Let's deal with simple concepts: sine and cosine and calculation cosine squared and sine squared.

Sine and cosine are studied in trigonometry (the study of right-angle triangles).

Therefore, first, let’s remember the basic concepts of a right triangle:

Hypotenuse- the side that always lies opposite right angle(90 degree angle). The hypotenuse is the longest side of a right angle triangle.

The remaining two sides in a right triangle are called legs.

You should also remember that three angles in a triangle always add up to 180°.

Now let's move on to cosine and sine of the angle alpha (∠α)(this can be called any indirect angle in a triangle or used as a designation x - "x", which does not change the essence).

Sine of angle alpha (sin ∠α)- this is an attitude opposite leg (the side opposite the corresponding angle) to the hypotenuse. If you look at the figure, then sin ∠ABC = AC / BC

Cosine of angle alpha (cos ∠α)- attitude adjacent to the angle of the leg to the hypotenuse. Looking again at the figure above, cos ∠ABC = AB / BC

And just as a reminder: cosine and sine will never be greater than one, since any roll is shorter than the hypotenuse (and the hypotenuse is the longest side of any triangle, because the longest side is located opposite the largest angle in the triangle).

Cosine squared, sine squared

Now let's move on to the basic trigonometric formulas: calculating cosine squared and sine squared.

To calculate them, you should remember the basic trigonometric identity:

sin 2 α + cos 2 α = 1(sine square plus cosine square of one angle always equals one).

From the trigonometric identity we draw conclusions about the sine:

sin 2 α = 1 - cos 2 α

sine square alpha equal to one minus cosine double angle alpha and divide it all by two.

sin 2 α = (1 – cos(2α)) / 2

​​​​​​​From the trigonometric identity we draw conclusions about the cosine:

cos 2 α = 1 - sin 2 α

or more difficult option formulas: cosine square alpha is equal to one plus the cosine of the double angle alpha and also divide everything by two.

cos 2 α = (1 + cos(2α)) / 2

These two are more complex formulas sine squared and cosine squared are also called “reducing the degree for squares of trigonometric functions.” Those. there was a second degree, they lowered it to the first and the calculations became more convenient.

Table of values ​​of trigonometric functions

Note. This table of trigonometric function values ​​uses the √ sign to indicate square root. To indicate a fraction, use the symbol "/".

See also useful materials:

For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, sine 30 degrees - we look for the column with the heading sin (sine) and find the intersection of this table column with the row “30 degrees”, at their intersection we read the result - one half. Similarly we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin column and the 60 degree line we find the value sin 60 = √3/2), etc. The values ​​of sines, cosines and tangents of other “popular” angles are found in the same way.

Sine pi, cosine pi, tangent pi and other angles in radians

The table below of cosines, sines and tangents is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the angle of 60 degrees in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi unambiguously expresses the dependence of the circumference on degree measure corner. Thus, pi radians are equal to 180 degrees.

Any number expressed in terms of pi (radians) can be easily converted to degrees by replacing pi (π) with 180.

Examples:
1. Sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and it is equal to zero.

2. Cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and it is equal to minus one.

3. Tangent pi
tg π = tg 180 = 0
thus, tangent pi is the same as tangent 180 degrees and it is equal to zero.

Table of sine, cosine, tangent values ​​for angles 0 - 360 degrees (common values)

If in the table of values ​​of trigonometric functions a dash is indicated instead of the function value (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle the function does not have a specific value. If there is no dash, the cell is empty, which means we have not yet entered desired value. We are interested in what queries users come to us for and supplement the table with new values, despite the fact that current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is quite sufficient to solve most problems.

Table of values ​​of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numeric values ​​“as per Bradis tables”)

One of the areas of mathematics that students struggle with the most is trigonometry. No wonder: in order to master this area of ​​knowledge fluently, you need to have spatial thinking, ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, be able to use pi in calculations. In addition, you need to be able to use trigonometry when proving theorems, and this requires either a developed mathematical memory, or the ability to derive complex logical chains.

Origins of trigonometry

Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.

Historically, the main object of study in this section mathematical science were right triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values ​​of all parameters of the figure in question using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy and even in art.

Initial stage

Initially, people talked about the relationship between angles and sides solely using the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in everyday life this branch of mathematics.

The study of trigonometry in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract trigonometric equations, which begin in high school.

