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