Spherical Trigonometry - Basic formulas

Spherical triangle - Sides and Angles

Take a sphere with radius = 1 centered at origin. The intersection of the sphere and a plane which passes through the center point of the sphere is called a great circle. An area on the sphere with radius one, bounded by three arcs of great circles is called a spherical triangle.

The sides of the spherical triangle ABC are a, b and c.

The measure of the side a, b and c are respectively the lengths of the arcs BC, CA and AB.

The angle A of the spherical triangle ABC is the angle between the tangent lines to the sides AC and AB in point A. The angles A, B and C are usually expressed in radians.

Cosine rule for spherical triangles

Relationship between the three sides and an angle. One can prove that 

 
cos a = cos b  cos c + sin b  sin c cos A   (1)
cos b = cos c  cos a + sin c  sin a cos B   (2)
cos c = cos a  cos b + sin a  sin b cos C   (3)

The polar triangle of a spherical triangle

Each circle on the sphere has two poles.

The great circle which contains B and C has two poles. Let A1 be the pole which is together with A in the same hemisphere. Define B1 and C1 analogously. The spherical triangle A1B1C1 is called the polar triangle of the spherical triangle ABC.

One can prove that each side of one of the triangles and the corresponding angle of the other triangle are supplementary. 

 
a + A1 = b + B1 = c + C1 = a1 + A = b1 + B = c1 + C = pi.     (4)

Relationship between the three angles and an side.

Apply the formula (1) to the polar triangle A1B1C1. Then 
 
cos a1 = cos b1  cos c1 + sin b1  sin c1 cos A1
By (4) and after simplification, we obtain 
 
cos A = - cos B cos C + sin B sin C cos a
We obtain a similar form with formulas (2) and (3). 

 
cos A = - cos B cos C + sin B sin C cos a
cos B = - cos C cos A + sin C sin A cos b
cos C = - cos A cos B + sin A sin B cos c

Sine rule for spherical triangles

We start from the formula cos a = cos b cos c + sin b sin c cos A. It follows 
 
          cos a  - cos b  cos c
cos A = ------------------------
             sin b  sin c

thus


sin2 A = 1- cos2 A

          sin2 b sin2 c - ( cos a  - cos b  cos c )2
        = ------------------------------------------------
                      sin2 b  sin2 c

          (1-cos2 b)(1-cos2 c) - ( cos a  - cos b  cos c )2
       = --------------------------------------------------------
                      sin2 b  sin2 c

          1 - cos2 a - cos2 b - cos2 c + 2 cos a cos b cos c
       = -------------------------------------------------------
                      sin2 b  sin2 c

sin2 A      1 - cos2 a - cos2 b - cos2 c + 2 cos a cos b cos c
--------- = --------------------------------------------------------
sin2 a             sin2 a  sin2 b  sin2 c
The right hand side is symmetrical to a, b and c. So, we have 
 
   sin2 A     sin2 B      sin2 C
  --------- = ---------  = ----------
   sin2 a     sin2 b      sin2 c
The values of the sides and the angles of a spherical triangle are between 0 and pi. Each sine is positive. So, we have 

 
   sin A       sin B         sin C
  --------- = ---------  = ----------
   sin a       sin b         sin c



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