# Finding the length of the sides, the measure of the angles and area of spherical triangles?

I'm trying to understand this problem in the textbook but I got lost in one part:

Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the angles and the area of the spherical triangle with vertices $A(70° N, 10° E), B(10° S, 100° E)$ and $C(50° S,80° W)$. The earth's radius will be used as the unit of length. The spherical coordinates $(r,v,u)$ of the three vertices are $(1,10,20),(1,100,100)$ and $(1,-80,140)$.

Their cartesian coordinates are:

\begin{align*} (\sin(20°)\cos(10°), \sin(20°)\sin(10°),\cos(20°))&=(0.3368,0.0594,0.9397)\\ (\sin(100°)\cos(100°), \sin(100°)\sin(100°),\cos(100°))&=(-0.1710,0.9698,-.1736)\\ (\sin(140°)\cos(-80°), \sin(140°)\sin(-80°),\cos(140°))&=(0.1116,-0.6330,-.7660) \end{align*}

THIS IS WHERE I BEGIN TO HAVE PROBLEMS UNDERSTANDING:

The cosines of the angles between radii $OA$, $OB$ and $OC$ are equal to the dot product of the corresponding position vectors. Thus

Where did they get those numbers? As in numbers, I mean $-0.1631,-0.5000,-0.7198$

• I suppose the angles were first given in degrees; and I suppose that $O$ is the center of your sphere? Do you know the relation between the dot product of two vectors and the cosine of the angle between’em? Since the vectors all are unit vectors, the dot products are simply equal to the cosines. Feb 3, 2015 at 4:43