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