All three sides of a spherical triangle
are arcs of great circles of the sphere.
For simplicity we consider the unit sphere
(r=1) and consider all measurements
to be in radians -- both the arc lengths
(a, b, c)
and the vertex angles
(A, B, C).
(Reminder; a right angle is
/2 radians.)
In this general case, in which the triangle is called oblique, none of the angles need be special.
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Sum of (vertex) angles <
A + B + C < 3
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Area formula area = ``spherical excess'' = A + B + C -
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Law of sines:
(sin a / sin A) =
(sin b / sin B) =
(sin c / sin C)
Law of cosines for sides:
cos c = cos a cos b + sin a sin b cos C
Law of cosines for angles:
cos C = -cos A cos B + sin A sin B cos c
A right spherical triangle has a
right angle vertex, i.e.,
one angle of measure /2.
For these formulae we name the right angle C,
and the other two angles are
A and B.
The formulae above still hold true but they
are simplified because of the special
value of C, since for
C=/2
we have
sin C = 1
and
cos C = 0.