Dans un vieux grimoire, on trouve la formule magique suivante:
\[ 
\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b).
\]
À l'aide de cette formule, donner la valeur exacte de \prodscal{BD}{BA}.
\begin{center}
\begin{tikzpicture}
\draw [shift={(5,0)}] (0,0) -- (119.0362434679265:0.9) arc (119.0362434679265:149.03624346792648:0.9) -- cycle;
\draw (2.178098575555627,4.209141521821144) -- (2.3840205631955795,3.838201991700114) -- (2.7549600933166105,4.044123979340066) -- (2.549038105676658,4.415063509461097) -- cycle; 
\draw(0.42426406871192857,0) -- (0.4242640687119286,0.42426406871192857) -- (0,0.42426406871192857) -- (0,0) -- cycle; 
\draw (0,3)-- (2.549038105676658,4.415063509461097);
\draw (2.549038105676658,4.415063509461097)-- (5,0);
\draw (0,0)-- (0,3);
\draw (0,0)-- (5,0);
\draw  (5,0)-- (0,3);
\draw (-0.17,-0.11) node {$A$};
\draw (5.19,-0.11) node {$B$};
\draw (-0.29,3.31) node {$C$};
\draw (4.07,0.95) node {$30^\circ$};
\draw (2.71,4.69) node {$D$};
\draw(-0.37,1.51) node {3};
\draw(2.57,-0.25) node {5};
\draw (2.35,2.09) node {$\sqrt{34}$};
\end{tikzpicture}
\end{center}

