So about a week ago I started brainstorming on how to rotate a function generally by an angle $\alpha$. After talking to my teacher about it we ended up on rotating a point instead and finding the coordinates of the point in terms of the original $x$, $y$ and the angle $\alpha$.
We found a simple way to write the rotated points coordinates $x'$ and $y'$:
\begin{align*} x'&=x\cos(\alpha)-y\sin(\alpha) \\ y'&=x\sin(\alpha)+y\cos(\alpha) \end{align*}
And so I started messing around with it. Obviously you can rotate any set of points on the $x$,$y$ plane not just functions. And even if you start out with a function it was likely that after the rotation it wasn't a function anymore.
So to get to my question. I noticed that you can rotate $\sin$ and $\cos$ up to a certain degree and it remains a function and it looked like the boundary was $\pi/4$.
And so I plugged it into the formula (changed the $y$ to $y'$ and the $x$ to $x'$ in $y=\sin(x)$)
And came up with:
$$\frac{x+y}{\sqrt 2}=\sin(\frac{x-y}{\sqrt 2})$$
And I couldn't solve this for $y$. (This is question number one.)
Is there a way to prove that the function (if it is one) is monotonically decreasing (because that would imply that it is a function) (This is question number two.)
My math knowledge only goes as far as calc 2.
I am grateful for any help I get with this.
And also any edits to make this question more accesable, easy to read are appreciated. (Sorry for the lack of MathJax)
My question differs from the one suggested in the fact that the other question does not give a solution for y in my first question.
And for the second question if I implicitly differentiate and find dy/dx being less than or equal to zero will that be it?
