[Math] Eliminating the parameter of triognometric parametric equations

Question

I'm new to parametric equations and I'm asked to eliminate the parameter to find a single Cartesian equation. I know how to do this with other problems but I am confused when it comes to trigonometric equations.

I have the following problem:

$$x = \sin\left(\frac{\theta}{2}\right),\; y = \cos\left(\frac{\theta}{2}\right) -\pi \le \theta \le \pi$$

I attempt to isolate $\theta$ from $x = \sin(\frac{1}{2})\theta$ however I am unsure of how to do this. I would assume that I use arcsin for this purpose but experimenting on Wolfram Alpha has yielded unexpected and frightening results.

Could somebody explain the process?

Answer 1

Note that $$x^2 + y^2 = \sin^2(\theta/2) + \cos^2(\theta/2) = 1$$