Hyperbolic Function

Hyperbolic Functions are similar to trigonometric functions but their graphs represent the rectangular hyperbola. These functions are defined using hyperbola instead of unit circles. Hyperbolic functions are expressed in terms of exponential functions e^x.

In this article, we will learn about the hyperbolic function in detail, including its definition, formula, and graphs.

Hyperbolic Definition

The six basic hyperbolic functions are,

• Hyperbolic sine or sinh x
• Hyperbolic cosine or cosh x
• Hyperbolic tangent or tanh x
• Hyperbolic cosecant or cosech x
• Hyperbolic secant or sech x
• Hyperbolic cotangent or coth x

Hyperbolic functions are defined using exponential functions. They are represented as, sinh x which is read as hyperbolic sinh x. Then the sinh x is defined as,

sinh x = (e^x + e^-x)/2

Similarly, other hyperbolic functions are defined.

Hyperbolic Functions Formulas

Various hyperbolic function formulas are,

sinh(x) = (e^x – e^-x)/2

Function	Definition
Hyperbolic Cosine (cosh x)	cosh(x) = (ex + e-x)/2​
Hyperbolic Sine (sinh x)
Hyperbolic Tangent (tanh x)	tanh(x) = sinhx/coshx = (ex – e-x)/(ex + e-x)​
Hyperbolic Cotangent (coth x)	coth(x) = cosh x/sin hx = (ex + e-x)/(ex – e-x)​
Hyperbolic Secant (sech x)	​ ech(x) = 1/cosh x = 2/(ex + e-x)
Hyperbolic Cosecant (csch x)	csch(x) = 1/sinh x = 2/(ex – e-x)

Domain and Range of Hyperbolic Functions

Domain and Range are the input and output of a function, respectively. The domain and range of various hyperbolic functions are added in the table below:

Hyperbolic Function	Domain	Range
sinh x	(-∞, +∞)	(-∞, +∞)
cosh x	(-∞, +∞)	[1, ∞)
tanh x	(-∞, +∞)	(-1, 1)
coth x	(-∞, 0) U (0, + ∞)	(-∞, -1) U (1, + ∞)
sech x	(-∞, + ∞)	(0, 1]
csch x	(-∞, 0) U (0, + ∞)	(-∞, 0) U (0, + ∞)

Learn, Domain and Range of a Function

Properties of Hyperbolic Functions

Various properties of the hyperbolic functions are added below,

• sinh (-x) = – sinh(x)
• cosh (-x) = cosh (x)
• tanh (-x) = – tanh x
• coth (-x) = – coth x
• sech (-x) = sech x
• csc (-x) = – csch x
• cosh 2x = 1 + 2 sinh²(x) = 2 cosh²x – 1
• cosh 2x = cosh²x + sinh²x
• sinh 2x = 2 sinh x cosh x

Hyperbolic functions are also derived from trigonometric functions using complex arguments. Such that,

• sinh x = – i sin(ix)
• cosh x = cos(ix)
• tanh x = – i tan(ix)
• coth x = i cot(ix)
• sech x = sec(ix)

Hyperbolic Trigonometric Identities

There are various identities that are related to hyperbolic functions. Some of the important hyperbolic trigonometric identities are,

• sinh(x ± y) = sinh x cosh y ± coshx 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)
• coth(x ± y) = (coth x coth y ± 1)/(coth y ± coth x)
• sinh x – sinh y = 2 cosh [(x+y)/2] sinh [(x-y)/2]
• sinh x + sinh y = 2 sinh [(x+y)/2] cosh[(x-y)/2]
• cosh x + cosh y = 2 cosh [(x+y)/2] cosh[(x-y)/2]
• cosh x – cosh y = 2 sinh [(x+y)/2] sinh [(x-y)/2])
• cosh²x – sinh²x = 1
• tanh²x + sech²x = 1
• coth²x – csch²x = 1
• 2 sinh x cosh y = sinh (x + y) + sinh (x – y)
• 2 cosh x sinh y = sinh (x + y) – sinh (x – y)
• 2 sinh x sinh y = cosh (x + y) – cosh (x – y)
• 2 cosh x cosh y = cosh (x + y) + cosh (x – y)

Hyperbolic Functions Derivative

Derivatives of Hyperbolic functions are used to solve various mathematical problems. The derivative of hyperbolic cos x is hyperbolic sin x, i.e.

d/dx (cosh x) = sinh x

= d(cosh(x))/dx

= d((e^x + e^-x)/2)/dx

= 1/2(d(e^x + e^-x)/dx)

= 1/2(e^x – e^-x)

= sinh(x)

Similarly, derivatives of other hyperbolic functions are found. The table added below shows the hyperbolic functions.

Derivatives of Hyperbolic Functions
Hyperbolic Function	Derivative
sinh x	cosh x
cosh x	sinh x
tanh x	sech2 x
coth x	-csch2 x
sech x	-sech x.tanh x
csch x	-csch x.coth x

Learn, Derivative in Maths

Integration of Hyperbolic Functions

Integral Hyperbolic functions are used to solve various mathematical problems. The integral of hyperbolic cos x is hyperbolic sin x, i.e.

