Import standard modules:


In [ ]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML 
HTML('../style/course.css') #apply general CSS

Import section specific modules:


In [ ]:
pass

2.13 Spherical Trigonometry

Most people have a basic understanding of planar trigonometry. In this section we explore how trigonometry can be extended into the spherical realm. Spherical trigonometry is a branch of spherical geometry in which we study the relationship between the sides and angles of spherical triangles. Spherical triangles are formed by the pairwise intersection of three great circlular arcs in three vertices. A great circular arc is an arc segment of a great circle. A great circle is formed by the intersection of a sphere and a plane that passes through the center of the sphere. It is possible to derive the following basic spherical trigonometric identities by studying arbitrary spherical triangles that are located on the unit sphere:

Spherical Trigonometric Identities

Spherical cosine rule: $\cos b = \cos a \cos c + \sin a \sin c \cos B$

Spherical sine rule: $\sin b \sin A = \sin B \sin a$

Five part rule: $\sin b \cos A = \cos a \sin c - \sin a\cos c\cos B$

The first two rules are analogous to the planar sine and cosine rule. The sides and angles used in the above expressions are graphically depicted in Fig. 2.13.1 ⤵ .

Advice: We use spherical trigonometry to convert between different astronomical coordinate systems (see [Appendix ➞](../0_Introduction/2_Appendix.ipynb)).

Figure 2.13.1: The spherical triangle $ABC$.