Hyperbolic tangent

Definition

The hyperbolic tangent, denoted as 'tanh', is a mathematical function that relates to hyperbolic functions and is defined as the ratio of the hyperbolic sine and hyperbolic cosine functions. It is expressed mathematically as $$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$$. This function has important applications in various fields, including engineering and physics, and plays a key role in solving differential equations and modeling hyperbolic geometry.

5 Must Know Facts For Your Next Test

1. The range of the hyperbolic tangent function is from -1 to 1, meaning it asymptotically approaches these values as x approaches positive or negative infinity.
2. The graph of the hyperbolic tangent function resembles the shape of an 'S' curve, indicating its increasing behavior across its domain.
3. The derivative of the hyperbolic tangent function is given by $$\frac{d}{dx} \tanh(x) = 1 - \tanh^2(x)$$, which is useful in calculus and differential equations.
4. The hyperbolic tangent function is odd, meaning that $$\tanh(-x) = -\tanh(x)$$ for all x, showcasing its symmetry about the origin.
5. In electrical engineering, the hyperbolic tangent function can describe voltage and current relationships in certain types of circuits involving nonlinear elements.

Review Questions

• How does the definition of hyperbolic tangent relate to its component functions, hyperbolic sine and cosine?
  • The hyperbolic tangent function is defined as the ratio of the hyperbolic sine and hyperbolic cosine functions, expressed as $$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$$. This relationship shows how tanh captures the behavior of both sine and cosine in a single function. As such, understanding how sinh and cosh work provides insight into how tanh behaves in different scenarios.
• What properties make the hyperbolic tangent function significant in calculus and physics?
  • The hyperbolic tangent function has several important properties that make it significant. It is an odd function, meaning it exhibits symmetry about the origin. Its derivative is useful in calculus for solving various differential equations. Additionally, its range between -1 and 1 allows it to model phenomena like signal attenuation in engineering, making it applicable in practical scenarios.
• Evaluate the implications of using the hyperbolic tangent function in modeling real-world systems compared to linear functions.
  • Using the hyperbolic tangent function for modeling real-world systems allows for capturing nonlinear behaviors that linear functions cannot represent effectively. For instance, while linear models may oversimplify relationships between variables, tanh can better describe systems that exhibit saturation effects or transition states. In fields like engineering or physics, this leads to more accurate predictions and analyses of complex behaviors compared to traditional linear approximations.

Related terms

hyperbolic sine: The hyperbolic sine function, denoted as 'sinh', is defined as $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$, representing the vertical coordinate of a point on a hyperbola.

hyperbolic cosine: The hyperbolic cosine function, denoted as 'cosh', is defined as $$\cosh(x) = \frac{e^x + e^{-x}}{2}$$, representing the horizontal coordinate of a point on a hyperbola.

inverse hyperbolic functions: Inverse hyperbolic functions are functions that reverse the action of hyperbolic functions, such as $$\text{artanh}(x)$$ for the hyperbolic tangent.