Hyperbolic Relations Math

Hyperbolic cotangent is a function c o t h.

Hyperbolic relations math. In mathematics hyperbolic functions are analogues of the ordinary trigonometric functions defined for the hyperbola rather than on the circle. For example two points uniquely define a. Q a for people studying math at any level and professionals in related fields stack exchange network stack exchange network consists of 176 q a communities including stack overflow the largest most trusted online community for developers to learn share their knowledge and build their careers. Just as the points form a circle with a unit radius the points form the right half of the equilateral hyperbola.

As we can see hyperbolic cotangent is an odd function meaning c o t h x c o t h x. The hyperbolic trigonometric functions extend the notion of the parametric equations for a unit circle. Tanh x kπi tanh x coth x kπi coth x. In mathematics hyperbolic geometry.

X cosh a dfrac e a e a 2 quad y sinh a dfrac e a e a 2. In the following k is any integer. Single lines in hyperbolic geometry have exactly the same properties as single straight lines in euclidean geometry. R 0 1 1 defined with.

Y sin t y sin t y sint to the parametric equations for a hyperbola which yield the following two fundamental hyperbolic equations. For please refer to the above negative angle formulas and make appropriate sign adjustments. X cosha 2ea e a. Hyperbolic definitions sinh x e x e x 2 csch x 1 sinh x 2 e x e x cosh x e x e x 2 sech x 1 cosh x 2 e x e x tanh x sinh x cosh x e x e x e x e x coth x 1 tanh x e x e x e x e x cosh 2 x sinh 2 x 1 tanh 2 x sech 2 x 1 coth 2 x csch 2 x 1 inverse hyperbolic defintions.

Sinh x 2kπi sinh x csch x 2kπi csch x. Hyperbolic functions also satisfy identities analogous to those of the ordinary trigonometric functions and have important physical applications. X cos t. Periodicity of hyperbolic functions.

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. For example the hyperbolic cosine function may be used to describe the shape of the curve formed by a high voltage line suspended between two towers see catenary. They also occur in the solutions of many linear differential equations cubic equations and laplace s equation in cartesian coordinates. Further because of the angle of parallelism hyperbolic geometry has an absolute scale a relation between distance and angle measurements.

X cosh a e a e a 2 y sinh a e a e a 2. Relationship between inverse hyperbolic and inverse trigonometric functions. X cos t x cost and. C o t h x c o s h x s i n h x e x e x e x e x.

Derivate of c o t h x is. Browse all wolfram community wolfram language demonstrations connected devices basic relations between hyperbolic functions. Definitions of hyperbolic functions and inverse hyperbolic functions links to the plots of hyperbolic inverse hyperbolic functions their basic relations formulas series expansions and their inter relations with trigonometric inverse trigonometric fun.