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Re: A triangle on the equator
From: Bill Lionheart
Date: 2025 Oct 7, 12:57 -0700
From: Bill Lionheart
Date: 2025 Oct 7, 12:57 -0700
Possibly this is obvious already to many of you. Folks probably know the spherical trig formulae
cos a = cos b cos c + sin b sin c cos A,
and its relatives. For the angular side lengths in radians this is approximately
1- a^2/2 = (1-b^2/2)(1 - c^2/2) + b c cos A
for a b and c small so this is only b^2c^2/4 from the plane trig cosine rule and in the right angle case
a^2 = b^2 +c^2 - b^2 c^2/2
to the same order of approximation.
A minute of arc is about 0.000290888 radians so on the scale of a few minutes the triangles are pretty close to plane triangles. Of course small right angle triangles will always fall a bit short of pythagoras in this way , a^2 > b^2 +c^2 and the angle sum will always be a bit more than 180 degrees.
Bill
