# Spherical Trigonometry question

• dpatte
In summary, in order to use spherical trigonometry to solve a problem, you need to know at least three out of the six parts (three sides, three vertex angles) of the spherical triangle. From this information, you can use the law of sines and the law of cosines to find the remaining three parts. In this case, if you already know the angles A, a, B, and b, you can use the law of cosines for angles to find the value of angle C, and then use the law of sines to find the value of side c.
dpatte
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

Last edited:
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. :(

you have two unknowns (C and c), and two equations …

where's the difficulty?

dpatte said:
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.

## 1. What is spherical trigonometry?

Spherical trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles on the surface of a sphere.

## 2. What are the main applications of spherical trigonometry?

Spherical trigonometry is used in a variety of fields, including astronomy, navigation, geodesy, and geography. It is also used in the study of the Earth's shape and the measurement of distances and angles on the surface of the Earth.

## 3. How is spherical trigonometry different from planar trigonometry?

The main difference between spherical trigonometry and planar trigonometry is that spherical trigonometry deals with triangles on the surface of a sphere, while planar trigonometry deals with triangles on a flat plane. This means that the rules and formulas used in spherical trigonometry are different from those used in planar trigonometry.

## 4. What are the basic elements of a spherical triangle?

A spherical triangle has three sides, three angles, and three vertices. The sides of a spherical triangle are arcs of great circles on the surface of the sphere, and the angles between these sides are measured in radians.

## 5. How is the law of cosines applied in spherical trigonometry?

The law of cosines is used to find the length of a side of a spherical triangle when the lengths of the other two sides and the included angle are known. It can also be used to find the angle between two sides when the lengths of the sides are known. In spherical trigonometry, the law of cosines is modified to take into account the curvature of the sphere.

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