Reflection of the graphics about 2 lines

Question

$y=\sin(x)$ is reflected first in the line $x=y$ and then in the line $y=2$ , what's the equation of the graphics ?

Answer 1

The first reflection results in $x=\sin y$.

The second one results in $x=\sin(4-y)$.

In general the reflection in line $ax+by+c=0$ results in the map: $$(x,y)\mapsto \left(\frac{x(b^{2}-a^{2})-2a(by+c)}{a^{2}+b^{2}},\frac{y(a^{2}-b^{2})-2b(ax+c)}{a^{2}+b^{2}}\right)$$ which means: $$\begin{align} a=1,b=-1, c=0:&\quad (x,y)\mapsto (y,x)\\ a=0,b=1,c=-2:&\quad (x,y)\mapsto (x,4-y) \end{align} $$