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If you hold a chain at both ends and let it hang loosely, the path of the chain follows the path of the hyperbolic cosine. (This is also the shape of the St. Lois Arch.) Also, the integrals of many useful functions. For example, if an object is falling in a constant gravitational field with air resistance, the velocity of the object as a function of time involves the inverse hyperbolic tangent.

Q: Use of hyperbolic functions

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The hyperbolic functions are related to a hyperbola is the same way the the circular functions are related to a circle. So, while the points with coordinates [cos(t), sin(t)] generate the unit circle, their hyperbolic counterparts, [cosh(t) , sinh(t)] generate the right half of the equilateral hyperbola. Other circular functions (tan, sec, cosec and cot) also have their hyperbolic counterparts, as do the inverse functions. An alternative, equivalent pair of definitions is: cosh(x) = (ex + e-x)/2 and sinh(x) = (ex - e-x)/2

There are three types of trigonometric functions, they are: 1- Plane Trigonometric Functions 2- Inverse Trigonometric Functions and 3- Hyperbolic Trigonometric Functions

There are infinitely many types of functions. For example: Discrete function, Continuous functions, Differentiable functions, Monotonic functions, Odd functions, Even functions, Invertible functions. Another way of classifying them gives: Logarithmic functions, Inverse functions, Algebraic functions, Trigonometric functions, Exponential functions, Hyperbolic functions.

An arc-hyperbolic function is an inverse hyperbolic function.

It can. And does, for example, in the hyperbolic trigonometric functions. It can make the solution harder but there is no law that says that solutions must be easy!

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Hyperbolic functions can be used to describe the position that heavy cable assumes when strung between two supports.

If f(x)=y, then the inverse function solves for y when x=f(y). You may have to restrict the domain for the inverse function to be a function. Use this concept when finding the inverse of hyperbolic functions.

The basic ones are: sine, cosine, tangent, cosecant, secant, cotangent; Less common ones are: arcsine, arccosine, arctangent, arccosecant, arcsecant, arccotangent; hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, hyperbolic cotangent; hyperbolic arcsine, hyperbolic arccosine, hyperbolic arctangent, hyperbolic arccosecant, hyperbolic arcsecant, hyperbolic arccotangent.

The inverses of hyperbolic function are the area hyperbolic functions. They are called area functions becasue they compute the area of a sector of the unit hyperbola x2 âˆ’ y2 = 1 This is similar to the inverse trig functions which correspond to arclength of a sector on the unit circle

The hyperbolic functions are related to a hyperbola is the same way the the circular functions are related to a circle. So, while the points with coordinates [cos(t), sin(t)] generate the unit circle, their hyperbolic counterparts, [cosh(t) , sinh(t)] generate the right half of the equilateral hyperbola. Other circular functions (tan, sec, cosec and cot) also have their hyperbolic counterparts, as do the inverse functions. An alternative, equivalent pair of definitions is: cosh(x) = (ex + e-x)/2 and sinh(x) = (ex - e-x)/2

There are three types of trigonometric functions, they are: 1- Plane Trigonometric Functions 2- Inverse Trigonometric Functions and 3- Hyperbolic Trigonometric Functions

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That's because like circular functions/trigonometric functions give the position(co-ordinates, technically) of a point on the circle, these give the position of points on a hyperbola.

The equation of a hyperbolic function is y = sinh(x) or y = cosh(x), where sinh(x) represents the hyperbolic sine function and cosh(x) represents the hyperbolic cosine function. Hyperbolic functions are similar to trigonometric functions but are defined in terms of exponentials.

Frederick Eugene Pernot has written: 'Abridged tables of hyperbolic functions'

There are infinitely many types of functions. For example: Discrete function, Continuous functions, Differentiable functions, Monotonic functions, Odd functions, Even functions, Invertible functions. Another way of classifying them gives: Logarithmic functions, Inverse functions, Algebraic functions, Trigonometric functions, Exponential functions, Hyperbolic functions.

* What are the exponential equivalents of hyperbolics? * How do hyperbolics relate to standard trig functions? * What shape does cosh produce? * Why does cosh grow faster than sinh? * What are the derivatives and integrals of various functions?