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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

1 Answer 1

up vote 1 down vote accepted

Below is the Spherical Law of Cosines as it appears in UCSMP Functions, Statistics, and Trigonometry, 3rd ed., copied here because the diagram is good and helps with clarity.

Spherical Law of Cosines (from UCSMP Functions, Statistics, and Trigonometry, 3rd ed.)

If $\triangle ABC$ is a spherical triangle with arcs $a$, $b$, and $c$ (meaning the measures of the arcs, not the lengths), then $\cos c=\cos a\cos b+\sin a\sin b\cos C$.

Now, to the specific problem at hand. Let's use the diagram below, also from UCSMP Functions, Statistics, and Trigonometry, 3rd ed., for reference.

globe (from UCSMP Functions, Statistics, and Trigonometry, 3rd ed.)

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.

(graphics from Lesson 5-10 of UCSMP Functions, Statistics, and Trigonometry, 3rd ed., © 2010 Wright Group/McGraw Hill)

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