Eliminate the parameter

Question

Given the parametric equations:

$x = sin(\frac{1}{2} \theta)$ $y = cos(\frac{1}{2} \theta)$

Eliminate the parameter. I am completely lost. Please help.

Answer 1

Another approach:

• Rewrite the parametric equations: \begin{align*} x&=\sin{ \frac{\theta}{2} }\\ y&=\cos{\frac{\theta}{2}} \end{align*}
• Square both sides and add up:

$$ x^2+y^2 = \sin^2{ \frac{\theta}{2} }+\cos^2{ \frac{\theta}{2} } = 1 $$

Answer 2

If you are completely lost then go step by step.

Solve for theta in the first equation, $\theta = 2 \sin^{-1} x.$
Plug this into the second equation $y = \cos (\sin^{-1} x)$

You have eliminated the parameter.

simplify. $y = \sqrt {1-x^2}\\ x^2 + y^2 = 1$

When you get more familiar with these. The answer will seem obvious. But, until then, you at least have a process to get you there.