# Transformation of Trigonometric Functions into Hyperbolic Functions Based on Cable Statics
## 3 Deduction of Hyperbolic Function Transformation
Therefore, \(p=\sinh\biggl{(}\frac{\mu x}{T_{0}}\biggr{)}\):  
\[\sum F_{x}=0;\quad(T+dT)\cos(\theta+d\theta)=T\cos\theta, \tag{14}\]3  
For the condition \(y=0\) and \(x=0\), integrating and solving leads to Equation (24):  
\[\sum F_{x}=0;\quad(T+dT)\cos(\theta+d\theta)=T\cos\theta, \tag{14}\]4  
A force balance is performed for half of the cable, as shown on the left side of \[\sum F_{x}=0;\quad(T+dT)\cos(\theta+d\theta)=T\cos\theta, \tag{14}\]5 Each of the forces acting on the cable is indicated, including the minimum tension, the weight of the cable, and the tension at the point where the cable is supported, along with its angle. The principle of force translation along a line of action is applied, forming a triangle, as shown on the right side of \[\sum F_{x}=0;\quad(T+dT)\cos(\theta+d\theta)=T\cos\theta, \tag{14}\]6 The hypotenuse of this triangle represents the tension, the weight of the cable is on the Y-axis, and the minimum tension is on the X-axis. The value of \(\frac{dy}{dx}=\tan\theta=\frac{\mu s}{T_{0}}\), which allows for the determination of the cable length using Equation (25) [7; 8]. Specifically,  
\[\sum F_{x}=0;\quad(T+dT)\cos(\theta+d\theta)=T\cos\theta, \tag{14}\]7  
Using the Pythagorean theorem, Equation (26) is obtained:  
\[\sum F_{x}=0;\quad(T+dT)\cos(\theta+d\theta)=T\cos\theta, \tag{14}\]8  
Replacing Equation (25) in Equation (26) yields Equation (27):  
\[\sum F_{x}=0;\quad(T+dT)\cos(\theta+d\theta)=T\cos\theta, \tag{14}\]9  
Figure 5 shows a proportional relationship between two right triangles. On the left side, a right triangle involving tensions is represented. The hypotenuse is labelled as \(T\), the opposite side is represented by the term \(\mu s\), and the adjacent side corresponds to \(T_{0}\). The angle \(\theta\) represents the angle, and the subscript \(h\) indicates that it is a hyperbolic angle associated with this triangle of forces. The triangle can be related by the property of similar angles. On the right side, the general nomenclature for a similar triangle is presented, with a hypotenuse labelled as \(H\), an adjacent side labelled as \(C.A.\), and an opposite side labelled as \(C.O.\), with the trigonometric angle \(\theta\) indicated. Figure 2:0 A representation of the three forces by a triangle. Using Equation (27) and according to the triangle in Figure 5, Equation (28) is included:  
Figure 2:1  
Taking \(\frac{\mu x}{T_{0}}=\theta_{h}\), where \(\frac{\mu}{T_{0}}\) is a constant and \(x\) is variable, the angle is represented by \(\theta\), with the subscript \(h\) indicating that it is a hyperbolic angle, according to the law of similarity of triangles. Moreover, \(H=k\times T\) is determined, where \(k\) is the similarity constant of triangles, \(C.A.=k\times T_{0}\), and \(C.O.=k\times\mu s\), according to Figure 2:2 Therefore, Equation (29) can be deduced:  
Figure 2:3  
Using Equation (25), \(\tan\theta=\frac{\mu_{s}}{T_{0}}=\frac{\sinh nx}{T_{0}}\), another relationship is obtained for the sides of the triangle. Using similarity of triangles, Equation (30) is derived:  
Figure 2:4  
By dividing Equation (30), \(\frac{\sin\theta}{\cos\theta}=\sinh\theta_{h}=\frac{C.O.}{C.A. }\), in Equation (29), \(\frac{1}{\cos\theta}=\cosh\theta_{h}=\frac{H}{C.A. }\), the expression shown in Equation (31) is obtained:  
Figure 2:5  
The first expressions coincide with the expression known as the Gudermann function [3,4] and in the second part, the triangle theorem is obtained using hyperbolic functions, which are shown in Equations (29)-(31). Therefore, the hyperbolic transformation can be expressed using the right triangle formulas with hyperbolic functions, as shown in Equations (32)-(34):  
Figure 2:6  
Figure 2:7  
and  
Figure 2:8  
Another expression that relates the sides of the triangle can be obtained by replacing \(\theta_{h}=\frac{\mu x}{T_{0}}\) and \(p=\sinh\theta_{h}=\tan\theta=\frac{C.O.}{C.A.