Solving of a right hyperbolic triangle

Question

Let us have a hyperbolic triangle, $\angle C = 90^{\circ}$, $\angle A$ and side $c$ are given.

How to compute other sides and angles?

I would especially like to know the answer for $\angle A = 30^{\circ}$ and for $\angle A =60^{\circ}$.

Answer 1

Consider the following triangle.

The law of sines is true in the following form:

$$\frac{\sin(A)}{\sinh(a)}= \frac{\sin(B)}{\sinh(b)}= \frac{\sin(C)}{\sinh(c)}. $$

If $C=90^{\circ}$ then $$\frac{\sin(A)}{\sinh(a)}= \frac{\sin(B)}{\sinh(b)}= \frac{1}{\sinh(c)}. $$

From here $$\sinh(a)=\sin(A)\sinh(c)$$

and

$$a=\operatorname{asinh}\left(\sin(A)\sinh(c)\right).$$

The law of sines is true in the following form:

$$\cosh(c)=\cosh(a)\cosh(b)-\sinh(a)\sinh(b)\cos(C).$$

If $C=90^{\circ}$ then $$\cosh(c)=\cosh(a)\cosh(b).$$

So,

$$\cosh(b)=\frac{\cosh(c)}{\cosh(a)}.$$ From here all the parts can be calculated.