2 [2 points] Let $ϕ, θ \in R$ . Use a linear transformation to prove that
$\cos (ϕ + θ) = \cos ϕ \cos θ - \sin ϕ \sin θ$
and
$\sin (ϕ + θ) = \sin ϕ \cos θ + \cos ϕ \sin θ$

Answer

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Answer

$\sin (ϕ + θ) = \sin ϕ \cos θ + \cos ϕ \sin θ$

Steps

Step 1 ： $[\begin{matrix} \cos ϕ & - \sin ϕ \\ \sin ϕ & \cos ϕ \end{matrix}] [\begin{matrix} \cos θ \\ \sin θ \end{matrix}] = [\begin{matrix} \cos ϕ \cos θ - \sin ϕ \sin θ \\ \sin ϕ \cos θ + \cos ϕ \sin θ \end{matrix}]$

Step 2 ： $\cos (ϕ + θ) = \cos ϕ \cos θ - \sin ϕ \sin θ$

Step 3 ： $\sin (ϕ + θ) = \sin ϕ \cos θ + \cos ϕ \sin θ$

2 [2 points] Let ϕ,θ∈R. Use a linear transformation to prove thatcos⁡(ϕ+θ)=cos⁡ϕcos⁡θ−sin⁡ϕsin⁡θandsin⁡(ϕ+θ)=sin⁡ϕcos⁡θ+cos⁡ϕsin⁡θ

Answer

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2 [2 points] Let $ϕ, θ \in R$ . Use a linear transformation to prove that
$\cos (ϕ + θ) = \cos ϕ \cos θ - \sin ϕ \sin θ$
and
$\sin (ϕ + θ) = \sin ϕ \cos θ + \cos ϕ \sin θ$