# Using the spherical law of cosines

Compute angular length c of the great-circle route between these two cities:

Daytona Beach (location A) 29º12'N., 81º1' W.

Sidi Ifni (location B) 29º23' N. 10º10' W.

Ok so I converted the latitudes and longitudes and I now have:

 Daytona Beach (location A) 29.20º N, 81.02º W

Sidi Ifni (location B) 29.38º N, 10.16º W

cosN = .18º


After using the law of cosines:

cosc = cos(81.02º)cos(10.16º) + sin(81.02º)sin(10.16º)cos(.18º)

   = .3279


arccos(.3279) = 70.86º = c

Am I on the right track?

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 In your work, I think you have the values for latitude (north/south) and longitude (east/west) swapped. I'd expect $\cos n=$ $$\cos(90°-29.20°)\cos(90°-29.38°)+\sin(90°-29.20°)\sin(90°-29.38°)\cos(81.02°-1‌​0.16°)$$ – Isaac Mar 31 '11 at 4:12 Thanks. I was wondering why I was getting 4900 instead of 4201 when I tried to calculate the distance. – user8640 Mar 31 '11 at 5:41

If $\triangle ABC$ is a spherical triangle with arcs $a$, $b$, and $c$ (meaning the measures of the arcs, not the lengths), then $\cos c=\cos a\cos b+\sin a\sin b\cos C$.
Let $A$ and $B$ be as you defined them. $N$ and $S$ are the north and south poles, respectively; $C$ and $D$ are the points on the equator that are on the same line of longitude as $A$ and $B$, respectively. Consider spherical $\triangle ABN$. $a=(90°-\text{latitude of point }B)$; $b=(90°-\text{latitude of point }A)$. $N=\text{positive difference in longitude between points }A\text{ and }B$. Use the Spherical Law of Cosines ($\cos n=\cdots$ form) to determine $n$, which is the shortest arc between the two points.