# Spherical Trigonometry question

Im trying to use spherical trig to solve a problem

In the standard method to compute the angles, I already have only the following four angles

A, a, B, b

but from these, how can I compute either C, or c?

Thanks

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## Answers and Replies

tiny-tim
Homework Helper
welcome to pf!

hi dpatte! welcome to pf!

i don't think there's any simple way to do this

if it was plane trigonometry, C would be 180° - (A + B), but in spherical trigonometry i think all you can do is use the cosine rule and the sine rule and try to solve simultaneously for C and c

the sin rule says sin c/sin C = sin b/sin B = sin a/sin A. Since I don't have C or c, i don't see how I can use it.

the cosine rule says cos c = cos a cos b + sin a sin b sin C. Since I don't have c or C, I don't see how to use it either. :(

tiny-tim
Homework Helper
you have two unknowns (C and c), and two equations …

where's the difficulty?

Im trying to use spherical trig to solve a problem

In the standard method to compute the angles, I already have only the following four angles

A, a, B, b

but from these, how can I compute either C, or c?

Thanks
You actually have MORE information than you need to solve the triangle. Every spherical triangle has six parts (three sides, three vertex angles) each measured in degrees. Knowing any THREE parts, you can find any of the remaining three. There are three sets of formulas that let you do this (1) the law of sines (2) the law of cosines for sides and (3) the law of cosines for angles, stated below.

(1) In any spherical triangle, the sines of the sides are proportional to the sines of the opposite angles.
(2) In any spherical triangle, the cosine of any side is equal to the product of the cosines of the other two sides plus the product of the sines of those sides times the cosine of their included angle.
(3) In any spherical triangle, the cosine of any angle is equal to minus the product of the cosines of the other two angles plus the product of the sines of those angles times the cosine of their included side.

Suppose the vertex angles are labeled A, B, and C and the opposite sides are labeled a, b, and c. If you know two sides and one opposing angle or two angles and one opposing side, you would use the law of sines to find the unknown opposite the known. For example, if you were given angle A and sides a and b, you would find angle B by solving the equation

sin(A)/sin(a) = sin(B)/sin(b). Then we approach your question: if you know angles A and B and sides a and b, how do you find angle C and side c ?

from (2)

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

from (3)
cos(C) = - cos(A) * cos(B) + sin(A)*sin(B)*cos(c)

Use the last equation above, substitute the right hand side for cos(C) in the next to last equation above. Collect the terms that involve cos(c) on the left hand side of this equation, all other terms on the right hand side. Factor out and solve for cos(c). Taking the inverse cosine, you have the value for c. Use the law of sines for A,a, c, C to find the value for C.

If you'd like to see it worked out, let me know.