# Solution of triangles

https://en.wikipedia.org/wiki/Solution_of_triangles

Solution of triangles ( Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.

## Solving plane triangles

A general form triangle has six main characteristics (see picture): three linear (side lengths a, b, c) and three angular (α, β, γ). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:  

• Three sides (SSS)
• Two sides and the included angle (SAS)
• Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length.
• A side and the two angles adjacent to it (ASA)
• A side, the angle opposite to it and an angle adjacent to it (AAS).

For all cases in the plane, at least one of the side lengths must be specified. If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution.

### Trigonomic relations

The standard method of solving the problem is to use fundamental relations.

Law of cosines
$a^{2}=b^{2}+c^{2}-2bc\cos \alpha$ $b^{2}=a^{2}+c^{2}-2ac\cos \beta$ $c^{2}=a^{2}+b^{2}-2ab\cos \gamma$ Law of sines
${\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}$ Sum of angles
$\alpha +\beta +\gamma =180^{\circ }$ Law of tangents
${\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.$ There are other (sometimes practically useful) universal relations: the law of cotangents and Mollweide's formula.