the reflection through the y-axis, preserves the y-axis, right? this means that the matrix for f sends (0,1) to (0,1) (since (0,1) lies on the y-axis). so all we need to know is what f does to (1,0), and we're done (almost).
but clearly, f sends (1,0) to it's "reflection", (1,0) in the "opposite direction", which is -(1,0) = (-1,0).
so let's say the matrix for f is:
and actually doing the matrix multiplication on the LHS gives us a = -1, c= 0.
similarly, we see that:
so b = 0, d = 1.
thus the matrix for f is:
which clearly sends the point (x,y) to (-x,y).
how does this relate to your qθ rule? note that the angle the y-axis forms with the x-axis is θ = π/2, so:
cos(2θ) = cos(π) = -1
sin(2θ) = sin(π) = 0
so we get the same matrix as above.