Spherical Geometry - Applications

TheoremSpherical Pythagorean Theorem

For a right spherical triangle (one angle is $\pi/2$ ) with legs $a$ , $b$ and hypotenuse $c$ (all measured as angles):

\cos c = \cos a \cos b

This generalizes the Euclidean Pythagorean theorem, reducing to it for small triangles where $\cos x \approx 1 - x^2/2$ .

Applications in celestial navigation use this and related formulas to compute positions from stellar observations. Modern GPS systems solve analogous spherical geometry problems to triangulate positions from satellite signals.