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Spherical Law of Cosines
From: Dan Allen
Date: 2002 Oct 18, 10:05 -0700

```On Friday, October 18, 2002, at 07:53 AM, David Weilacher wrote:

> Please define little (a), (b), (c), and (ab), for me.

With pleasure!

cos(c) = cos(a) * cos(b) + sin(a) * sin(b) * cos(ab)

a is the length of the first side of the spherical triangle.
b is the length of the second side of the spherical triangle.
ab is the angle between the two sides a and b.
c is the length of the side opposite the angle ab.

Here is an example of a great circle distance between
San Francisco (SF) and Salt Lake City (SLC).  All angles
and lengths are expressed in degrees for this example,
and North and West are positive.

SF:
lat1 = 37 degrees
lon1 = 122 degrees

SLC:
lat2 = 40 degrees
lon2 = 112 degrees

So,

a is the co-latitude of lat1, or 90-lat1 or 53 degrees.
b is the co-latitude of lat2, or 90-lat2 or 50 degrees.
ab is the difference of the longitudes, or lon1-lon2 or 10 degrees.

Solving for c one learns the distance in degrees between SF and SLC,
which is about 8.375 degrees.  To get a distance we understand,
multiply degrees times 60 nmi per deg and you get 502.5 nmi.

One then can re-substitute back in and use the formula again to
determine the initial great circle course.  In this case we
use a as co-latitude of lat1, b is the distance we just
computed (8.375 degrees -- make sure to use degrees), c is
the co-latitude of lat2, and the angle ab is the initial
course.  We have rotated the whole spherical triangle around
to put the sides a,b, and c where we know their values,
leaving ab to be solved for.  Substituting and solving one
determines that the initial course is 65.9588 degrees.

---

In sight reduction:

a is the co-latitude of the assumed position
b is the declination of the body (say the sun)
ab is the hour angle of the body
c is the altitude of the body

---

Note too that there are equivalent ways of writing the spherical
law of cosines.  One can write it as

cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)

where the pairs of sin/cos of a/b are switched.  This allows one to do
great circle calculations using latitudes directly,
without using co-latitudes.  The origin is moved from the pole to
the equator, so to speak.  This form is often handier but the first
version is the canonical version.

Hope this helps.

Dan

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