Why doesn't a 20 degree rotation change the slopes of $y=x$ and $y=\frac{x}{2}$ by the same amount?

Question

It seems that if I rotate different lines (lying in the same quadrant) the same number of degrees they move different amounts (in terms of their slope). (where the rotation is such that all the lines do not enter a different quadrant)

Can someone give me intuition why this is?

My guess was that maybe it has something to do with the nature of a circle.

Answer 1

Just reposting Matt Samuel's comment as an answer:

The slope is the tangent of the angle. The tangent is a nonlinear function (so going from, for example, 5 to 25 degrees will yield a different change in slope from going from 10-30 degrees, even though the change in degrees is the same)

Answer 2

Slope is $\tan x$ , $x$ being the angle the line makes with $x$ axis. $x$ is 45 in the line $y=x$ and around 26 when $2y=x$. As you know that $\tan$ approaches infinity when $x$ approaches 90, so as we add 20 to 45, the value of $\tan$ will increase at a greater rate than when we add 20 to ~26.