Editing Hyperbolic triangle (section)

==Trigonometry==
In all the formulas stated below the sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} must be measured in [[Hyperbolic geometry#Standardized Gaussian curvature|absolute length]], a unit so that the [[Gaussian curvature]] {{mvar|K}} of the plane is −1. In other words, the quantity {{mvar|R}} in the paragraph above is supposed to be equal to 1.

Trigonometric formulas for hyperbolic triangles depend on the [[hyperbolic function]]s sinh, cosh, and tanh.

===Trigonometry of right triangles===
If ''C'' is a [[right angle]] then:
*The '''sine''' of angle ''A'' is the '''hyperbolic sine''' of the side opposite the angle divided by the '''hyperbolic sine''' of the [[hypotenuse]].
::<math>\sin A=\frac{\textrm{sinh(opposite)}}{\textrm{sinh(hypotenuse)}}=\frac{\sinh a}{\,\sinh c\,}.\,</math>

*The '''cosine''' of angle ''A'' is the '''hyperbolic tangent''' of the adjacent leg divided by the '''hyperbolic tangent''' of the hypotenuse.
::<math>\cos A=\frac{\textrm{tanh(adjacent)}}{\textrm{tanh(hypotenuse)}}=\frac{\tanh b}{\,\tanh c\,}.\,</math>

*The '''tangent''' of angle ''A'' is the '''hyperbolic tangent''' of the opposite leg divided by the '''hyperbolic sine''' of the adjacent leg.
::<math>\tan A=\frac{\textrm{tanh(opposite)}}{\textrm{sinh(adjacent)}} = \frac{\tanh a}{\,\sinh b\,}</math>.

*The '''hyperbolic cosine''' of the adjacent leg to angle A is the '''cosine''' of angle B divided by the '''sine''' of angle A.
::<math>\textrm{cosh(adjacent)}= \frac{\cos B}{\sin A}</math>.

*The '''hyperbolic cosine''' of the hypotenuse is the product of the '''hyperbolic cosines ''' of the legs.
::<math>\textrm{cosh(hypotenuse)}= \textrm{cosh(adjacent)} \textrm{cosh(opposite)}</math>.

*The '''hyperbolic cosine''' of the hypotenuse is also the product of the '''cosines''' of the angles divided by the product of their '''sines'''.<ref>{{cite book|last1=Martin|first1=George E.|title=The foundations of geometry and the non-Euclidean plane|url=https://archive.org/details/foundationsofgeo0000mart|url-access=registration|date=1998|publisher=Springer|location=New York, NY|isbn=0-387-90694-0|page=[https://archive.org/details/foundationsofgeo0000mart/page/433 433]|edition=Corrected 4. print.}}</ref>
::<math>\textrm{cosh(hypotenuse)}= \frac{\cos A \cos B}{\sin A\sin B} = \cot A \cot B</math>

====Relations between angles====
We also have the following equations:<ref>{{cite book|last1=Smogorzhevski|first1=A.S.|title=Lobachevskian geometry|publisher=Mir Publishers|location=Moscow 1982|page=63}}</ref>

:<math> \cos A = \cosh a \sin B</math>

:<math> \sin A = \frac{\cos B}{\cosh b}</math>

:<math> \tan A = \frac{\cot B}{\cosh c}</math>

:<math> \cos B = \cosh b \sin A</math>

:<math> \cosh c = \cot A \cot B</math>

====Area====
The area of a right angled triangle is:
:<math>\textrm{Area} = \frac{\pi}{2} - \angle A - \angle B</math>
also
:<math>\textrm{Area}= 2 \arctan (\tanh (\frac{a}{2})\tanh (\frac{b}{2}) )</math>{{citation needed|date=October 2015}}<ref>{{cite web|title=Area of a right angled hyperbolic triangle as function of side lengths|url=https://math.stackexchange.com/q/1462778 |website=[[Stack Exchange]] Mathematics|accessdate=11 October 2015}}</ref>
The area for any other triangle is:
:<math>\textrm{Area} = {\pi} - \angle A - \angle B - \angle C</math>

====Angle of parallelism====
The instance of an [[omega triangle]] with a right angle provides the configuration to examine the [[angle of parallelism]] in the triangle.

In this case angle ''B'' = 0, a = c = <math> \infty </math> and <math>\textrm{tanh}(\infty )= 1</math>, resulting in <math>\cos A= \textrm{tanh(adjacent)}</math>.

====Equilateral triangle====
The trigonometry formulas of right triangles also give the relations between the sides ''s'' and the angles ''A'' of an [[equilateral triangle]] (a triangle where all sides have the same length and all angles are equal).

The relations are:

:<math>\cos A= \frac{\textrm{tanh}(\frac12 s) }{\textrm{tanh} (s)}</math>

:<math>\cosh( \frac12 s)= \frac{\cos(\frac12 A)}{\sin( A)}= \frac{1}{2 \sin(\frac12 A)}</math>

===General trigonometry===
Whether ''C'' is a right angle or not, the following relationships hold:
The [[hyperbolic law of cosines]] is as follows:
:<math>\cosh c=\cosh a\cosh b-\sinh a\sinh b \cos C,</math>

Its [[duality (projective geometry)|dual theorem]] is
:<math>\cos C= -\cos A\cos B+\sin A\sin B \cosh c,</math>

There is also a ''law of sines'':
:<math>\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c},</math>

and a four-parts formula:
:<math>\cos C\cosh a=\sinh a\coth b-\sin C\cot B</math>
which is derived in the same way as the [[Spherical_trigonometry#Cotangent_four-part_formulae|analogous formula in spherical trigonometry]].

<!--- still in development
====Solving Hyperbolic triangles====
see also [[Solving triangles]]
*'''''Angle - Angle - Angle''''' use the dual form of the hyperbolic law of cosines
*'''''Angle - Angle - Side''''' use hyperbolic law of sines to get to Angle - Angle - Side -side
*'''''Angle - Angle - Side -side ''''' use the four-parts formula
*'''''Angle - Side - Angle'''''
*'''''Angle - Side - side''''' use hyperbolic law of sines to get to Angle - Angle - Side -side
*'''''Side - Angle - Side'''''
*'''''Side - Side - Side''''' use the hyperbolic law of cosines
end of still in development ---->