Find catesian coordinate of T-point $P(-\frac{65\pi}{2}) $

Question

Find the Cartesian coordinates of T-point $P(-\frac{65\pi}{2}) $.

It is easy when there is no negative sign. I don' t know how to do with negative sign.

Answer 1

Since we can go around the unit circle as many times as we want without changing the values of the sine and the cosine, we can add $c\cdot2\pi$ for some $c\in\mathbb{Z}$:

$\theta=-\frac{65\pi}{2}=-\frac{65\pi}{2}+17\cdot2\pi=\frac{3}{2}\pi$.

Now find the cosine and sine of the angle $\frac32\pi$.

$\cos\big(\frac32\pi\big)=0$, $\sin\big(\frac32\pi\big)=-1$, so the coordinate pair is $(0,-1)$.