Consider the differential equation $\frac{d^{2}y}{dx^{2}}+y=0$ with initial conditions $y(0)=0$ and $y'(0)=1$. The solution is well known - $y=\sin(x)$. I know how to derive this solution, since the given equation is a linear differential equation with constant coefficients, and characteristic equation $z^{2}+1=0$.
I also know that this identity, combined with the initial conditions, allows us to compute $y^{(n)}(0)$ and thus the Maclaurin series of $y$, which coincides with the Maclaurin series of $\sin(x)$. Neither of these proofs appear to use any property of $\sin(x)$ other than its oscillating derivatives.
Does there exist a proof of the solution to this equation which uses some other properties to $\sin(x)$? If not, is there a way of visualising it, considering the connection of $\sin$ to the unit circle?