Finding the height of circle as a function of an angle

Question

I'm trying to find the height of a circle based on an angle, I have a diagram below.

I can find y(theta) up to 90 degrees using simple trig, however after that I'm stumped.

I would appreciate any assistance.

Answer 1

You are trying to find the distance between two horizontal lines. The further line (which is fixed) is at $y = -r$. The other line depends on the angle, and has an equation of $y = -r\cos\theta$. So we have: $$ \Delta y = r(1 - \cos \theta) $$

Answer 2

To get the 'height' of the circle you first need to know the radius $\rho$ of the circle. After that by definition of the trigonometric functions you have : $$y = -\rho \cos(\theta)$$ We can see in particular $$ y(0)=-\rho $$ $$ y\left(\frac{\pi}{2}\right)=0 $$ $$ y(\pi)=\rho $$ $$ y\left(\frac{3\pi}{2}\right)=0 $$