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    The Cosine Theorem
    From: Herbert Prinz
    Date: 2002 Oct 19, 12:48 +0000

    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
    "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@AOL.COM 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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