Spherical trigonometry

Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence at least because earth's surface, and the surface of any other planet is convex, which means that any surface marking will be in three-dimensional space"arc-shaped".

Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.

Right triangle

Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.

The first step is to understand the concepts related to right triangle. First, the hypotenuse is the side opposite the 90 degree angle. It is the longest. We remember that according to the Pythagorean theorem, its numerical value equal to the root of the sum of the squares of the other two sides.

For example, if the two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.

The two remaining sides, which form a right angle, are called legs. In addition, we must remember that the sum of the angles in a triangle is rectangular system coordinates is 180 degrees.

Definition

Finally, with a firm understanding of the geometric basis, one can turn to the definition of sine, cosine and tangent of an angle.

The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio adjacent leg to the hypotenuse.

Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.

Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.

Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.

So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.

The simplest formulas

In trigonometry you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.

The first formula you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you need to know the size of the angle rather than the side.

Many students cannot remember the second formula, which is also very popular when solving school tasks: the sum of one and the square of the tangent of the angle is equal to one divided by the square of the cosine of the angle. Take a closer look: this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation does trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, conversion rules and several basic formulas You can at any time derive the required more complex formulas on a sheet of paper yourself.

Formulas for double angles and addition of arguments

Two more formulas that you need to learn are related to the values ​​of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of sine and cosine is added.

There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a training try to get them yourself by taking the alpha angle equal to the angle beta.

Finally, note that double angle formulas can be rearranged to reduce the power of sine, cosine, tangent alpha.

Theorems

The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of ​​the figure, and the size of each side, etc.

The sine theorem states that by dividing the length of each side of a triangle by the opposite angle, we get same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a given triangle.

The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the adjacent angle - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.

Careless mistakes

Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.

Firstly, you shouldn't convert fractions to decimals until you get the final result - you can leave the answer as common fraction, unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will be wasting your time on unnecessary mathematical operations. This is especially true for values ​​such as the root of three or the root of two, because they are found in problems at every step. The same goes for rounding “ugly” numbers.

Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but you will also demonstrate a complete lack of understanding of the subject. This is worse than a careless mistake.

Thirdly, do not confuse the values ​​for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because sine is 30 degrees equal to cosine 60, and vice versa. It is easy to confuse them, as a result of which you will inevitably get an erroneous result.

Application

Many students are in no hurry to start studying trigonometry because they do not understand its practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.

In conclusion

So you're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.

The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: length three sides and the sizes of the three angles. The only difference in the tasks lies in the fact that different input data are given.

You now know how to find sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a ratio, and a ratio is a fraction, main goal trigonometric problem is finding the roots of an ordinary equation or a system of equations. And here regular school mathematics will help you.

In this article we will take a comprehensive look. Basic trigonometric identities are equalities that establish a connection between the sine, cosine, tangent and cotangent of one angle, and allow one to find any of these trigonometric functions through a known other.

Let us immediately list the main trigonometric identities that we will analyze in this article. Let's write them down in a table, and below we'll give the output of these formulas and provide the necessary explanations.

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Relationship between sine and cosine of one angle

Sometimes they do not talk about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind . The explanation for this fact is quite simple: the equalities are obtained from the main trigonometric identity after dividing both of its parts by and respectively, and the equalities And follow from the definitions of sine, cosine, tangent and cotangent. We'll talk about this in more detail in the following paragraphs.

That is, special interest represents precisely the equality, which was given the name of the main trigonometric identity.

Before proving the main trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.

The basic trigonometric identity is very often used when transformation trigonometric expressions . It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often the basic trigonometric identity is used in reverse order: unit is replaced by the sum of the squares of the sine and cosine of any angle.

Tangent and cotangent through sine and cosine

Identities connecting tangent and cotangent with sine and cosine of one angle of view and follow immediately from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, sine is the ordinate of y, cosine is the abscissa of x, tangent is the ratio of the ordinate to the abscissa, that is, , and the cotangent is the ratio of the abscissa to the ordinate, that is, .

Thanks to such obviousness of the identities and Tangent and cotangent are often defined not through the ratio of abscissa and ordinate, but through the ratio of sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.

In conclusion of this paragraph, it should be noted that the identities and take place for all angles at which the trigonometric functions included in them make sense. So the formula is valid for any , other than (otherwise the denominator will have zero, and we did not define division by zero), and the formula - for all , different from , where z is any .

