Consider planar coordinates, given (say in the first quadrant to avoid any range issues) by
$$\left\lbrace \begin{matrix}x=r\cos\theta\\y=r\sin\theta\\\end{matrix}\right.\qquad\text{and}\qquad\left\lbrace \begin{matrix}r=\sqrt{x^2+y^2}\\\theta=\arctan \frac yx\ \ \\\end{matrix}\right.$$
One straightforwardly computes that ${\partial x\over\partial r}=\cos\theta$ and ${\partial y\over\partial r}=\sin\theta$. It turns out that computing the derivatives the other way around yields the same results: $${\partial r\over\partial x}={\partial \sqrt{x^2+y^2}\over\partial x}=\frac12\frac{2x}{\sqrt{x^2+y^2}} =\cos\theta\ \ \left(={\partial x\over\partial r}\right)$$ and similarly ${\partial r\over\partial y}={\partial y\over\partial r}$.
Apart from direct computation, one could also argue that the inverse of the jacobian matrix $\left(\matrix{\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta\\}\right)$ is $\left(\matrix{\cos\theta&\sin\theta\\-\frac{\sin\theta}{r}&\frac{\cos\theta}{r}\\}\right)$ where one can see that the entries $\cos\theta$ and $\sin\theta$ appear twice.
Is there yet a deeper, possibly geometrical, explanation for the identities $\displaystyle{\partial x\over\partial r}={\partial r\over\partial x}$ and $\displaystyle{\partial y\over\partial r}={\partial r\over\partial y}$?
To which extent can one regard this as an intrinsic natural definition of $\cos$ and $\sin$?