# Using the spherical law of cosines

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

• Daytona Beach (location A): $29^\circ12'\ N, 81^\circ1' \ W$.

• Sidi Ifni (location B): $29^\circ23' \ N. 10^\circ10' \ W$.

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

• Daytona Beach (location A): $29.20^\circ N, 81.02^\circ W$

• Sidi Ifni (location B): $29.38^\circ N, 10.16^\circ W$

$$\cos N = .18º$$

After using the law of cosines:

$$\cos c = \cos(81.02^\circ)\cos(10.16^\circ) + \sin(81.02^\circ)\sin(10.16^\circ)\cos(.18^\circ) = 0.3279$$ $$\arccos(0.3279) = 70.86^\circ = c$$

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.