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 ex.
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 = (ex + e-x)/2
Similarly, other hyperbolic functions are defined.
Various hyperbolic function formulas are,
sinh(x) = (ex - 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)
Also, Check Trigonometric Identities
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((ex + e-x)/2)/dx
= 1/2(d(ex + e-x)/dx)
= 1/2(ex - 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:
Hyperbolic Functions Examples
Example 1: Find the value of x solving, 4sinh x - 6cosh x - 2 = 0.
Solution:
We know that,
- sinh x = (ex - e-x)/2
- cosh x = (ex + e-x)/2
Given,
4sinh x - 6cosh x + 2 = 0
⇒ 4[(ex - e-x)/2] - 6[(ex + e-x)/2] + 6 = 0
⇒ 2(ex - e-x) - 3(ex + e-x) + 6 = 0
⇒ 2ex - 2e-x - 3ex - 3e-x + 6 = 0
⇒ -ex - 5e-x + 6 = 0
⇒ -e2x - 5 + 6ex = 0
⇒ e2x - 6ex + 5 = 0
⇒ e2x - 5ex - ex + 5 = 0
⇒ ex(ex - 5) - 1(ex - 5) = 0
⇒ (ex - 1)(ex - 5) = 0
(ex - 1) = 0
ex = 1
x = 0
(ex - 5) = 0
ex = 5
x = ln 5
Example 2: Prove, cosh x + sinh x = ex
Solution:
LHS
= cosh x + sinh x
= (ex - e-x)/2 + (ex + e-x)/2
= (ex - e-x + ex + e-x)/2
= 2ex / 2
= ex
= 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
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