2 [2 points] Let \( \phi, \theta \in \mathbb{R} \). Use a linear transformation to prove that
\[
\cos (\phi+\theta)=\cos \phi \cos \theta-\sin \phi \sin \theta
\]
and
\[
\sin (\phi+\theta)=\sin \phi \cos \theta+\cos \phi \sin \theta
\]

Answer

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Answer

\[\sin (\phi+\theta) = \sin \phi \cos \theta + \cos \phi \sin \theta\]

Steps

Step 1 ：\[\begin{bmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix} = \begin{bmatrix} \cos\phi\cos\theta - \sin\phi\sin\theta \\ \sin\phi\cos\theta + \cos\phi\sin\theta \end{bmatrix}\]

Step 2 ：\[\cos (\phi+\theta) = \cos \phi \cos \theta - \sin \phi \sin \theta\]

Step 3 ：\[\sin (\phi+\theta) = \sin \phi \cos \theta + \cos \phi \sin \theta\]

2 [2 points] Let \( \phi, \theta \in \mathbb{R} \). Use a linear transformation to prove that\[\cos (\phi+\theta)=\cos \phi \cos \theta-\sin \phi \sin \theta\]and\[\sin (\phi+\theta)=\sin \phi \cos \theta+\cos \phi \sin \theta\]

Answer

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2 [2 points] Let \( \phi, \theta \in \mathbb{R} \). Use a linear transformation to prove that
\[
\cos (\phi+\theta)=\cos \phi \cos \theta-\sin \phi \sin \theta
\]
and
\[
\sin (\phi+\theta)=\sin \phi \cos \theta+\cos \phi \sin \theta
\]