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