Closed form for sine graphic rotated by 45 degrees? Is there a non-parametric closed form for a function looking like a sine rotated 45 degrees?

I have encountered also a similar question but it asks for a function resembling the rotated sine, but not necessarily exact.
Let $(X_0,Y_0)$ be a point on the rotated graph. Then, converting to polar coordinates, the point is $(\sqrt{X_0^2+Y_0^2},arctan \frac{Y_0}{X_0})$. Rotating 45 deg. gives  ($\sqrt{X_0^2+Y_0^2},arctan \frac{Y_0}{X_0}$+45 deg.$)$. That is $(\sqrt{X_0^2+Y_0^2}cos(arctan \frac{Y_0}{X_0}+45$ deg.$),\sqrt{X_0^2+Y_0^2}sin(arctan \frac{Y_0}{X_0}+45$ deg.$))$ in Cartesian coordinates. If this point is $(x,y)$, then $y=sin$ $x$, so the closed form $\sqrt{x^2+y^2}sin(arctan \frac{y}{x}+45$ deg.$)=sin(\sqrt{x^2+y^2}cos(arctan \frac{y}{x}+45$ deg.$)$ works.
For $45$ deg. rotation clockwise instead, change $+45$ deg. to $+45$ deg.
Let $(X_0,Y_0)$ be a point on the rotated graph. Then, converting to polar coordinates, the point is $(\sqrt{X_0^2+Y_0^2},arctan \frac{Y_0}{X_0})$. Rotating 45 deg. gives  ($\sqrt{X_0^2+Y_0^2},arctan \frac{Y_0}{X_0}$+45 deg.$)$. That is $(\sqrt{X_0^2+Y_0^2}cos(arctan \frac{Y_0}{X_0}+45$ deg.$),\sqrt{X_0^2+Y_0^2}sin(arctan \frac{Y_0}{X_0}+45$ deg.$))$ in Cartesian coordinates. If this point is $(x,y)$, then $y=sin$ $x$, so the closed form $\sqrt{x^2+y^2}sin(arctan \frac{y}{x}+45$ deg.$)=sin(\sqrt{x^2+y^2}cos(arctan \frac{y}{x}+45$ deg.$)$ works.
For $45$ deg. rotation clockwise instead, change $+45$ deg. to $+45$ deg.