Hyperbolic function formulas are similar to trigonometric functions but use natural exponents (ex). The normal trigonometric functions are referred to as circular functions, and their graph is similar to a circle. Further, the graph of a hyperbolic function synonymous with its name represents a rectangular hyperbola and the hyperbolic function formula can often be seen in the formulas of a hyperbola.
What are the Hyperbolic Function Formulas?
The hyperbolic functions of hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, hyperbolic cosecant use the Euler's constant 'e' to form the respective hyperbolic function formulas.
Basic hyperbolic function formulas are expressed by making use of exponential function, ex and inverse exponential function e-x.
Sinhx = (ex - e-x)/2
Coshx = (ex + e-x)/2
Tanhx = (ex - e-x)/(ex + e-x)
Cothx = (ex + e-x)/(ex - e-x)
Sechx = 2/(ex + e-x)
Cosechx = 2/(ex - e-x)
Relationship between Hyperbolic Functions is similar to the normal trigonometric functions but has a small variation.
Sinhx = 1/Cosechx
Coshx = 1/Sechx
Tanhx = 1/Cothx
Cothx = 1/Tanhx
Cosh2x - Sinh2x = 1
Tanh2x + Sech2x = 1
Coth2x - Cosech2x = 1
Sum and difference of hyperbolic function formulas, follow the same expressions as the formulas of the sum of angles of the trigonometric formulas.
Sinh(x + y) = Sinhx.Coshy + Sinhy.Coshx
Sinh(x - y) = Sinhx.Coshy - Sinhy.Coshx
Cosh(x + y) = Coshx.Coshy - Sinhx.Sinhy
Cosh(x - y) = Coshx.Coshy + Sinhx.Sinhy
Tanh(x + y) = (Tanhx + Tanhy)/(1 - Tanhx.Tanhy)
Tanh(x - y) = (Tanhx - Tanhy)/(1 + Tanhx.Tanhy)
Derivatives of hyperbolic functions is similar to the derivation of the normal trigonometric functions.Unlike the derivative of trigonometric functions, we can observe the change in sign in the derivative of hyperbolic secant function.
d/dx. Sinhx = Coshx
d/dx. Coshx = Sinhx
d/dx. Tanhx = Sech2x
d/dx. Cothx = - Cosech2x
d/dx.Sechs = - Sechx.Tanhx
d/dx.Cosechx = -Cosechx.Cothx
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