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Solution of Triangles (Ambiguous Cases)

The three sides a, b, c and the three angles A, B, C are called the elements of the triangle ABC. When any three of these six elements (except all the three angles) of a triangle are given, the triangle is known completely; that is, the other three elements can be expressed in terms of the given elements and can be evaluated. This process is called the solution of triangles.
  • If the three sides a, b, c are given, angle A is obtained from tan70059.png = 70053.png or cos A = 70047.png. B and C can be obtained in the similar way.
  • If two sides b and c and the included angle A are given, then 70041.png = 70034.pngcot70027.png gives 70021.png Also 70015.png, so that B and C can be evaluated. The third side is given by a = 70007.png or a2 = b2 + c2 – 2bc cos A.
  • If two sides b and c and the angle B (opposite to side bare given, then sin C = 70001.png, A = 180° – (B + C) and a = 69995.png give the remaining elements.

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