Spherical Trigonometry

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In astronomy we must have some basics about the spherical trigonometry, as we know that all heavenly  bodies we deal in astronomy are spherical, like, The Sun, The Moon, Stars, Earth & amp; planets etc.


Spherical trigonometry is a branch of spherical geometry which deals with polygons (especially triangles) on thesphere and the relationships between the sides and the angles. This is of great importance for calculations in astronomyand earth-surface, orbital and space navigation.


Spherical trigonometry was studied by early Greek mathematicians such as Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem.



Spherical triangles satisfy a spherical law of cosines

\cos c= \cos a \cos b + \sin a \sin b \cos C. \!
The identity may be derived by considering the triangles formed by the tangent lines to the spherical triangle subtending angle C and using the plane law of cosines. Moreover, it reduces to the plane law in the small area limit.


They also satisfy an analogue of the law of sines

\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}.