COTH

Application:

The hyperbolic cotangent function, often written as coth(x), is a mathematical function defined as the ratio of the hyperbolic cosine to the hyperbolic sine:

$\coth (x)=\frac{\cosh (x)}{\sinh (x)}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$

An application of the coth function can be found in the shape of a hanging cable or chain, known as a catenary. While the catenary curve is most famously described by the cosh function, the coth function appears in the equations that describe the physical properties of the catenary. For instance, the slope of a hanging cable at any point is directly related to the coth function.

Imagine a power line hanging between two poles. The shape of this power line is a catenary. Let's set up a coordinate system where the y-axis passes through the lowest point of the cable. The equation for the curve is given by y=a⋅cosh(x/a), where 'a' is a parameter related to the tension and weight of the cable.

The slope of the cable at any point x is given by the derivative of the curve's equation, which is:

$\frac{dy}{dx}=a\cdot \sinh \left(\frac{x}{a}\right)\cdot \left(\frac{1}{a}\right)=\sinh \left(\frac{x}{a}\right)$

However, the more interesting application of coth comes when we consider the horizontal and vertical components of the tension in the cable. Let TH​ be the constant horizontal tension and TV​ be the vertical tension at any point. The vertical tension at a given point x is equal to the weight of the cable segment from the lowest point to x. The total tension T at any point on the cable is the vector sum of TH​ and TV​.

The angle θ that the tangent to the cable makes with the horizontal at a point x is given by:

$\tan (\theta )=\frac{dy}{dx}=\sinh \left(\frac{x}{a}\right)$

The total tension T at any point is given by T=TH​⋅cosh(x/a).

The vertical component of the tension is TV​=TH​⋅sinh(x/a).

The ratio of the total tension to the horizontal tension is:

T/TH​=cosh(x/a)

The ratio of the vertical tension to the horizontal tension at any point is:

TV​/TH​=sinh(x/a)=tan(θ)

The coth function emerges when we consider the ratio of the vertical component of the tension to the horizontal component of the tension relative to the slope, but it is more direct to use the coth function to describe the ratio of the total tension to the vertical tension, or to describe the ratio of the horizontal distance to the vertical height in a specific configuration.

Let's consider a practical application in electrical engineering. A transmission line conductor's sag, and therefore its tension, are critical for safety and performance. Engineers must calculate the sag to ensure the line does not come too close to the ground or other objects.

Let's create a simplified scenario with a power line and use a table to show how coth values might be used. We'll use coth(x/a) where x represents the horizontal distance from the lowest point of the sag and 'a' is a constant specific to the line's properties. In this case, coth(x/a) can be used to describe the ratio of certain physical properties. For example, let's say we have a specific configuration where the ratio of a particular force to another is governed by the coth function.