I'd like to describe all of the points on the Earth's surface that are exactly 4000 miles from the North Pole. I know that this will eventually give me an equation for a circle; I want to find that equation assuming that the Earth's radius is 3960 miles and that the center of the Earth is at $(0,0)$. I started by drawing some triangles: from the center of the Earth to the North Pole is one side, from the center to the surface is another, and from the North Pole to the surface is the third. The lengths of these sides are 3960, 3960, and 4000 respectively.
Using the law of cosines should let me find the size of each angle, but I'm not getting angles that add up to 180. I get approximately $60.9908^\circ$ for the angle opposite the 4000 side and about $59.6653^\circ$ for the other two angles, which adds up to $180.321^\circ$. I didn't do any rounding; is this just an issue with my calculator making some internal rounding errors? Usually copy-pasting the long decimals keeps everything working well through the end but this is a pretty significant error (at least in my mind).
When I try to find the radius (which I think should be equal to the altitude running from the surface to the side connecting the center and the North Pole), I get different values depending on which angles I use, which I don't really like. Is there any way to resolve this? Am I thinking about it in the wrong way?
Edit: It turns out that instead of getting 60.9908$^\circ$ I should have been getting 60.669$^\circ$, which resolves all of my problems. I hope the problem proves of some interest to anyone else!
Best Answer
When I do the law of cosines on a $3960-3960-4000$ triangle, I get the angle opposite $4000$ to be $\arccos 1-\frac{4000^2}{2\cdot 3960^2}\approx \arccos 0.48984798 \approx 60.669410^\circ$ and the angle opposite $3960$ to be $\arccos \frac {4000}{2 \cdot 3960} \approx \arccos 0.5050505051 \approx 59.665295^\circ$. The three angles add to $180^\circ$ within rounding error. You didn't show how you calculated the angles, so I can't guess where the error is.