I have the following homework problem and I just don't know how to go about starting it. Is it asking me to find a unique value of ϕ? I just can't see any other solution apart from when ϕ = θ. So my attempt to solve was to start by looking at A of the first type. I graphed out the standard cosine and sine functions to see if there existed two values such that Sinθ = Sinϕ and Cosθ = Cosϕ. However, graphically no such angles existed, thus I figured that the first type of A can only be true when ϕ = θ. And similarly for the type 2 case, it looks to be only true when ϕ = -θ
Let $A$ be a $2\times2$ matrix with real entries having one of the following forms: $$ A = \left[ \begin{array}{cc} r \cos{\theta} & -r\sin{\theta}\\ r\sin{\theta} & r\cos{\theta} \end{array} \right] \qquad \text{or} \qquad A = \left[ \begin{array}{cc} r \cos{\theta} & r\sin{\theta}\\ r\sin{\theta} & -r\cos{\theta} \end{array} \right] $$ for some $r>0$ and $0 \leq \theta < 2 \pi$. Prove that for some $0 \leq \varphi < 2\pi$ we can write $$ A = r R_{\varphi} \qquad \text{or} \qquad A = r R R_{\varphi} $$ where $$ R = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right] $$ is a reflection and $$ R_{\varphi} = \left[ \begin{array}{cc} \cos{\varphi} & -\sin{\varphi} \\ \sin{\varphi} & \cos{\varphi} \end{array} \right] $$ is a rotation. Note it is not necessarily the case that $\varphi = \theta$.
If $A$ is of the first type you can just choose $\phi=\theta$. In the second case, just take $\phi=-\theta$.