
trouble with reflections
I am having a lot of trouble understanding reflections and formulas. These two problems really have me confused. Any sort of guidance would help. Thank you.
1)By explicit comparison of formulas, verify that if R is the reflection across the xaxis and $\displaystyle p_{\theta}$ is the counterclockwise rotation about (0, 0) through the angle $\displaystyle \theta$, then $\displaystyle p_{\theta} \circ R$ is the reflection $\displaystyle r_{L}$ across the line L making an angle $\displaystyle \frac{\theta}{2}$ with the positive xaxis, for any $\displaystyle \theta , 0\leq \theta \leq 2 \pi $. Thus, every reflection across a line L passing through (0, 0) equals $\displaystyle p_{\theta} \circ R$ for some angle $\displaystyle \theta $.
so $\displaystyle p_{\theta}(x,y) = (xcos(\theta)  ysin(\theta), xsin(\theta) + ycos(\theta)) $ but what would be the formula for R so I can apply this to $\displaystyle p_{\theta}(R)$
2)Let R be the reflection across the xaxis and let $\displaystyle p_{\theta}$ be the counterclockwise rotation about (0,0) through the angle $\displaystyle \theta$
a) prove: $\displaystyle R\circ p_{\theta} = p_{\theta}^{1} \circ R $
b) prove: if $\displaystyle R_{L}$ is any reflection across a line L passing through (0, 0), then $\displaystyle R_{L} \circ p_{\theta} = p^{1}_{\theta} \circ R_{L} $
a) i kinda understand this issue of inverses, am I gonna have to actually apply the formula for p theta? again, not sure about what R is.
b) same thing, but what is the difference between R and R_L??

The reflection R across the xaxis is:
R(x,y) = (x,y).