Find parametric equations for the path a particle that moves along the circle
$$x^2+(y-1)^2=4.$$
In the manner describe
a) One around clockwise starting at $(2,1)$
b) Three times around counterclockwise starting at $(2,1)$
c) halfway around counterclockwise starting at $(0,3)$
The answers:
a) $y=1-2\sin t, x=2\cos t, 0 \leq t \leq 2\pi$
b) $x=2\cos t, y=2\sin t+1, 0 \leq t \leq 6\pi$
c) $x=2\cos t, y=2\sin t+1, \frac{\pi}{2} \leq t \leq \frac{3\pi}{2}$
I know why there is $\sin(t)$ and $\cos(t)$ but why when its move in clockwise the $\sin(t)$ will be with minus ?
Best Answer
HINT : Compare $$\left(\frac{x}{2}\right)^2+\left(\frac{y-1}{2}\right)^2=1$$ with $$\cos^2 t+\sin^2 t=1.$$
EDIT :
Drawing the circle will help. In your case, since one around clockwise, $y$-coordinate of the point has to be decreasing first. This is the reason why the $\sin t$ is with minus. Setting $t=0, \frac{\pi}{6},\frac{\pi}{4},\cdots$ will also help.