A defining feature of radians as a unit of measurement is that if an angle $\theta$ is expressed in radians, the height of a point on the unit circle at this angle is $\sin(\theta)$.
Is it possible to choose a different unit of measurement for the angle so that we get something other than a sinusoid?
Why doesn't the height of a point on the unit circle as a function of the angle give us a well-shaped half-circle on each half-period rather than a sinusoid? Wouldn't such a circle shape be more fundamental than a sinusoid?
If not, why must the coordinates of a point on the unit circle be sinusoidal as a function of the angle?
I may have a very basic error in my thought, maybe sines were not taught properly to me, but when I learned them they were something that suddenly came out of nowhere. I know it's like the movement of a wriggling snake or the time diagram of a spring that is bouncing, but I still have this question in my head. I would be thankful if I hear your input on it.