Relationship between tangent and cotangent

Even more obvious trigonometric identity than the previous two, is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it holds for any angles other than , otherwise either the tangent or the cotangent are not defined.

Proof of the formula very simple. By definition and from where . The proof could have been carried out a little differently. Since , That .

So, the tangent and cotangent of the same angle at which they make sense are .

If we construct a unit circle with the center at the origin, and set an arbitrary value for the argument x 0 and count from the axis Ox corner x 0, then this corner on unit circle corresponds to some point A(Fig. 1) and its projection onto the axis Oh there will be a point M. Section length OM equal to absolute value abscissa dots A. Given argument value x 0 function value mapped y=cos x 0 like abscissa dots A. Accordingly, point IN(x 0 ;at 0) belongs to the graph of the function at=cos X(Fig. 2). If the point A is to the right of the axis Oh, The current sine will be positive, but if to the left it will be negative. But anyway, period A cannot leave the circle. Therefore, the cosine lies in the range from –1 to 1:

–1 = cos x = 1.

Additional rotation at any angle, multiple of 2 p, returns point A to the same place. Therefore the function y = cos xp:

cos( x+ 2p) = cos x.

If we take two values ​​of the argument, equal in absolute value, but opposite in sign, x And - x, find the corresponding points on the circle A x And A -x. As can be seen in Fig. 3 their projection onto the axis Oh is the same point M. That's why

cos(– x) = cos ( x),

those. cosine – even function, f(–x) = f(x).

This means we can explore the properties of the function y=cos X on the segment , and then take into account its parity and periodicity.

At X= 0 point A lies on the axis Oh, its abscissa is 1, and therefore cos 0 = 1. With increasing X dot A moves around the circle up and to the left, its projection, naturally, is only to the left, and at x = p/2 cosine becomes equal to 0. Point A at this moment rises to maximum height, and then continues to move to the left, but already descending. Its abscissa keeps decreasing until it reaches lowest value, equal to –1 at X= p. Thus, on the interval the function at=cos X decreases monotonically from 1 to –1 (Fig. 4, 5).

From the parity of the cosine it follows that on the interval [– p, 0] the function increases monotonically from –1 to 1, taking a zero value at x =–p/2. If you take several periods, you get a wavy curve (Fig. 6).

So the function y=cos x takes zero values ​​at points X= p/2 + kp, Where k – any integer. Maximums equal to 1 are achieved at points X= 2kp, i.e. in steps of 2 p, and minimums equal to –1 at points X= p + 2kp.

Function y = sin x.

On a unit circle angle x 0 corresponds to a dot A(Fig. 7), and its projection onto the axis Oh there will be a point N.Z function value y 0 = sin x 0 defined as the ordinate of a point A. Dot IN(corner x 0 ,at 0) belongs to the graph of the function y= sin x(Fig. 8). It is clear that the function y= sin x periodic, its period is 2 p:

sin( x+ 2p) = sin ( x).

For two argument values, X And - , projections of their corresponding points A x And A -x per axis Oh located symmetrically relative to the point ABOUT. That's why

sin(– x) = –sin ( x),

those. sine is an odd function, f(– x) = –f( x) (Fig. 9).

If the point A rotate relative to a point ABOUT at an angle p/2 counterclockwise (in other words, if the angle X increase by p/2), then its ordinate in the new position will be equal to the abscissa in the old one. Which means

sin( x+ p/2) = cos x.

Otherwise, the sine is a cosine “late” by p/2, since any cosine value will be “repeated” in the sine when the argument increases by p/2. And to build a sine graph, it is enough to shift the cosine graph by p/2 to the right (Fig. 10). Extremely important property sine is expressed by equality

The geometric meaning of equality can be seen from Fig. 11. Here X - this is half an arc AB, a sin X - half of the corresponding chord. It is obvious that as the points get closer A And IN the length of the chord is increasingly approaching the length of the arc. From the same figure it is easy to derive the inequality

|sin x| x|, true for any X.

Mathematicians call formula (*) remarkable limit. From it, in particular, it follows that sin X» X at small X.

Functions at= tg x, y=ctg X. The other two trigonometric functions, tangent and cotangent, are most easily defined as the ratios of the sine and cosine already known to us:

Like sine and cosine, tangent and cotangent are periodic functions, but their periods are equal p, i.e. they are half the size of sine and cosine. The reason for this is clear: if sine and cosine both change signs, then their ratio will not change.

