- Spherical triangle - Sides and Angles

- Cosine rule for spherical triangles

- The polar triangle of a spherical triangle

- Relationship between the three angles and an side.

- Sine rule for spherical triangles

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.

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 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 A_{1}B_{1}C_{1} 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 + A_{1}= b + B_{1}= c + C_{1}= a_{1}+ A = b_{1}+ B = c_{1}+ C = pi. (4)

cos aBy (4) and after simplification, we obtain_{1}= cos b_{1}cos c_{1}+ sin b_{1}sin c_{1}cos A_{1}

cos A = - cos B cos C + sin B sin C cos aWe 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 |

cos a - cos b cos c cos A = ------------------------ sin b sin c thus sinThe right hand side is symmetrical to a, b and c. So, we have^{2}A = 1- cos^{2}A sin^{2}b sin^{2}c - ( cos a - cos b cos c )^{2}= ------------------------------------------------ sin^{2}b sin^{2}c (1-cos^{2}b)(1-cos^{2}c) - ( cos a - cos b cos c )^{2}= -------------------------------------------------------- sin^{2}b sin^{2}c 1 - cos^{2}a - cos^{2}b - cos^{2}c + 2 cos a cos b cos c = ------------------------------------------------------- sin^{2}b sin^{2}c sin^{2}A 1 - cos^{2}a - cos^{2}b - cos^{2}c + 2 cos a cos b cos c --------- = -------------------------------------------------------- sin^{2}a sin^{2}a sin^{2}b sin^{2}c

sinThe values of the sides and the angles of a spherical triangle are between 0 and pi. Each sine is positive. So, we have^{2}A sin^{2}B sin^{2}C --------- = --------- = ---------- sin^{2}a sin^{2}b sin^{2}c

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

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