In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions defined for the hyperbola rather than on the circle: just as the points form a circle with a unit radius, the points form the right half of the equilateral hyperbola.

1. Topic : Hyperbolic
Functions
PRESENTER NAME :
OMER SHAIKH
SUBJECT : MATH'S A

2. Introduction
Definition of
Hyperbolic Function
Discovery of
Hyperbolic
Basic function of
Hyperbolic
Hyperbolic
Graph of Hyperbolic
Curves

3. What is hyperbolic function
The hyperbolic functions are analogs of the circular
function or the trigonometric functions. The hyperbolic
function occurs in the solutions of linear differential
equations, calculation of distance and angles in the
hyperbolic geometry, Laplace’s equations in the
cartesian coordinates. Generally, the hyperbolic
function takes place in the real argument called the
hyperbolic angle.

4. Hyperbolic Functions
Vincenzo Riccati (1707 -
1775) is given credit for
introducing
the hyperbolic functions.
Hyperbolic functions are
very useful in both
mathematics and physics.
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5.  The hyperbolic trigonometric functions extend the notion of the parametric
equations for a unit circle [x=cos t and y=sin t ] to the parametric equations for a
hyperbola.
 cosh2 (x) – sinh2 (x) = 1
 tanh2 (x) + sech2 (x) = 1
 coth2 (x) – cosech2 (x) = 1
 hyperbolic sine "sinh"
 hyperbolic cosine "cosh"
 hyperbolic tangent "tanh“
 (hyperbolic cosecant "csch" or "cosech“
 hyperbolic secant "sech"
 hyperbolic cotangent "coth"
Basic functions of Hyperbolic function

6. Graph

7. Hyperbolic functions show up in
many real-life situations. For
example, they are related to the
curve one traces out when chasing
an object that is moving linearly.
They also define the shape of a chain
being held by its endpoints and are
used to design arches that will
provide stability to structures. This
shape, defined as the graph of the
functions also referred to as a
catenary.
hyperbolas which are closely related
to the hyperbolic functions, also
define the shape of the path a
spaceship takes when it uses the
"gravitational slingshot" effect to
alter its course via a planet's
gravitational pull propelling it away
from that planet at high velocity.

8. Curves :
Mathematically, the catenary
curve is the graph of the
hyperbolic cosine function. The
surface of revolution of the
catenary curve, the catenoid, is
a minimal surface, specifically
a minimal surface of
revolution. A hanging chain
will assume a shape of least
potential energy which is a
Catenary

9. APPLICATION OF HYPERBOLIC
Applications of Hyperbolic functions
Hyperbolic functions occur in the
solutions of some important linear
differential equations, for example the
equation defining a catenary, and
Laplaces equation in Cartesian
coordinates. The latter is important in
many areas of physics, including
electromagnetic theory, heat transfer,
fluid dynamics, and special relativity.
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10. Thank You