Since the denominator of the tangent contains a cosine, the tangent is not defined at those points where the cosine is 0 - when X= p/2 +kp. At all other points it increases monotonically. Direct X= p/2 + kp for tangent are vertical asymptotes. At points kp tangent and slope are 0 and 1, respectively (Fig. 12).

The cotangent is not defined where the sine is 0 (when x = kp). At other points it decreases monotonically, and straight lines x = kp – his vertical asymptotes. At points x = p/2 +kp the cotangent becomes 0, and the slope at these points is equal to –1 (Fig. 13).

Parity and periodicity.

A function is called even if f(–x) = f(x). The cosine and secant functions are even, and the sine, tangent, cotangent and cosecant functions are odd:

Parity properties follow from the symmetry of points P a and R-a (Fig. 14) relative to the axis X. With such symmetry, the ordinate of the point changes sign (( X;at) goes to ( X; –у)). All functions - periodic, sine, cosine, secant and cosecant have a period of 2 p, and tangent and cotangent - p:

The periodicity of sine and cosine follows from the fact that all points P a+2 kp, Where k= 0, ±1, ±2,…, coincide, and the periodicity of the tangent and cotangent is due to the fact that the points P a+ kp alternately fall into two diametrically opposite points of the circle, giving the same point on the tangent axis.

The main properties of trigonometric functions can be summarized in a table:

Reduction formulas.

According to these formulas, the value of the trigonometric function of the argument a, where p/2 a p , can be reduced to the value of the argument function a , where 0 a p /2, either the same or complementary to it.

Argument b	-a	+a	p-a	p+a	+a	+a	2p-a
sin b	cos a	cos a	sin a	–sin a	–cos a	–cos a	–sin a
cos b	sin a	–sin a	–cos a	–cos a	–sin a	sin a	cos a

Therefore, in the tables of trigonometric functions, values ​​are given only for sharp corners, and it is enough to limit ourselves, for example, to sine and tangent. The table shows only the most commonly used formulas for sine and cosine. From these it is easy to obtain formulas for tangent and cotangent. When casting a function from an argument of the form kp/2 ± a, where k– an integer, to a function of the argument a:

1) the function name is saved if k even, and changes to "complementary" if k odd;

2) the sign on the right side coincides with the sign of the reducible function at the point kp/2 ± a if angle a is acute.

For example, when casting ctg (a – p/2) we make sure that a – p/2 at 0 a p /2 lies in the fourth quadrant, where the cotangent is negative, and, according to rule 1, we change the name of the function: ctg (a – p/2) = –tg a .

Addition formulas.

Formulas for multiple angles.

These formulas are derived directly from the addition formulas:

sin 2a = 2 sin a cos a ;

cos 2a = cos 2 a – sin 2 a = 2 cos 2 a – 1 = 1 – 2 sin 2 a ;

sin 3a = 3 sin a – 4 sin 3 a ;

cos 3a = 4 cos 3 a – 3 cos a;

The formula for cos 3a was used by François Viète when solving cubic equation. He was the first to find expressions for cos n a and sin n a, which were later obtained in a simpler way from Moivre's formula.

If you replace a with a /2 in double argument formulas, they can be converted to half angle formulas:

Universal substitution formulas.

Using these formulas, an expression involving different trigonometric functions of the same argument can be rewritten as rational expression from one function tg (a /2), this can be useful when solving some equations:

Formulas for converting sums into products and products into sums.

Before the advent of computers, these formulas were used to simplify calculations. Calculations were made using logarithmic tables, and later - slide rule, because logarithms are best suited for multiplying numbers, so all the original expressions were brought to a form convenient for logarithmization, i.e. to works, for example:

2 sin a sin b = cos ( a–b) – cos ( a+b);

2cos a cos b=cos( a–b) + cos ( a+b);

2 sin a cos b= sin( a–b) + sin ( a+b).

Formulas for the tangent and cotangent functions can be obtained from the above.

Degree reduction formulas.

From the multiple argument formulas the following formulas are derived:

Using these formulas trigonometric equations can be reduced to equations of lower degrees. In the same way, we can derive reduction formulas for more high degrees sine and cosine.

