Tangent, cotangent

Definitions of hyperbolic functions, their domains of definitions and values

sh x- hyperbolic sine
, -∞ < x < +∞; -∞ < y < +∞ .
ch x- hyperbolic cosine
, -∞ < x < +∞; 1 ≤ y< +∞ .
th x- hyperbolic tangent
, -∞ < x < +∞; - 1 < y < +1 .
cth x- hyperbolic cotangent
, x ≠ 0; y< -1 или y > +1 .

Graphs of hyperbolic functions

Plot of the hyperbolic sine y = sh x

Plot of the hyperbolic cosine y = ch x

Plot of the hyperbolic tangent y= th x

Plot of the hyperbolic cotangent y = cth x

Formulas with hyperbolic functions

Relationship with trigonometric functions

sin iz = i sh z ; cos iz = ch z
sh iz = i sin z ; ch iz = cos z
tgiz = i th z ; ctg iz = - i cth z
th iz = i tg z ; cth iz = - i ctg z
Here i is an imaginary unit, i 2 = - 1 .

Applying these formulas to trigonometric functions, we obtain formulas relating hyperbolic functions.

Parity

sh(-x) = - sh x; ch(-x) = ch x.
th(-x) = -th x; cth(-x) = - cth x.

Function ch(x)- even. Functions sh(x), th(x), cth(x)- odd.

Difference of squares

ch 2 x - sh 2 x = 1.

Formulas for sum and difference of arguments

sh(x y) = sh x ch y ch x sh y,
ch(x y) = ch x ch y sh x sh y,
,
,

sh 2 x = 2 sh x ch x,
ch 2 x = ch 2 x + sh 2 x = 2 ch 2 x - 1 = 1 + 2 sh 2 x,
.

Formulas for products of hyperbolic sine and cosine

,
,
,

,
,
.

Formulas for the sum and difference of hyperbolic functions

,
,
,
,
.

Relation of hyperbolic sine and cosine with tangent and cotangent

, ,
, .

Derivatives

,

Integrals of sh x, ch x, th x, cth x

,
,
.

Expansions into series

Inverse functions

Areasine

At - ∞< x < ∞ и - ∞ < y < ∞ имеют место формулы:
,
.

Areacosine

At 1 ≤ x< ∞ And 0 ≤ y< ∞ there are formulas:
,
.

The second branch of the areacosine is located at 1 ≤ x< ∞ and - ∞< y ≤ 0 :
.

Areatangent

At - 1 < x < 1 and - ∞< y < ∞ имеют место формулы:
,

Introduction

In mathematics and its applications to natural science and technology, exponential functions are widely used. This, in particular, is explained by the fact that many phenomena studied in natural science are among the so-called processes of organic growth, in which the rates of change of the functions participating in them are proportional to the values ​​of the functions themselves.

If denoted by a function, and by an argument, then the differential law of the process of organic growth can be written in the form where is some constant coefficient of proportionality.

Integration of this equation leads to common decision as an exponential function

If you set initial condition when, then it is possible to determine an arbitrary constant and, thus, to find a particular solution, which is an integral law of the process under consideration.

The processes of organic growth include, under some simplifying assumptions, such phenomena as, for example, changes in atmospheric pressure depending on the height above the Earth's surface, radioactive decay, cooling or heating of the body in environment constant temperature, unimolecular chemical reaction(for example, the dissolution of a substance in water), in which the law of mass action takes place (the reaction rate is proportional to the amount of the reactant), the reproduction of microorganisms, and many others.

The increase in the amount of money due to the accrual of compound interest on it (interest on interest) is also a process of organic growth.

These examples could be continued.

Along with individual exponential functions in mathematics and its applications, various combinations of exponential functions are used, among which certain linear and fractional-linear combinations of functions and the so-called hyperbolic functions are of particular importance. There are six of these functions, the following special names and designations have been introduced for them:

(hyperbolic sine),

(hyperbolic cosine),

(hyperbolic tangent),

(hyperbolic cotangent),

(hyperbolic secant),

(hyperbolic secant).

The question arises why exactly such names are given, and here is a hyperbole and the names of functions known from trigonometry: sine, cosine, etc.? It turns out that the relations connecting trigonometric functions with the coordinates of points of a circle of unit radius are similar to the relations connecting hyperbolic functions with the coordinates of points of an equilateral hyperbola with a unit semiaxis. This justifies the name of hyperbolic functions.

Hyperbolic functions

The functions given by formulas are called hyperbolic cosine and hyperbolic sine, respectively.

These functions are defined and continuous on, and is an even function and is an odd function.

Figure 1.1 - Graphs of functions

From the definition of hyperbolic functions it follows that:

By analogy with trigonometric functions, the hyperbolic tangent and cotangent are defined, respectively, by the formulas

A function is defined and continuous on, and a function is defined and continuous on a set with a punctured point; both functions are odd, their graphs are shown in the figures below.

Figure 1.2 - Graph of the function

Figure 1.3 - Graph of the function

It can be shown that the functions and are strictly increasing, while the function is strictly decreasing. Therefore, these functions are reversible. Denote the functions inverse to them, respectively, by.

Consider a function inverse to a function, i.e. function. We express it in terms of elementary ones. Solving the equation with respect to, we get Since, then, from where

Replacing with and with, we find the formula for the inverse function for the hyperbolic sine.

Along with the connection between trigonometric and exponential functions(Euler formulas)

in the complex domain there is such a very simple connection between trigonometric and hyperbolic functions.

Recall that, according to the definition:

If in identity (3) we replace by then on the right side we get the same expression that is on the right side of the identity, from which the equality of the left sides follows. The same holds for identities (4) and (2).

By dividing both parts of identity (6) into the corresponding parts of identity (5) and vice versa (5) by (6), we obtain:

A similar replacement in identities (1) and (2) and a comparison with identities (3) and (4) gives:

Finally, from identities (9) and (10) we find:

If we put in identities (5)-(12) where x is a real number, i.e. consider the argument purely imaginary, then we get eight more identities between the trigonometric functions of a purely imaginary argument and the corresponding hyperbolic functions of a real argument, as well as between hyperbolic functions of a purely imaginary imaginary Argument and the corresponding trigonometric functions of the real argument:

The obtained relations make it possible to pass from trigonometric functions to hyperbolic and from

hyperbolic functions to trigonometric ones with the replacement of the imaginary argument by the real one. They can be formulated as the following rule:

To move from trigonometric functions of an imaginary argument to hyperbolic ones or, conversely, from hyperbolic functions of an imaginary argument to trigonometric ones, one should take the imaginary unit out of the function sign for the sine and tangent, and discard it altogether for the cosine.

The connection established is remarkable, in particular, in that it makes it possible to obtain all relations between hyperbolic functions from known relations between trigonometric functions by replacing the latter by hyperbolic functions

Let's show how it is. is being done.

Take for example the basic trigonometric identity

and put in it where x is a real number; we get:

If in this identity we replace the sine and cosine by the hyperbolic sine and cosine according to the formulas, then we get or and this is the basic identity between the previously derived in a different way.

Similarly, you can derive all other formulas, including formulas for the hyperbolic functions of the sum and difference of arguments, double and half arguments, etc., thus, from ordinary trigonometry, get "hyperbolic trigonometry".