How can we transform these parametric equations to Cartesian form?
$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$
How can we transform these parametric equations to Cartesian form?
$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$
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.
$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$
$x^2+y^2=(\sin \frac{t}{2})^2+(\cos \frac{t}{2})^2=1$ so $x^2+y^2=1$ is equation of some circle