Sinh

Definition

The hyperbolic sine function, denoted as sinh, is defined as $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$. This function is one of the fundamental hyperbolic functions, closely related to the geometry of hyperbolas in the same way that sine is related to circles. The hyperbolic sine function arises naturally in various areas, including physics and engineering, where hyperbolic functions model certain types of wave equations and phenomena.

5 Must Know Facts For Your Next Test

1. The graph of sinh is similar in shape to the graph of the regular sine function but has a different range and domain; it extends infinitely in both directions along the x-axis and y-axis.
2. The value of sinh at zero is zero: $$\sinh(0) = 0$$, which makes it an odd function since $$\sinh(-x) = -\sinh(x)$$.
3. For large values of x, sinh approximates the exponential function: $$\sinh(x) \approx \frac{e^x}{2}$$.
4. The derivative of sinh is cosh: $$\frac{d}{dx}(\sinh(x)) = \cosh(x)$$, which shows how they are interconnected in calculus.
5. Hyperbolic functions like sinh are essential for solving differential equations that arise in physics, particularly in problems involving hyperbolic geometry.

Review Questions

• Compare and contrast sinh with regular sine. How do their definitions and properties differ?
  • The hyperbolic sine function, sinh, is defined using exponential functions: $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$, while the regular sine function is based on circular geometry. One key difference is that sinh can take any real number as input and extends infinitely in both directions, whereas sine oscillates between -1 and 1. Additionally, sinh is an odd function, meaning $$\sinh(-x) = -\sinh(x)$$, while sine is periodic with a specific range.
• Explain how sinh and its relationship to cosh leads to applications in calculus. What role does this play in evaluating integrals?
  • The relationship between sinh and cosh allows for efficient evaluation of integrals involving hyperbolic functions. Since the derivative of sinh is cosh, integrating functions like sinh often leads to simpler calculations. Additionally, these relationships enable the application of integration techniques such as substitution or parts when solving problems involving hyperbolic functions. This interplay between derivatives and integrals helps streamline complex calculus problems encountered in various scientific fields.
• Evaluate the significance of hyperbolic functions like sinh in modeling real-world phenomena. How do they contribute to our understanding in fields like physics?
  • Hyperbolic functions like sinh are significant in modeling real-world phenomena because they represent relationships that arise in various natural contexts. For example, they are used in physics to describe waveforms and oscillations where hyperbolic geometry applies, such as in relativistic scenarios. Furthermore, they provide solutions to differential equations that model physical systems like springs or electrical circuits. Understanding these functions enhances our ability to analyze systems governed by hyperbolic relationships, leading to deeper insights into physical behaviors.

Related terms

cosh: The hyperbolic cosine function, defined as $$\cosh(x) = \frac{e^x + e^{-x}}{2}$$, it is closely related to sinh and often appears in conjunction with it in mathematical expressions.

tanh: The hyperbolic tangent function, defined as $$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$$, represents the ratio of the hyperbolic sine and cosine functions.

Inverse Hyperbolic Functions: Functions that give the angle whose hyperbolic sine (or cosine) is a given number; for sinh, the inverse is called arcsinh.