Each trigonometric function at each point of its domain of definition is continuous and infinitely differentiable. Moreover, the derivatives of trigonometric functions are trigonometric functions, and when integrated, trigonometric functions or their logarithms are also obtained. Integrals from rational combinations of trigonometric functions are always elementary functions.

Representation of trigonometric functions in the form of power series and infinite products.

All trigonometric functions can be expanded in power series. In this case, the functions sin x bcos x are presented in rows. convergent for all values x:

These series can be used to obtain approximate expressions for sin x and cos x at small values x:

at | x| p/2;

at 0 x| p

(B n – Bernoulli numbers).

sin functions x and cos x can be represented in the form of infinite products:

Trigonometric system 1, cos x,sin x, cos 2 x, sin 2 x,¼,cos nx,sin nx, ¼, forms on the segment [– p, p] orthogonal system functions, which makes it possible to represent functions in the form of trigonometric series.

are defined as analytic continuations of the corresponding trigonometric functions of the real argument in complex plane. Yes, sin z and cos z can be defined using series for sin x and cos x, if instead x put z:

These series converge over the entire plane, so sin z and cos z- entire functions.

Tangent and cotangent are determined by the formulas:

tg functions z and ctg z– meromorphic functions. tg poles z and sec z– simple (1st order) and located at points z = p/2 + pn, poles ctg z and cosec z– also simple and located at points z = p n, n = 0, ±1, ±2,…

All formulas that are valid for trigonometric functions of a real argument are also valid for a complex one. In particular,

sin(– z) = –sin z,

cos(– z) = cos z,

tg(– z) = –tg z,

ctg(– z) = –ctg z,

those. even and odd parity are preserved. Formulas are also saved

sin( z + 2p) = sin z, (z + 2p) = cos z, (z + p) = tg z, (z + p) = ctg z,

those. periodicity is also preserved, and the periods are the same as for functions of a real argument.

Trigonometric functions can be expressed through an exponential function of a purely imaginary argument:

Back, e iz expressed in terms of cos z and sin z according to the formula:

e iz=cos z + i sin z

These formulas are called Euler's formulas. Leonhard Euler developed them in 1743.

Trigonometric functions can also be expressed in terms of hyperbolic functions:

z = –i sh iz, cos z = ch iz, z = –i th iz.

where sh, ch and th – hyperbolic sine, cosine and tangent.

Trigonometric functions of complex argument z = x + iy, Where x And y – real numbers, can be expressed through trigonometric and hyperbolic functions of real arguments, for example:

sin( x + iy) = sin x ch y + i cos x sh y;

cos( x + iy) = cos x ch y + i sin x sh y.

The sine and cosine of a complex argument can take real values, exceeding 1 in absolute value. For example:

If an unknown angle enters an equation as an argument of trigonometric functions, then the equation is called trigonometric. Such equations are so common that their methods the solutions are very detailed and carefully designed. WITH with help various techniques and formulas reduce trigonometric equations to equations of the form f(x)=a, Where f– any of the simplest trigonometric functions: sine, cosine, tangent or cotangent. Then express the argument x this function through its known value A.

Since trigonometric functions are periodic, the same A from the range of values ​​there are infinitely many values ​​of the argument, and the solutions to the equation cannot be written as a single function of A. Therefore, in the domain of definition of each of the main trigonometric functions, a section is selected in which it takes all its values, each only once, and the function inverse to it is found in this section. Such functions are denoted by adding the prefix arc (arc) to the name of the original function, and are called inverse trigonometric functions or simply arc functions.

Inverse trigonometric functions.

For sin X, cos X, tg X and ctg X inverse functions can be defined. They are denoted accordingly by arcsin X(read "arcsine" x"), arcos x, arctan x and arcctg x. By definition, arcsin X there is such a number y, What

sin at = X.

Similarly for other inverse trigonometric functions. But this definition suffers from some inaccuracy.

