach \P in {A,B,C,L1,L2,S} \draw[fill=black] (\P) circle (1.5pt); \fi % AnnotationS - Task \ifnum\anno=1%%%%%%%%%%%%% \begin{scope}[shift={($(C)+(-\u cm-7mm,-7mm)$)}, ] \draw[|-|] (0,0) -- (\u,0) node[midway, above]{1}; \end{scope} \begin{scope}[shift={($(C)+(-\u cm-7mm,-17mm)$)}, ] \pgfmathsetmacro{\Winkel}{\Alpha} \pgfmathsetmacro{\WinkelXShift}{\Winkel > 90 ? -cos(\Winkel) : 0} % \draw[] (\Winkel:1) coordinate(P) -- (0,0) coordinate(Q) -- (1,0) coordinate(R); \draw pic [draw, angle radius=5mm, %angle eccentricity=1.3, % pic text={$\Winkel$}, pic text options={}, "$\alpha$", ] {angle =R--Q--P}; \end{scope} \begin{scope}[shift={($(C)+(-\u cm-7mm+15mm,-17mm)$)}, ] \pgfmathsetmacro{\Winkel}{\Beta} \pgfmathsetmacro{\WinkelXShift}{\Winkel > 90 ? -cos(\Winkel) : 0} % \draw[] (\Winkel:1) coordinate(P) -- (0,0) coordinate(Q) -- (1,0) coordinate(R); \draw pic [draw, angle radius=5mm, %angle eccentricity=1.3, % pic text={$\Winkel$}, pic text options={}, "$\beta$", ] {angle =R--Q--P}; \end{scope} \fi%%%%%%%%%%%%%%%%%% \end{tikzpicture}</math>$

Man entliest: $\boxed{
\tan(\alpha+\beta)
= \dfrac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)}
}.$