Relationship between 2 definitions of sin. One using differential equations and one using the unit circle.

Question

I know of 2 ways to define the sine function:

1. The height of a point on the unit circle given the arc length.
2. The solution to the differential equation $f''(\theta)=-f(\theta)$

I know it can also be defined in terms of complex exponentials, but that seems to me to be a different way of writing method #2.

Method #1 is defined using the fact that sine and cosine (a shifted sine) can describe the y and x coordinates of the unit circle, respectively. That means that $sin^2(\theta) + sin^2(\theta+\pi/4) = 1$

My question is how the seemingly unrelated definition of sine as the solution to a differential equation (#2), comes out to be such that it forms a circle as described above.