The cosine theorem on a sphere is an interesting relation that holds for a spherical triangle. One of it’s main applications is the computation of the air distance between two geographical points given by their latitude – longitude coordinates (I’ll show how in the next post).
Consider the spherical triangle (see figure). The cosine theorem express a relation between the angles and which I’ll prove with basic geometry knowledge.
First of all note that and have nothing to do with the problem, they are just an auxiliary construction used for the proof. So first look only at the sphere and ignore .
Then let’s consider the tangent to the arc in the point , and , the tangent to the arc in the point . Thus we’ll have and .
To prove the spherical cosine theorem the length of the segment can be written in two different ways: using cosine theorem in the triangle and using cosine theorem in the triangle .
Note that is a right triangle( being the right angle) because is a tangent to the sphere. Thus and . Note that . Take it slowly to understand it, and make sure you visualise correctly the 3D figure :). (*)
Same for triangle we get: and . (**)
From the cosine theorem in triangle we get:
Note that the angle is . From generalised cosine theorem in we get:
It follows that:
Replacing with the values computed in (*) and (**) leads to:
Now add the fractions on both sides of the equality by making them have the same denominator, and divide both sides by :
Because for every :
Then (***) become:
By dividing with we get:
that can be written in a final form as:
A remarkable thing is that when is a right angle, , and we get the Pithagora theorem on the sphere: