triangle solving
Let us consider skewangled triangles^{}. If one knows three parts of a triangle, among which at least one side, then the other parts may be calculated by using the law of sines and the law of cosines. We distinguish four cases:

1.
ASA. Known two angles and one side, e.g. $\alpha $, $\beta $, $c$. Other parts:
$$\gamma ={180}^{\circ}(\alpha +\beta ),a=\frac{c\mathrm{sin}\alpha}{\mathrm{sin}\gamma},b=\frac{c\mathrm{sin}\beta}{\mathrm{sin}\gamma}$$ 
2.
SSS. Known all sides $a$, $b$, $c$. The angles are obtained from
$$\mathrm{cos}\alpha =\frac{{b}^{2}+{c}^{2}{a}^{2}}{2bc},\mathrm{cos}\beta =\frac{{c}^{2}+{a}^{2}{b}^{2}}{2ca},\mathrm{cos}\gamma =\frac{{a}^{2}+{b}^{2}{c}^{2}}{2ab}.$$ 
3.
SAS. Known two sides and the angle between them, e.g. $b$, $c$, $\alpha $. Other parts from
$${a}^{2}={b}^{2}+{c}^{2}2bc\mathrm{cos}\alpha ,\mathrm{sin}\beta =\frac{b\mathrm{sin}\alpha}{a},\mathrm{sin}\gamma =\frac{c\mathrm{sin}\alpha}{a}$$ 
4.
SSA. Known two sides and the angle of one of them, e.g. $a$, $b$, $\alpha $. Other parts are gotten from
$$\mathrm{sin}\beta =\frac{b\mathrm{sin}\alpha}{a},\gamma ={180}^{\circ}(\alpha +\beta ),c=\frac{a\mathrm{sin}\gamma}{\mathrm{sin}\alpha}.$$ Since the SSA criterion alone does not prove congruence^{}, it is not surprising that there may not always be a single solution for $\beta $ here. In fact, if the first equation gives $\mathrm{sin}\beta >1$, then the situation is impossible and the triangle does not exist. If the equation gives $$, one gets two distinct values of $\beta $; an acute ${\beta}_{1}$ and an obtuse ${\beta}_{2}={180}^{\circ}{\beta}_{1}$. If in this case ${\beta}_{1}>\alpha $, then there are two different triangles as , but if ${\beta}_{1}\le \alpha $, then there is only one triangle.

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http://svn.goldsaucer.org/repos/PlanetMath/TriangleSolving/triangle.mpMetaPost source code for the above diagrams
Title  triangle solving 

Canonical name  TriangleSolving 
Date of creation  20130322 15:44:02 
Last modified on  20130322 15:44:02 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  11 
Author  stevecheng (10074) 
Entry type  Definition 
Classification  msc 5100 
Related topic  Congruence 