How to find endpoint? How do you find the coordinates of endpoint of arc (pi + s) when the endpoint of arc s on the unit circle is (-sqrt (21)/5, 2/5)?
 A: For your particular problem, with $\pi$, there is an almost no work way of doing things.  You want to travel counterclockwise from $(1,0)$ to $(-\sqrt{21}/5, 2/5)$, and then travel through a further angle $\pi$, which is halfway around the circle.  
You end up at the point which is $(-\sqrt{21}/5,2/5)$ reflected through a point mirror at the origin.  This changes the sign of both the $x$ coordinate and the $y$-coordinate, so you end up at $(\sqrt{21}/5,-2/5)$.
If a point mirror at the origin sounds unintuitive to you, you can think instead of $(-\sqrt{21}/5,2/5)$  rotated about the origin (counterclockwise, though with $\pi$ it doesn't matter) through an angle $\pi$. That takes you from the second quadrant to the fourth quadrant, but it is clear that the absolute values of the $x$-coordinate and $y$-coordinate don't change, so you end up at $(\sqrt{21}/5,-2/5)$.  
More generally, suppose that you have a circle of radius $1$, and starting at $(1,0)$ you travel through an angle $\alpha$ (counterclockwise) and then through a further angle $\beta$.
Then you end up at the point with coordinates
$$(\cos(\alpha+\beta),\sin(\alpha+\beta))$$
Now use the identities
$$\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$$
$$\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)$$
In your particular example, $\cos(\alpha)=-\sqrt{21}/5$ and $\sin(\alpha)=2/5$,
and $\beta=\pi$ so $\cos(\beta)=-1$ and $\sin(\beta)=0$.
So we find using the identities above that $\cos(\alpha+\beta)=\sqrt{21}/5$ and $\sin(\alpha+\beta)=-2/5$.  
So that's another way of finding the answer, overkill for a simple angle like $\beta=\pi$, but potentially useful in more complicated situations.  For instance, you might want to compute where you end up if instead of $\beta=\pi$ you have $\beta=\pi/4$.
A: I'm not sure what arc $s$ and arc $(\pi+s)$ mean, but maybe it means find the point on the unit circle you get to if you start at $(-\sqrt{21}/5,2/5)$ and go around a further $\pi$ radians, in which case, can you see that going $\pi$ radians from any point $(x,y)$ on the unit circle gets you to the point on the circle diametrically opposite to $(x,y)$? and, do you know what that point would be?
