Hyperbolic Functions Formula

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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 \[\sqrt{c^{2}-a^{2}}\].

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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-line 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 \[\frac{\sqrt{c^{2}-a^{2}}}{a}\]


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

The eccentricity of a Hyperbola:

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For a Hyperbola, the value of eccentricity is:

\[\frac{\sqrt{c^{2}-a^{2}}}{a}\]


Definition of Hyperbolic Functions Formula

Let’s Discuss the Trigonometric Hyperbolic Formulas.

The Hyperbolic sine of x

Sinh x : \[(e^{x} - e^{-x})\]/2

The Hyperbolic cosine of x

Cosh x :\[(e^{x} + e^{-x})\]/2

The Hyperbolic tangent of x

Tanh x: sinh x/ cosh x 

 =\[(e^{x} - e^{-x})\] / \[(e^{x} + e^{-x})\]


The Hyperbolic cotangent of x

Coth x: cosh x/ sinh x 

 =\[(e^{x} + e^{-x})\]/\[(e^{x} - e^{-x})\], where x is not equal to 0.

The Hyperbolic secant of x

Sech x: 1/ cosh x 

= 2/ \[(e^{x} + e^{-x})\]

The Hyperbolic cosecant of x

Csch x: 1/ sinh x 

= 2/ \[(e^{x} - e^{-x})\], where x is not equal to 0.


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)

  1. Tanh x :

Sinh x / Cosh x 

  1. Coth x:

1/tanh x = Cosh x / Sinh x 

  1. Sech x:

1/cosh x

  1. Csch x:

1/sinh x

  1. \[cosh^{2}x - sinh^{2}x\]

1

  1. \[sech^{2}x - tanh^{2}x\]

1

  1. \[coth^{2}x - csch^{2}x\]

1

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)

And

  • tanh(−x) = −tanh(x)

  • coth(−x) = −coth(x)

  • sech(−x) = sech(x)

  • csch(−x) = −csch(x)


Questions To Be Solved

Question 1) Derive addition 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,

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

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

FAQ (Frequently Asked Questions)

Question 1) What are Cosh x and Sinh x?

Answer) Hyperbolic functions in Mathematics can generally be defined as analogs of the trigonometric functions in mathematics that are defined for the hyperbola rather than on the circle (unit circle): just as the points (cos t, sin t) and we use a circle with a unit radius, the points generally (cosh t, sinh t)these form the right half of the equilateral hyperbola.

Question 2) What is the Trigonometry Formula?

Answer) The six trigonometric functions are sine, cosine, secant, co-secant, tangent, and co-tangent. We can use a right-angled triangle as a reference for this, the trigonometric functions or identities that are derived: sin θ = Opposite Side/Hypotenuse. sec θ = Hypotenuse/Adjacent Side. cos θ = Adjacent Side/Hypotenuse.

Question 3) Where are Hyperbolic functions used?

Answer) 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). Hyperbolic functions can also be used to define a measure of distance in certain types or kinds of non-Euclidean geometry.

Question 4) What are SOH, COH, and TOA?

Answer) SOH, COH and TOA is a mnemonic way or trick to remember the three basic trigonometric ratios defined by the trigonometric ratio definition.

SOH stands for Sine equals Opposite over Hypotenuse. (Sin (θ) = Opposite/Hypotenuse)

CAH stands for Cosine equals Adjacent over Hypotenuse. (Cos (θ) = Adjacent/Hypotenuse)

TOA stands for Tangent equals Opposite over Adjacent. (Tan (θ) = Opposite/Adjacent)