Hyperbolic Identities

Definition

Hyperbolic identities are mathematical relationships that hold true for the hyperbolic functions, which are the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic secant, hyperbolic cosecant, and hyperbolic cotangent. These identities are analogous to the trigonometric identities, but they involve the hyperbolic functions instead of the trigonometric functions.

5 Must Know Facts For Your Next Test

1. Hyperbolic identities are used to simplify and manipulate expressions involving hyperbolic functions, just as trigonometric identities are used for trigonometric functions.
2. The hyperbolic identities are derived from the definitions of the hyperbolic functions and their relationships to the exponential function.
3. The most fundamental hyperbolic identities are the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent identities, which are analogous to the Pythagorean identities for trigonometric functions.
4. Hyperbolic identities can be used to solve problems involving hyperbolic functions, such as finding the value of a hyperbolic function given the values of other hyperbolic functions.
5. Understanding and applying hyperbolic identities is essential for working with the calculus of the hyperbolic functions, which is covered in the 2.9 Calculus of the Hyperbolic Functions topic.

Review Questions

• Explain the relationship between the hyperbolic functions and the exponential function, and how this relationship is used to derive the hyperbolic identities.
  • The hyperbolic functions are defined in terms of the exponential function, with the hyperbolic sine and hyperbolic cosine being expressed as combinations of the exponential function. This relationship allows the hyperbolic identities to be derived directly from the properties of the exponential function. For example, the fundamental hyperbolic identity $\cosh^2(x) - \sinh^2(x) = 1$ can be proven by manipulating the definitions of the hyperbolic sine and hyperbolic cosine in terms of the exponential function.
• Describe how hyperbolic identities can be used to simplify and manipulate expressions involving hyperbolic functions, and provide an example.
  • Hyperbolic identities can be used to simplify and manipulate expressions involving hyperbolic functions in a similar way to how trigonometric identities are used for trigonometric functions. For example, the identity $\sinh(x) + \cosh(x) = e^x$ can be used to rewrite the expression $2\cosh(x)$ as $e^x + e^{-x}$, which may be a more convenient form for certain calculations or applications. Applying hyperbolic identities can often lead to more compact and easier-to-work-with expressions, making them an essential tool in the calculus of the hyperbolic functions.
• Analyze how the understanding and application of hyperbolic identities is crucial for the study of the calculus of the hyperbolic functions, and explain how this knowledge can be used to solve problems in this topic.
  • The calculus of the hyperbolic functions, covered in the 2.9 Calculus of the Hyperbolic Functions topic, relies heavily on the understanding and application of hyperbolic identities. These identities are used to differentiate and integrate expressions involving hyperbolic functions, as well as to solve problems involving the properties of the hyperbolic functions. For example, the identity $\frac{d}{dx}\cosh(x) = \sinh(x)$ is derived using the hyperbolic identities and is a fundamental result in the calculus of the hyperbolic functions. Mastering the hyperbolic identities allows students to manipulate expressions, apply differentiation and integration rules, and solve a wide range of problems involving the hyperbolic functions, which is essential for success in this topic.

Related terms

Hyperbolic Functions: The hyperbolic functions are a set of functions that are defined in terms of the exponential function, similar to how the trigonometric functions are defined in terms of the circular functions.

Exponential Function: The exponential function is a fundamental mathematical function that describes exponential growth or decay, and it is used in the definition of the hyperbolic functions.

Inverse Hyperbolic Functions: The inverse hyperbolic functions are the functions that undo the effects of the hyperbolic functions, just as the inverse trigonometric functions undo the effects of the trigonometric functions.