Sinh⁻¹

Definition

sinh⁻¹, also known as the inverse hyperbolic sine function, is a mathematical function that is the inverse of the hyperbolic sine function, sinh(x). It is used to find the angle whose hyperbolic sine is a given value.

5 Must Know Facts For Your Next Test

1. The sinh⁻¹ function is used to find the angle whose hyperbolic sine is a given value.
2. The domain of the sinh⁻¹ function is the set of all real numbers, and the range is the set of all real numbers.
3. The graph of the sinh⁻¹ function is an increasing function that passes through the origin and has an asymptotic behavior as the input approaches positive or negative infinity.
4. The sinh⁻¹ function is often used in physics and engineering applications, such as in the analysis of electrical circuits and the study of special relativity.
5. The sinh⁻¹ function can be expressed in terms of the natural logarithm function as: sinh⁻¹(x) = ln(x + √(x^2 + 1)).

Review Questions

• Explain the relationship between the hyperbolic sine function and the inverse hyperbolic sine function.
  • The hyperbolic sine function, sinh(x), and the inverse hyperbolic sine function, sinh⁻¹(x), are inverse functions. This means that if y = sinh(x), then x = sinh⁻¹(y), and vice versa. The inverse hyperbolic sine function is used to find the angle whose hyperbolic sine is a given value, just as the inverse trigonometric functions are used to find the angle whose trigonometric function is a given value.
• Describe the properties of the graph of the sinh⁻¹ function.
  • The graph of the sinh⁻¹ function is an increasing function that passes through the origin and has an asymptotic behavior as the input approaches positive or negative infinity. The function is defined for all real numbers, and its range is also the set of all real numbers. The graph of the sinh⁻¹ function is similar to the graph of the inverse trigonometric functions, but it is not periodic like the trigonometric functions.
• Explain how the sinh⁻¹ function is used in physics and engineering applications.
  • The sinh⁻¹ function is often used in physics and engineering applications, such as in the analysis of electrical circuits and the study of special relativity. In electrical circuits, the sinh⁻¹ function can be used to describe the behavior of certain types of nonlinear components, such as diodes and transistors. In special relativity, the sinh⁻¹ function is used to describe the relationship between the proper time and the coordinate time of an object moving at a relativistic speed. The ability of the sinh⁻¹ function to model these types of phenomena makes it a valuable tool in these fields.

Related terms

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

Inverse Hyperbolic Functions:

The inverse hyperbolic functions are the inverse of the hyperbolic functions, and they are used to find the angle whose hyperbolic function is a given value.

Exponential Function: The exponential function is a mathematical function that describes a relationship in which a quantity grows or decays at a rate proportional to its current value.