User:WillemienH/Hyperbolic trigonometry

Draft page for Trigonometry for hyperbolic geometry

PS not hyperbolic trigonometry see there

Split of from Hyperbolic triangle

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In all the formulas stated below the sides $a$, $b$, and $c$ must be measured in a unit so that the Gaussian curvature $K$ of the plane is −1. In other words, $R$ is supposed to be equal to 1.

Trigonometric formulas for hyperbolic triangles depend on the hyperbolic functions sinh, cosh, and tanh.

Trigonometry of triangles with a right angle
If C is a right angle then:

Angle formulas

 * The sine of angle A is the ratio of the hyperbolic sine of the side opposite the angle to the hyperbolic sine of the hypotenuse.
 * $$\sin A=\frac{\textrm{sinh(opposite)}}{\textrm{sinh(hypotenuse)}}=\frac{\sinh a}{\,\sinh c\,}.\,$$


 * The cosine of angle A is the ratio of the hyperbolic tangent of the adjacent leg to the hyperbolic tangent of the hypotenuse.
 * $$\cos A=\frac{\textrm{tanh(adjacent)}}{\textrm{tanh(hypotenuse)}}=\frac{\tanh b}{\,\tanh c\,}.\,$$


 * The tangent of angle A is the ratio of the hyperbolic tangent of the opposite leg to the hyperbolic sine of the adjacent leg.
 * $$\tan A=\frac{\textrm{tanh(opposite)}}{\textrm{sinh(adjacent)}} = \frac{\tanh a}{\,\sinh b\,}.$$

Side formulas

 * The hyperbolic cosine of the hypotenuse is the product of hyperbolic cosine  of the adjacent leg and the hyperbolic cosine of the opposite leg.
 * $$\textrm{cosh(hypotenuse)}= \textrm{cosh(adjacent)} \textrm{cosh(opposite)}.$$


 * The hyperbolic cosine of the adjacent leg to angle A is the ratio of the cosine of angle B to the sine of angle A.


 * $$\textrm{cosh(adjacent)}= \frac{\cos B}{\sin A}.$$


 * The hyperbolic cosine of the hypotenuse is the ratio of the product of the cosines of the angles to the product of their sines.


 * $$\textrm{cosh(hypotenuse)}= \frac{\cos A \cos B}{\sin A\sin B}.$$

Relations between angles
We also have the following equations:


 * $$ \cos \alpha = \cosh a \sin \beta $$


 * $$ \cos \beta = \cosh b \sin \alpha $$


 * $$ \cosh c = \cot \alpha \cot \beta $$

Area
The area of a right angled triangle is:


 * $$\textrm{Area} = \frac{\pi}{2} - \angle A - \angle B $$

also


 * $$\textrm{Area}= 2 \arctan \left( \tanh (\frac{a}{2})\tanh (\frac{b}{2}) \right)$$

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 = $$ \infty $$ and $$\textrm{tanh}(\infty )= 1 $$, resulting in $$\cos A= \textrm{tanh(adjacent)}.$$

More formulas of triangles with a right angle
see martin page ??? about formulas including \Pi needs rewriting

General trigonometry
Whether C is a right angle or not, the following relationships hold: The hyperbolic law of cosines is as follows:


 * $$\cosh c=\cosh a\cosh b-\sinh a\sinh b \cos C,$$

By duality we also have the dual hyperbolic law of cosines theorem:


 * $$\cos C= -\cos A\cos B+\sin A\sin B \cosh c,$$

There is also a law of sines:


 * $$\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c},$$

and a four-parts formula:


 * $$\cos C\cosh a=\sinh a\coth b-\sin C\cot B.$$

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 sinus 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 sinus to get to  Angle - Angle - Side -side
 * Side - Angle - Side
 * Side - Side - Side use the  hyperbolic law of cosines

more

 * lambert quadrilateral
 * saccheri quadrilateral