Converting $x = \sin \frac{t}{2}, y = \cos \frac{t}{2}$ to Cartesian form

Question

How can we transform these parametric equations to Cartesian form?

$$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$$

Answer 1

If $-\pi\leq t\leq \pi$ then $-\pi/2\leq t/2\leq \pi/2$. Also $x^2+y^2=1$.

Here is the animated curve for $0\leq t\leq \pi$. Try to imagine what happens for $t$ negative.

Answer 2

$$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$$

$$x^2+y^2=(\sin \frac{t}{2})^2+(\cos \frac{t}{2})^2=1$$ so $$x^2+y^2=1$$ is equation of some circle