Hyperbolic functions

Definition

Hyperbolic functions are a set of mathematical functions that are analogs of the ordinary trigonometric functions but are based on hyperbolas instead of circles. They include hyperbolic sine ($$\sinh$$), hyperbolic cosine ($$\cosh$$), and others, which are essential in various calculus applications such as integrals, differential equations, and trigonometric substitution. These functions exhibit properties similar to trigonometric functions but have distinct geometric interpretations related to hyperbolas.

5 Must Know Facts For Your Next Test

1. Hyperbolic functions can be defined using exponential functions, with $$\sinh(x)$$ and $$\cosh(x)$$ being derived from combinations of the exponential function.
2. The derivatives of hyperbolic functions mirror those of trigonometric functions: $$\frac{d}{dx} \sinh(x) = \cosh(x)$$ and $$\frac{d}{dx} \cosh(x) = \sinh(x)$$.
3. The identities for hyperbolic functions are analogous to trigonometric identities; for example, $$\cosh^2(x) - \sinh^2(x) = 1$$ is similar to the Pythagorean identity for sine and cosine.
4. Hyperbolic functions are useful in solving differential equations, especially in applications related to physics and engineering where hyperbolic trajectories are involved.
5. They also appear in calculus when evaluating integrals that require substitutions or transformations involving hyperbolic identities.

Review Questions

• Compare and contrast hyperbolic functions with trigonometric functions in terms of their definitions and key properties.
  • Hyperbolic functions like $$\sinh$$ and $$\cosh$$ are defined using exponential functions and exhibit properties similar to trigonometric functions. For instance, both have derivatives that relate them to one another: the derivative of $$\sinh(x)$$ is $$\cosh(x)$$ just like the derivative of sine is cosine. However, while trigonometric functions are based on circular geometry, hyperbolic functions stem from hyperbolic geometry, leading to different identities and values across their ranges.
• Explain how hyperbolic functions can be applied in integral calculus and give an example of such an application.
  • Hyperbolic functions often simplify integrals involving square roots or exponential expressions. For example, when integrating $$\int \sqrt{x^2 + a^2} \, dx$$, a substitution using the hyperbolic function $$x = a \, \sinh(t)$$ can transform the integral into a simpler form. This substitution exploits the relationship between hyperbolas and areas under curves, making it easier to evaluate integrals involving complex algebraic expressions.
• Evaluate the significance of understanding hyperbolic functions in solving real-world problems, particularly in physics and engineering contexts.
  • Understanding hyperbolic functions is crucial in fields like physics and engineering as they describe real-world phenomena involving hyperbolic trajectories, such as in special relativity or certain types of wave equations. They model situations where variables grow exponentially or follow specific paths dictated by the laws of physics. Recognizing these patterns allows engineers and scientists to apply mathematical models effectively, leading to innovations in technology and deeper insights into natural processes.

Related terms

Inverse Hyperbolic Functions: Functions that provide the angle whose hyperbolic sine, cosine, or tangent gives a specific value; these are denoted as $$\text{arsinh}$$, $$\text{arcosh}$$, and $$\text{artanh}$$.

Exponential Functions: Functions of the form $$f(x) = e^x$$, which serve as the foundation for defining hyperbolic functions; specifically, $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$ and $$\cosh(x) = \frac{e^x + e^{-x}}{2}$$.

Trigonometric Functions: Functions like sine, cosine, and tangent that relate angles to side lengths in right triangles; hyperbolic functions can be seen as their counterparts for hyperbolas.