Hyperbolic functions are analogues of ordinary trigonometric and circular functions in mathematics, but they are defined using the hyperbola rather than the circle.
In hyperbolic geometry, these functions are used to calculate distances and angles. They can also be found in the solutions of many linear differential equations, cubic equations, and Laplace’s equation in Cartesian coordinates. We will discuss in detail the basic hyperbolic function, its properties, formulas, and hyperbolic function identities. The free online Hyperbolic function calculator can be used to compute the value of hyperbolic functions.
Hyperbolic Functions Formulas
Below are the basic formulas for hyperbolic functions and their graph functions:
Hyperbolic Sine Function
The formula defines the hyperbolic function \( f(x) = sinh \space x \)
\( sinh \space x = \dfrac{e^x – e^{-x}}{2} \)
The function meets these two conditions:
\( sinh \space 0 = 0 \) and \( sinh(−x) = − sinh \space x \)
The graph of sinh x is consistently between the graphs of \( \dfrac{e^x}{2} \) and \( \dfrac{e^{-x}}{2} \).
Hyperbolic Cosine Function
The hyperbolic Cosine function is defined by the formula
\( cosh \space x = \dfrac{e^x + e^{-x}}{2} \)
The function meets these two conditions:
\( cosh \space 0 = 1 \) and \( cosh \space x = cosh(−x) \)
The graph of cosh x is consistently above the graphs of \( \dfrac{e^x}{2} \) and \( \dfrac{e^{-x}}{2} \).
Hyperbolic Tangent Function
The formula defines the hyperbolic function f(x) = tanh x.
\( tanh \space x = \dfrac{e^x – e^{-x}}{2} ÷ \dfrac{e^x + e^{-x}}{2} \)
\( = \dfrac{e^x – e^{-x}}{e^x + e^{-x}} \)
Hyperbolic Function Identities
Hyperbolic Function Identities are:
\( cosh^2 \space x – sinh^2 \space x = 1 \)
\( sinh ( – x) = – sinh \space x\)
\( cosh ( – x) = cosh \space x \)
\( tanh ( – x) = – tanh \space x \)
\( coth (– x) = – coth \space x\)
FAQs
What are Inverse Hyperbolic Functions?
Inverse hyperbolic functions, also known as area hyperbolic functions, are multivalued functions that are inverse functions of hyperbolic functions. These functions return the hyperbolic angles corresponding to the given hyperbolic function value. For inverse hyperbolic function in complex plane we use these formulas:
\(sinh^{-1} \space x = ln(x + \sqrt{1+x^2}) \)
\( cosh^{-1} \space x = ln(x + \sqrt{x^2 –1}) \)
\( tanh^{-1} \space x = (\dfrac{1}{2})(ln(1 + x) – ln(1 – x)) \)
What are Hyperbolic Trig Functions?
Trigonometric functions relate the angle of a circle to its sides. The trig function captures the oscillatory movement of an angle around the circle, while the hyperbolic functions capture the oscillatory movements around a hyperbola. The basic trig functions include sin(x), tan(x), & cos(x). Their hyperbolic counterparts have sinh(x), cosh (x), & tanh(x).
Can't find your query?
Fill out the form below with your query and we will get back to you in 24 hours.
MOST POPULAR
Most commonly used free online Calculators
Learning Center
The most Popular Blogs among people
Cartesian Coordinates System – Dimensions, Formula...
Cartesian coordinates, sometimes called rectangular coordinates, are t...
Math Playground: A Hands-On Approach to Learning M...
Step into the math playground, where the wonders of numbers come to li...
How to Graph a Parabola – Formula, Properties, and...
When you discharge an arrow or throw a stone, it arcs into the air and...