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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?

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

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  • $\begingroup$ So should I attempt to remove the /2 from sin and cosine and wind up with $x/2 + y/2 = 1$? Please be patient, I've been trying to figure this out for a while. $\endgroup$ – Irresponsible Newb Jun 14 '14 at 16:40
  • $\begingroup$ No: You have the equation $x^2 + y^2 = 1$. For any angle $\alpha$, $\sin^2 \alpha + \cos^2 \alpha = 1$. In your case, $\alpha$ just happens to be given by $\theta/2$. $\endgroup$ – amWhy Jun 14 '14 at 16:42
  • $\begingroup$ Oh thanks a ton. $\endgroup$ – Irresponsible Newb Jun 14 '14 at 16:43
  • $\begingroup$ You're welcome! $\endgroup$ – amWhy Jun 14 '14 at 16:47

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