Sinh

Definition

Sinh, also known as the hyperbolic sine function, is a mathematical function that is part of the family of hyperbolic functions. It is defined as the ratio of the opposite side to the adjacent side of a right-angled triangle in the hyperbolic plane, similar to how the sine function is defined in the Euclidean plane.

5 Must Know Facts For Your Next Test

1. The sinh function is defined as $\sinh(x) = \frac{e^x - e^{-x}}{2}$, where $e$ is the base of the natural logarithm.
2. The sinh function is an odd function, meaning that $\sinh(-x) = -\sinh(x)$.
3. The derivative of the sinh function is $\frac{d}{dx}\sinh(x) = \cosh(x)$.
4. The integral of the sinh function is $\int\sinh(x)dx = -\cosh(x) + C$, where $C$ is the constant of integration.
5. The sinh function has many applications in physics, engineering, and other scientific fields, such as in the analysis of hyperbolic geometry, special relativity, and electrical engineering.

Review Questions

• Explain the relationship between the sinh function and the exponential function.
  • The sinh function is defined in terms of the exponential function, as $\sinh(x) = \frac{e^x - e^{-x}}{2}$. This relationship shows that the sinh function can be expressed as the difference between two exponential functions, which is a fundamental property of the hyperbolic functions. The connection between the sinh function and the exponential function is important for understanding the properties and applications of the hyperbolic functions in various mathematical and scientific contexts.
• Describe the key properties of the sinh function, such as its symmetry, derivative, and integral.
  • The sinh function has several important properties that are crucial to understand in the context of the calculus of hyperbolic functions. First, the sinh function is an odd function, meaning that $\sinh(-x) = -\sinh(x)$, which reflects its symmetry about the origin. Additionally, the derivative of the sinh function is the cosh function, as $\frac{d}{dx}\sinh(x) = \cosh(x)$. The integral of the sinh function is the negative of the cosh function, as $\int\sinh(x)dx = -\cosh(x) + C$. These properties of the sinh function, along with its definition in terms of the exponential function, are essential for solving problems and understanding the applications of hyperbolic functions in various fields.
• Explain how the sinh function and other hyperbolic functions are used in the analysis of hyperbolic geometry and special relativity.
  • The hyperbolic functions, including the sinh function, play a crucial role in the analysis of hyperbolic geometry and special relativity. In hyperbolic geometry, the sinh function is used to describe the properties of the hyperbolic plane, such as the lengths of hyperbolic lines and the areas of hyperbolic triangles. Similarly, in special relativity, the sinh function is used to describe the relationships between the spatial and temporal coordinates of an object moving at a relativistic speed. The properties of the sinh function, such as its connection to the exponential function and its derivative and integral relationships, are fundamental to the mathematical formulations and applications of these important areas of mathematics and physics.

Related terms

Hyperbolic Functions: The hyperbolic functions, including sinh, cosh, and tanh, are analogs of the trigonometric functions, but they are defined using the hyperbolic plane instead of the Euclidean plane.

Cosh:

The hyperbolic cosine function, which is the ratio of the adjacent side to the opposite side of a right-angled triangle in the hyperbolic plane.

Tanh: The hyperbolic tangent function, which is the ratio of the opposite side to the adjacent side of a right-angled triangle in the hyperbolic plane, divided by the ratio of the adjacent side to the opposite side.