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Subject: The Cosine Theorem
From: Herbert Prinz (hprinz@XXX.XXX)
Date: Sat Oct 19 2002 - 08:48:10 EDT
Hello Bill,
Regiomontanus stated the cosine theorem for the spherical triangle for
the first time in its general form in his "Five books about triangles of
each kind" in 1533. Algebraic notation was not yet invented, so the
theorem was expressed in plain language in form of a proportion. Not
only is the proof rather hard to follow, even the statement itself is
not easily recognizable as the law of cosines. Barnabas Hughes
translates it in his book "Regiomontanus On Triangles", 1967, as
follows:
"In every spherical triangle that is constructed from the arcs of great
circles, the ratio of the versed sine of any angle to the difference of
two versed sines, of which one is [the versed sine] of the side
subtending this angle while the other is [the versed sine] of the
difference of the two arcs including this angle, is as the ratio of the
square of the whole right sine to the rectangular product of the sines
of the arcs placed around the mentioned angle."
Since then we have come a long way. Using algebra, the cosine theorem
for the spherical triangle is a trivial consequence of the equality of
the scalar product of two unit vectors with the cosine of the angle
between them. Convert two positions on the unit sphere given in
spherical coordinates into rectangular coordinates of the corresponding
position vectors (sometimes called "direction cosines"), multiply the
vectors, and you are done. It's actually much more work to write it all
down than to "see" it:
Without sacrificing general validity, we may assume that one point is at
longitude 0, because we can rotate them both together without changing
their relative position. In that case, Lon_b represents the difference
in longitude between the two points, i.e. their hour angle. We have
a = (cos Lat_a, 0, sin Lat_a)
b = (cos Lat_b * cos Lon_b, cos Lat_b * sin Lon_b, sin Lat_b)
and therefore
cos angle(a,b) = a . b = cos Lat_a * cos Lat_b * cos Lon_b + 0 +
sin Lat_a * sin Lat_b
That's really all.
The beauty of Smart's purely geometric derivation is that it shows how
all "spherical" trigonometry is in fact plane trigonometry in disguise.
The law of cosines for spherical triangles is actually a formula about
the relation of four angles in an irregular tetrahedron (see fig. 3 in
his textbook). Three of those angles are between the three edges of the
tetrahedron meeting in the center of the sphere and one is at the corner
where the tangential plane touches the sphere. One can completely leave
the sphere out of the game and still proof the law of cosines. This is
one possible key to understanding why "spherical" trigonometry works for
celestial navigation on a non-spherical Earth. It works, because the
navigational triangle is about the mutual angles between three
verticals, and not about distances on the surface of the Earth.
WSMurdoch@XXX.XXX wrote:
> Chapter 1 section 5 derives the cosine formula from simple plane trig.
> I have used it in USPS JN courses to answer the question, "Where did
> that formula come from?" It is not too hard to pick your way through.
>
> Bill Murdoch
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