Determine the angle

Let $A$ be the center of the 90 mm driving gear, $B$ be the center of the 70 mm driving gear, and C be the center of the 30 mm driving gear. We know that
\[|AB|=c=99.78, \]
\[ |AC|=b=45+15=60\]
and
\[|BC|=a=35+15=50.\]
By law of cosines we have
\[c^2=a^2+b^2-2ab \cos \theta    \]
\[\Rightarrow    (99.78)^2=60^2+50^2-2 \times 60 \times 50 \cos \theta  \]
\[\Rightarrow \cos \theta =\frac{60^2+50^2-(99.78)^2}{2 \times 60 \times 50}=-0.642675.\]
Hence
\[\theta = \cos^{-1}(-0.642675)= 2.26878104 rad = 2.26878104*180/\pi=123   \text{degrees}.\]