# Transformation of Trigonometric Functions into Hyperbolic Functions Based on Cable Statics
## 2 Background  
The Background Section provides a general literature review to demonstrate the novelty of the proposed approach. Various mathematical tools have been developed since the 17 th century with great discoveries and scientific contributions. Scientists and mathematicians observed that certain combinations of exponential functions were frequently presented in applications such as the decay of entities such as light, electricity, speed, radioactivity, the fall of a body through a viscous medium [8], the diffusion of a gas in a porous medium, the transmission of heat [3], geodesic triangles [11] and so on. At the beginning of the 18 th century, in an attempt to broaden the concept of imaginary and complex, the functions \(\sin\varphi\) and \(\cos\varphi\) were defined by complex exponential addition and subtraction, and [PERSON] obtained the following expressions for \(\cos\varphi\) and \(\sin\varphi\) as shown in [2,12]:  
\[\cos\varphi=\frac{e^{i\varphi}+e^{-i\varphi}}{2};\sin\varphi=\frac{e^{i\varphi }-e^{-i\varphi}}{2i}. \tag{1}\]  
The success and recognition obtained by [PERSON] for rediscovering the Cotes formula is not entirely undeserved [2]. These combinations of addition and subtraction of exponentials can be deduced from a hyperbole [7,8]; this is how real numbers are related to imaginary and complex numbers. These identities, shown in Equation (2) as (hyperbolic sine of an angle) \(\sinh i\varphi\) and (hyperbolic cosine of an angle) \(\cosh i\varphi\)[3,11,12], have been related in an analogous way to the trigonometric functions for a hyperbolic angle \(x\) with identities such as (hyperbolic sine) \(\sinh x\), (hyperbolic cosine) \(\cosh x\), (hyperbolic tangent) \(\tanh x\), (hyperbolic cosecant) \(\cosh x\), (hyperbolic secant) \(x\), and (hyperbolic cotangent) \(\coth x\)[3,11,13,14,15,16]. Equation (2) establishes a relationship with Equation (1) using hyperbolic functions. Thus, the similarity between trigonometric and hyperbolic functions is proven [11,15,17,18,19], as shown below:  
\[\cosh i\varphi=\frac{e^{i\varphi}+e^{-i\varphi}}{2};\quad\sinh i\varphi=\frac{ e^{i\varphi}-e^{-i\varphi}}{2}. \tag{2}\]  
Hyperbolic functions have their name because the geometry with which they are built is defined on a hyperbole with trigonometric functions, also called circular functions, where their geometry is described with the shape of a circle [14,15]. Hyperbolic functions describe one side of the hyperbola; they are also called aperiodic functions, and the angle of the hyperbola can vary from 0 to \(\infty\)[14]. Hyperbolic functions can be derived because they are related to curve geometries such as hyperbolas, ellipses, and catenaries and are frequently used to describe real life applications based on engineering concepts. There are several identities for relating hyperbolic functions; one of the most fundamental is shown in [11,13,14,15,17]:  
\[\cosh^{2}x-\sinh^{2}x=1. \tag{3}\]  
Since hyperbolic functions are derived from the sum of exponentials, their inverse functions can be formulated using logarithmic functions, as shown in Equations (4) and (5) [11,13,17,18,19]. In these equations, the intervals indicate the domain of the inverse functions, i.e., the set of x values for which the function is defined, not the codomain. Specifically, Equation (4) applies for all \(x\in(-\infty,\infty)\), and Equation (5) applies for \(x\in[1,\infty)\)[11,13,17,18,19], as illustrated bellow:  
\[\sinh^{-1}x=\ln\Big{(}x+\sqrt{1+x^{2}}\Big{)},\quad x\in(-\infty,\infty), \tag{4}\]  
and  
\[\cosh^{-1}x=\ln\Big{(}x+\sqrt{x^{2}-1}\Big{)},\quad x\in[1,\infty). \tag{5}\]Derivatives and integrals can be calculated by writing the hyperbolic functions in terms of exponential functions. The derivatives and integrals generally yield another hyperbolic function, as presented in Equations (6) and (7) [13; 14; 16].