Question

How do you eliminate the parameter theta to find a Cartesian equation of the curve: x=sin(1/2 theta), y=cos(1/2 theta), 0 is less than or equal to theta and theta is less than or equal to 4pi

Answer 1

First, we'll find the intervals of x and y. As $0\leq\theta\leq4\pi\iff 0\leq\dfrac{\theta}{2}\leq2\pi$, the angle of x and y makes a complete turn. Hence, $0\leq x\leq1$ and $0\leq y\leq1$.

You must remember the identity: $\sin^2\alpha+\cos^2\alpha=1$. Then:

$x^2+y^2=\sin^2\left(\dfrac{\theta}{2}\right)+\cos^2\left(\dfrac{\theta}{2}\right)=1\\\\\Longrightarrow \boxed{x^2+y^2=1}~\text{with}~0\leq x\leq1~\text{and}~0\leq y\leq1~$