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If from an hyperbolic triangle $ \triangle ABC$ the angels $\angle A$, $\angle B$ and side $c$ are given. (ASA triangle, a side and both adjacent angles are given)

How can I calculate the remaining angles and sides? (or at least one of them)

In Euclidean geometry you start with calculating $\angle C$ by substracting $\angle A$ and $\angle B$ from a straight angle, but in hyperbolic geometry that obviously doesn't work.

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You can use the cosine rule for hyperbolic triangles:
$\sin A \sin B \cosh c = \cos C + \cos A \cos B$ or
$\cos C = \sin A \sin B \cosh c - \cos A \cos B $

and then use the law of sines to find the other two sides:
$\frac{\sin C}{\sinh c}=\frac{\sin B}{\sinh b}=\frac{\sin A}{\sinh a}$

Sources:
http://en.wikipedia.org/wiki/Hyperbolic_triangle
http://www.maths.gla.ac.uk/wws/cabripages/hyperbolic/hypertrig.html

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