Hyperbolic Functions Formula

Hyperbolic functions refer to the exponential functions that share similar properties to trigonometric functions. These functions are analogous trigonometric functions in that they are named the same as trigonometric functions with the letter ‘h’ appended to each name.  These have the same relationship to the hyperbola that trigonometric functions have to the circle. Thus, they are collectively known as hyperbolic functions and are individually called hyperbolic sine, hyperbolic cosine, and so on. In addition to modeling, they can be used as solutions to some types of partial differential equations. In this article, we are going to discuss the hyperbolic functions formula, general equation of hyperbola, standard equation of hyperbola, hyperbola formula, trigonometric hyperbolic formulas.

General Equation of Hyperbola / Standard Equation of Hyperbola

A hyperbola is a plane curve that is generated by a point so moving that the difference of the distances from two fixed points is constant. The two fixed points are the foci and the mid-point of the line segment joining the foci is the center of the hyperbola. The transverse axis is the line through the foci. The conjugate axis is the line through the center and perpendicular to the transverse axis.

The vertices of the hyperbola are the points at which the hyperbola intersects the transverse axis. 2c is the distance between the two foci. The distance between the two vertices is 2a. 2a s is also the length of the transverse axis. The length of the conjugate axis is 2b. The value of b is.

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Eccentricity of Hyperbola

• A hyperbola is defined as the set of all points in a plane where the difference of whose distances from two fixed points is constant. 

• In simpler words, the distance from the fixed point in a plane bears a constant ratio greater than the distance from the fixed-point in a plane.

Therefore, the eccentricity of the Hyperbola is always greater than 1. i.e. e > 1

The general equation of Hyperbola or standard equation of the Hyperbola is denoted as

For any Hyperbola, the values a and b are the lengths of the semi-major and semi-minor axes respectively.

Definition of Hyperbolic Functions Formula

Let’s discuss the trigonometric hyperbolic formulas.

Graph showing the Difference between Sin(x) and Sinh (x) Functions

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Graph showing the Difference between Cos(x) and Cosh (x) Functions

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Graph showing the Difference between Tan(x) and Tanh (x) Functions

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Relationship Among Hyperbolic Functions( Trigonometric Hyperbolic Formulas)

Addition Formulas

• sinh (x+y)=sinh x cosh y + cosh x sinh y

• cosh (x+y)= cosh x cosh y + sinh x sinh y

• tanh (x+y) = (tanh x + tanh y) / 1+tanh x. tanh y

• tanh (x-y) = (tanh x - tanh y) / 1-tanh x. tanh y

• coth (x+y) =(coth x. coth y+1) / coth y. coth x

• coth (x-y) =(coth x. coth y-1) / coth y. coth x

Trigonometric Identities

• sinh(−x) = −sinh(x)

• cosh(−x) = cosh(x)

• tanh(−x) = −tanh(x)

• coth(−x) = −coth(x)

• sech(−x) = sech(x)

• csch(−x) = −csch(x)

Questions To Be Solved

1.Derive additional identities for sin h (x+y) and cos h (x+y).

In the identity tanh (x+y)= (tanh x + tanh y)/1+tanh x. tanh y

Solution: tanh x(x+y) = sinh (x+y)/ cosh (x+y)

Since sinh (x+y) = sinh x cosh y +cosh x sinh y

cosh (x+y) equals cosh x cosh y + sinh x sinh y

=\[\frac{sinh x\ cosh y + sinh y\ cosh x}{cosh x\ cosh y + sinh x\ sinh y}\]

Dividing numerator and denominator by cosh x and cosh y,

= \[\frac{sinh x / cosh x\ +\ sinh y / cosh x}{1 + sinh x / cosh x\ .\ sinh y / cosh y}\]

=\[\frac{tanh x\ +\ tanh y}{1 + tanh x\ tanh y}\]

2. What are Hyperbolic Functions?

We all know about trigonometric functions which are defined on or for a circle in a similar way to defied hyperbola we use the hyperbolic function. Commonly we use sine, cosine and other functions in trigonometry. For hyperbole in a similar way, we use csch, sech, sinh, tan h, coth, and cosh. Again in normal trigonometry, we know that the points of coordinates on the circle unit are (sinΦ, cos Φ) in a similar way(sinnΦ, cosΦ) forms the right half of the equilateral hyperbole for hyperbolic functions.

Laplace's equation, differential equations and cubic equations have various solutions because of hyperbolic function. There are six of those functions which are given below:

• Sinh x or hyperbolic sine

• Cosech x or hyperbolic cosecant

• Cosh X or hyperbolic cosine

• Tanh x or, hyperbolic tangent

• Coth x or hyperbolic cotangent

• Sech x or hyperbolic secant

Hyperbolic Functions Meaning

Analogously hyperbole functions are defined as trigonometric functions. Namely sinh x, tan h x, coth x, sech x, cosech x, and cosh x are the main six functions of hyperbole. The combination of exponential functions is expressed for hyperbolic functions. Just like trigonometric functions are derived using the circle of the unit the hyperbolic functions are derived using the hyperbola.

Hyperbolic Functions Formulas

The algebraic expressions include the exponential functions \[e^{x}\] and its inverse exponential \[e^{-x}\] where it is known as Euler's constant; through this we can define hyperbolic functions.

Hyperbolic Functions Identities

As we know trigonometric functions identities are similar to the hyperbolic function identities and can be understood better from the explanation below. Osborn's rule states that when expanded properly and completely in terms of integral powers of cosines and sines which included position changing of cosine to cosh and again sine to h trigonometric function identities can be changed or converted into hyperbolic function identities. A sign of each term that contains the product of two sinh should be exchanged.

Differentiation and Integration of Hyperbolic Functions

The derivative and integrals of trigonometric functions are similar to the derivative and integral of hyperbolic functions. Unlike the change in sign in the derivative of the hyperbolic secant function, we can observe the same in the derivative of trigonometric functions.

Inverse Hyperbolic Functions

The inverse of the hyperbola is known as the inverse of hyperbolic functions. For example, if \[y = sinh^{-1} x\], then x = sinh y is the inverse of the sine of hyperbolic functions. We express the inverse of the hyperbolic functions into the terms of function logarithm. Just as the inverse of the hyperbole is used in another way on the contrary trigonometric functions are useful in certain integration, calculus. There are some restrictions on the domain to make functions into one to one of each and the domains resulting and inverse functions of their ranges.

Important Notes on Hyperbolic Functions

• Cosh x, coth x, csch x, sinh x, sech x, and tanh x are the six hyperbolic functions.

• For hyperbola, we define a hyperbolic function.

• The trigonometric function identities are similar to the hyperbolic functions identities.