Angle addition formula
Task number: 2564
Derive formuli for \(\sin(\alpha+\beta)\) and \(\cos(\alpha+\beta)\) by using matrices of linear maps.
Hint
Determine the matrix of the composed mapping.
Resolution
When \(f\) and \(g\) are the rotations by angle \(\alpha\) and \(\beta\), then their matrices are:
\([f]_{KK}= \begin{pmatrix} \cos \alpha & -\sin\alpha \\ \sin \alpha & \cos\alpha \\ \end{pmatrix}\) a \([g]_{KK}= \begin{pmatrix} \cos \beta & -\sin\beta \\ \sin \beta & \cos\beta \\ \end{pmatrix}\)
The composition of \(f\) and \(g\) is the rotation by angle \(\alpha+\beta\). Its matrix is:
\([g\circ f]_{KK}= \begin{pmatrix} \cos (\alpha+\beta) & -\sin(\alpha+\beta) \\ \sin (\alpha+\beta) & \cos(\alpha+\beta) \\ \end{pmatrix}\)
Simultaneously, the following must hold:
\([g\circ f]_{KK}=[g]_{KK}[f]_{KK}= \begin{pmatrix} \cos \beta & -\sin\beta \\ \sin \beta & \cos\beta \\ \end{pmatrix} \begin{pmatrix} \cos \alpha & -\sin\alpha \\ \sin \alpha & \cos\alpha \\ \end{pmatrix} =\\= \begin{pmatrix} \cos\beta\cos\alpha -\sin\beta\sin\alpha & -\cos\beta\sin\alpha -\sin\beta\cos\alpha \\ \sin\beta\cos\alpha +\cos\beta\sin\alpha & -\sin\beta\sin\alpha +\cos\beta\cos\alpha \\ \end{pmatrix} \)
As both matrices should be qual, we get the desired formuli:
\(\sin (\alpha+\beta) = \sin\beta\cos\alpha +\cos\beta\sin\alpha\),
\(\cos(\alpha+\beta) = -\sin\beta\sin\alpha +\cos\beta\cos\alpha\).
