Hyperbolic sinusoidal functions, also known as hyperbolic trigonometric functions, are a set of mathematical functions that are analogous to the regular trigonometric functions but are defined in terms of the hyperbola rather than the unit circle. These functions are commonly used in various branches of mathematics, physics, and engineering to model a wide range of phenomena.
The hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions are the most fundamental hyperbolic functions:
Hyperbolic Sine (sinh):
The hyperbolic sine of a real number 'x' is defined as:
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sinh(x) = (e^x - e^(-x)) / 2
where 'e' is the base of the natural logarithm.
Hyperbolic Cosine (cosh):
The hyperbolic cosine of a real number 'x' is defined as:
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cosh(x) = (e^x + e^(-x)) / 2
Just like the regular trigonometric functions, these hyperbolic functions have a range of properties and identities. Some important properties include:
Odd and Even Properties:
sinh(-x) = -sinh(x)
cosh(-x) = cosh(x)
Relationship with Exponential Function:
cosh(x) = (e^x + e^(-x)) / 2
sinh(x) = (e^x - e^(-x)) / 2
Hyperbolic Pythagorean Identity:
cosh^2(x) - sinh^2(x) = 1
Derivatives:
d/dx(cosh(x)) = sinh(x)
d/dx(sinh(x)) = cosh(x)
Addition Formulas:
sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y)
cosh(x + y) = cosh(x)cosh(y) + sinh(x)sinh(y)
Hyperbolic functions find applications in various fields such as physics, engineering, and mathematics. They often appear in problems involving exponential growth, heat conduction, wave propagation, and more.
Additionally, there are other hyperbolic functions like hyperbolic tangent (tanh), hyperbolic cotangent (coth), hyperbolic secant (sech), and hyperbolic cosecant (csch) that are defined similarly in terms of the hyperbola. These functions have their own set of properties and applications.
In summary, hyperbolic sinusoidal functions are a set of functions that mirror the behavior of trigonometric functions but are based on the hyperbola. They have a wide range of applications and play an important role in various mathematical and scientific contexts.