∫ (cosh x).dx = sinh x + C

The table added below shows the integration of various hyperbolic functions.

Integral of Hyperbolic Functions
Hyperbolic Function	Integral
sinh x	cosh x + C
cosh x	sinh x + C
tanh x	ln (cosh x) + C
coth x	ln (sinh x) + C
sech x	arctan (sinh x) + C
csch x	ln (tanh(x/2)) + C

Learn, Integration

Inverse Hyperbolic Functions

Inverse hyperbolic functions are found by taking the inverse of the hyperbolic function, i.e. if y = sinh x then, x = sinh^-1 (y) this represents the inverse hyperbolic sin function. Now the inverse of various hyperbolic function are,

• sinh^-1x = ln (x + √(x² + 1))
• cosh^-1x = ln (x + √(x² – 1))
• tanh^-1x = ln [(1 + x)/(1 – x)]
• coth^-1x = ln [(x + 1)/(x – 1)]
• sech^-1x = ln [{1 + √(1 – x²)}/x]
• csch^-1x = ln [{1 + √(1 + x²)}/x]

Learn More:

• Inverse Trigonometric Identities

Hyperbolic Functions Examples

Example 1: Find the value of x solving, 4sinh x – 6cosh x – 2 = 0.

Solution:

We know that,

• sinh x = (e^x – e^-x)/2
• cosh x = (e^x + e^-x)/2

Given,

4sinh x – 6cosh x + 2 = 0

⇒ 4[(e^x – e^-x)/2] – 6[(e^x + e^-x)/2] + 6 = 0

⇒ 2(e^x – e^-x) – 3(e^x + e^-x) + 6 = 0

⇒ 2e^x – 2e^-x – 3e^x – 3e^-x + 6 = 0

⇒ -e^x – 5e^-x + 6 = 0

⇒ -e^2x – 5 + 6e^x = 0

⇒ e^2x – 6e^x + 5 = 0

⇒ e^2x – 5e^x – e^x + 5 = 0

⇒ e^x(e^x – 5) – 1(e^x – 5) = 0

⇒ (e^x – 1)(e^x – 5) = 0

(e^x – 1) = 0

e^x = 1

x = 0

(e^x – 5) = 0

e^x = 5

x = ln 5

Example 2: Prove, cosh x + sinh x = e^x

Solution:

LHS

= cosh x + sinh x

= (e^x – e^-x)/2 + (e^x + e^-x)/2

= (e^x – e^-x + e^x + e^-x)/2

= 2e^x / 2

= e^x

= RHS

Hyperbolic Functions Practice Questions

Q1. Find the value of x solving, sinh x + 5cosh x – 4 = 0

Q2. Find the value of x solving, 2sinh x – 6cosh x – 5 = 0

Q3. Find the value of x solving, 9sinh x + 6cosh x + 11 = 0

Q4. Find the value of x solving, sinh x – cosh x – 3 = 0

Related :

• Hyperbola
• Trigonometric Functions
• Unit Circle
• Differentiation Formulas
• Integration Formulas
• Trigonometric Formulas

FAQs on Hyperbolic Functions

What is Hyperbolic in Calculus?

Hyperbolic function in calculus are functions that are defined using exponents and their graphs represents rectangular hyperbola and hence the name hyperbolic functions. These functions are defined on hyperbola.

What is Hyperbolic Function Formula?

The formulas for various hyperbolic functions are,

• sinh x = (e^x – e^-x)/2
• cosh x = (e^x + e^-x)/2
• tanh x = (e^x – e^-x)/(e^x + e^-x)
• coth x = (e^x + e^-x)/(e^x – e^-x)
• sech x = 2/(e^x + e^-x)
• csch x = 2/(e^x – e^-x)

What are the Six Hyperbolic Functions?

Thesix hyperbolic functions are,

• Hyperbolic Sine (sinh x)
• Hyperbolic Cosine (cosh x)
• Hyperbolic Tangent (tanh x)
• Hyperbolic Cotangent (coth x)
• Hyperbolic Secant (sech x)
• Hyperbolic Cosecant (csch x)

What is Differentiation of Hyperbolic Functions?

The derivatives of hyperbolic functions are added below,

• d(sinh x)/dx = cosh x
• d(cosh x)/dx = sinh x
• d(tanh x)/dx = sech²x
• d(coth x)/dx = – cosech²x
• d(sech x)/dx = – sech x.tanh x
• d(cosech x)/dx = – cosech x.coth x

What are Inverse Hyperbolic Functions?

The formula for inverse hyperbolic functions are,

• sinh^-1x = ln (x + √(x² + 1))
• cosh^-1x = ln (x + √(x² – 1))
• tanh^-1x = ln [(1 + x)/(1 – x)]
• coth^-1x = ln [(x + 1)/(x – 1)]
• sech^-1x = ln [{1 + √(1 – x²)}/x]
• csch^-1x = ln [{1 + √(1 + x²)}/x]