If you reflect sin X, cos X, tg X and ctg X relative to the bisector of the first and third quadrants coordinate plane, then the functions, due to their periodicity, become ambiguous: the same sine (cosine, tangent, cotangent) corresponds to infinite number corners

To get rid of ambiguity, a section of the curve with a width of p, in this case it is necessary that a one-to-one correspondence be maintained between the argument and the value of the function. Areas near the origin of coordinates are selected. For sine in As a “one-to-one interval” we take the segment [– p/2, p/2], on which the sine monotonically increases from –1 to 1, for the cosine – the segment, for the tangent and cotangent, respectively, the intervals (– p/2, p/2) and (0, p). Each curve on the interval is reflected relative to the bisector and now inverse trigonometric functions can be determined. For example, let the argument value be given x 0 , such that 0 Ј x 0 Ј 1. Then the value of the function y 0 = arcsin x 0 there will be only one meaning at 0 , such that - p/2 Ј at 0 Ј p/2 and x 0 = sin y 0 .

Thus, arcsine is a function of arcsin A, defined on the interval [–1, 1] and equal for each A to such a value a , – p/2 a p /2 that sin a = A. It is very convenient to represent it using a unit circle (Fig. 15). When | a| 1 on a circle there are two points with ordinate a, symmetrical about the axis u. One of them corresponds to the angle a= arcsin A, and the other is the corner p - a. WITH taking into account the periodicity of the sine, the solution sin equations x= A is written as follows:

x =(–1)n arcsin a + 2p n,

Where n= 0, ±1, ±2,...

Other simple trigonometric equations can be solved in the same way:

cos x = a, –1 =a= 1;

x =±arcos a + 2p n,

Where n= 0, ±1, ±2,... (Fig. 16);

tg X = a;

x= arctan a + p n,

Where n = 0, ±1, ±2,... (Fig. 17);

ctg X= A;

X= arcctg a + p n,

Where n = 0, ±1, ±2,... (Fig. 18).

Basic properties of inverse trigonometric functions:

arcsin X(Fig. 19): domain of definition – segment [–1, 1]; range – [– p/2, p/2], monotonically increasing function;

arccos X(Fig. 20): domain of definition – segment [–1, 1]; range – ; monotonically decreasing function;

arctg X(Fig. 21): domain of definition – all real numbers; range of values ​​– interval (– p/2, p/2); monotonically increasing function; straight at= –p/2 and y = p /2 – horizontal asymptotes;

arcctg X(Fig. 22): domain of definition – all real numbers; range of values ​​– interval (0, p); monotonically decreasing function; straight y= 0 and y = p– horizontal asymptotes.

For anyone z = x + iy, Where x And y are real numbers, inequalities apply

½| e\e y–e-y| ≤|sin z|≤½( e y +e-y),

½| e y–e-y| ≤|cos z|≤½( e y +e -y),

of which at y® Ґ asymptotic formulas follow (uniformly with respect to x)

|sin z| » 1/2 e |y| ,

|cos z| » 1/2 e |y| .

Trigonometric functions first appeared in connection with research in astronomy and geometry. The ratios of segments in a triangle and a circle, which are essentially trigonometric functions, are found already in the 3rd century. BC e. in the works of mathematicians of Ancient Greece – Euclid, Archimedes, Apollonius of Perga and others, however, these relations were not an independent object of study, so they did not study trigonometric functions as such. They were initially considered as segments and in this form were used by Aristarchus (late 4th - 2nd half of the 3rd centuries BC), Hipparchus (2nd century BC), Menelaus (1st century AD). ) and Ptolemy (2nd century AD) when solving spherical triangles. Ptolemy compiled the first table of chords for acute angles every 30" with an accuracy of 10 -6. This was the first table of sines. As a ratio sin function a is found already in Aryabhata (late 5th century). The functions tg a and ctg a are found in al-Battani (2nd half of the 9th – early 10th centuries) and Abul-Wef (10th century), who also uses sec a and cosec a. Aryabhata already knew the formula (sin 2 a + cos 2 a) = 1, and also sin formulas and cos of a half angle, with the help of which I built tables of sines for angles every 3°45"; based on known values trigonometric functions for the simplest arguments. Bhaskara (12th century) gave a method for constructing tables in terms of 1 using addition formulas. Formulas for converting the sum and difference of trigonometric functions of various arguments into a product were derived by Regiomontanus (15th century) and J. Napier in connection with the latter’s invention of logarithms (1614). Regiomontan gave a table of sine values ​​in 1". The expansion of trigonometric functions into power series was obtained by I. Newton (1669). In modern form the theory of trigonometric functions was introduced by L. Euler (18th century). He owns their definition for real and complex arguments, the currently accepted symbolism, establishing a connection with exponential function and orthogonality of the system of sines and cosines.