Trigonometric functions hyperbolic

The ordinary trigonometric functions include the sine, the cosine, the tangent, the cotangent, the secant, and the cosecant. These are sometimes called the circular trigonometric Junctions to distinguish them from the hyperbolic trigonometric functions discussed briefly in the next section of this chapter. 

The other hyperbolic trigonometric functions are the hyperbolic tangent, denoted by tanh(x) the hyperbolic cotangent, denoted by coth(x) the hyperbolic secant, denoted by sech(x) and the hyperbolic cosecant, denoted by csch(x). These functions are given by the equations... 

In this chapter we have introduced symbolic mathematics, which involves the manipulation of symbols instead of performing numerical operations. We have presented the algebraic tools needed to manipulate expressions containing real scalar variables, real vector variables, and complex scalar variables. We have also introduced ordinary and hyperbolic trigonometric functions, exponentials, and logarithms. A brief introduction to the techniques of problem solving was included. 

Hyperbolic trigonometric functions sinh, cosh, tanh.369... 

Substituting these values for Cj and C2 into equation (ix) and writing the exponential terms as hyperbolic trigonometric functions gives... 

The physical property options are labeled as thermo, fluid package, property package, or databank in common process simulators. There are pure-conponent and mixture sections, as well as a databank. For temperature-dependent properties, different functional forms are used (from extended Antione equation to polynomial to hyperbolic trigonometric functions). The equation appears on the physical property screen or in the help utility. 

Figure 2.10 shows the hyperbolic sine and hyperbolic cosine for values of x from 0 to 3. Note that the values of the hyperbolic sin and the hyperbolic cosine do not necessarily lie between -1 and 1 as do the values of the circular sine and cosine functions and that both functions approach e /2 for large values of x. The hyperbolic trigonometric functions are available in Excel. 

Directed Angles 27. Basic Trigonometric Functions 28. Radian Measure 28. Trigonometric Properties 29. Hyperbolic Functions 33. Polar Coordinate System 34. 

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

The trigonometric functions developed in the previous seefidtiare lef pd to as circidar functions, as they are related to the circle shown in Fig, 11, somewhat less familiar family of functions, the hyperbolic funefens, c also be derived from the exponential. They are analogous to the circular iom considered above and can be defined bv the relations... 

Special mathematical functions exponential, logarithm (base 10 and base e), trigonometric, reciprocal trigonometric and hyperbolic trigonometric. 

Hyperbolic functions are combinations of exponentials. They are given in Table A1.4, and these functions are plotted in Fig. A1.4. Since they are continuous functions, with continuous derivatives obtained in the same way as normal trigonometric functions, that is... 

To handle this equation, it is necessary, as above, to introduce the customary approximations into the hyperbolic and trigonometric functions appearing in it. Then, after long calculations, Eq. (17.168) can be written as... 

Note once more that the suppression of the inertial terms is equivalent to linearizing the hyperbolic and trigonometric functions appearing in the problem. 

Equations (1.80) and (1.81) provide relationships between complex variables and trigonometric functions. These can be manipulated to find relationships with hyperbolic function. Some important definitions and identities are presented in Table 1.6. ... 

The remaining four hyperbolic functions, analogous to the remaining four trigonometrical functions, are tanh u, cosech u, sech u and coth u. Values for each of these functions may be deduced from their relations with sinh u and cosh u. Thus,... 

In the following formulas u, v, w represent functions of x, while a, c, n represent fixed real numbers. All arguments in the trigonometric functions are measured in radians, and all inverse trigonometric and hyperbolic functions represent principal values Let y = f(x) and = f (x) define, respectively, a function and its derivative for any value x in their common domain. 

Hyperbolic functions are copycats of the corresponding trigonometric functions, in which the complex exponentials in Eqs. (4.51) and (4.52) are replaced by real exponential functions. The hyperbolic sine and hyperbolic cosine are defined, respectively, by... 

Analogous constmctions in Fig. 4.14 can then be used to represent the trigonometric functions sin 9 and cos 6 and the hyperbolic functions sinh t and cosh t. 

In [182] the authors presented an algorithm and a MATHEMATICA code for the conversion of formulae expressed in terms of the trigonometric functions sin(wx), cos(wx) or hyperbolic functions sinh(ct)x), cosh(ct)x) to formulae expressed in terms of functions. 

Similarly to the definition of the usual trigonometric functions, the hyperbolic functions are given by... 

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