Cosine three x. What are sine and cosine

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 the unit circle corner 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, 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 –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 acute angles, 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 of 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 as 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 determined 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, CTG poles 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 in terms of 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 developed. 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 can be determined inverse functions. 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, – 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 of values ​​– ; 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 hold

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 he built tables of sines for angles through 3°45"; based on the known values ​​of trigonometric functions for the simplest arguments. Bhaskara (12th century) gave a method for constructing tables through 1 using addition formulas. Formulas for converting the sum and the differences 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 Regiomontanus gave a table of sine values ​​in 1" increments). 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.

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.

Page navigation.

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 type . The explanation for this fact is quite simple: the equalities are obtained from the basic trigonometric identity after dividing both its parts by and respectively, and the equality 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

An 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 .

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 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 main ones trigonometric formulas: Calculate 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.

Trigonometry - section mathematical science, which explores trigonometric functions and their use in geometry. The development of trigonometry began back in the days ancient Greece. During the Middle Ages, scientists from the Middle East and India made important contributions to the development of this science.

This article is dedicated to basic concepts and definitions of trigonometry. It discusses the definitions of the basic trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.

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Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of the ratio of the sides of a right triangle.

Definitions of trigonometric functions

The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.

Cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.

Angle tangent (t g α) - ratio opposite side to the adjacent one.

Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.

These definitions are given for acute angle right triangle!

Let's give an illustration.

IN triangle ABC with right angle C sine of angle A equal to the ratio leg BC to hypotenuse AB.

The definitions of sine, cosine, tangent and cotangent allow you to calculate the values ​​of these functions from the known lengths of the sides of the triangle.

Important to remember!

The range of values ​​of sine and cosine is from -1 to 1. In other words, sine and cosine take values ​​from -1 to 1. The range of values ​​of tangent and cotangent is the entire number line, that is, these functions can take on any values.

The definitions given above apply to acute angles. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to 0 to 90 degrees. The rotation angle in degrees or radians is expressed by any real number from - ∞ to + ∞.

IN in this context You can define sine, cosine, tangent and cotangent of an angle of arbitrary size. Let us imagine a unit circle with its center at the origin of the Cartesian coordinate system.

The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).

Sine (sin) of the rotation angle

The sine of the rotation angle α is the ordinate of point A 1 (x, y). sin α = y

Cosine (cos) of the rotation angle

The cosine of the rotation angle α is the abscissa of point A 1 (x, y). cos α = x

Tangent (tg) of the rotation angle

The tangent of the angle of rotation α is the ratio of the ordinate of point A 1 (x, y) to its abscissa. t g α = y x

Cotangent (ctg) of the rotation angle

The cotangent of the rotation angle α is the ratio of the abscissa of point A 1 (x, y) to its ordinate. c t g α = x y

Sine and cosine are defined for any rotation angle. This is logical, because the abscissa and ordinate of a point after rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is undefined when a point after rotation goes to a point with a zero abscissa (0, 1) and (0, - 1). In such cases, the expression for tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with cotangent. The difference is that the cotangent is not defined in cases where the ordinate of a point goes to zero.

Important to remember!

Sine and cosine are defined for any angles α.

Tangent is defined for all angles except α = 90° + 180° k, k ∈ Z (α = π 2 + π k, k ∈ Z)

Cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)

When deciding practical examples do not say "sine of the angle of rotation α". The words “angle of rotation” are simply omitted, implying that it is already clear from the context what is being discussed.

Numbers

What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?

Sine, cosine, tangent, cotangent of a number

Sine, cosine, tangent and cotangent of a number t is a number that is respectively equal to sine, cosine, tangent and cotangent in t radian.

For example, the sine of the number 10 π is equal to the sine of the rotation angle of 10 π rad.

There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's take a closer look at it.

Anyone real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.

The starting point on the circle is point A with coordinates (1, 0).

Positive number t

Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and will go the way t.

Now that the connection between a number and a point on a circle has been established, we move on to the definition of sine, cosine, tangent and cotangent.

Sine (sin) of t

Sine of a number t- ordinate of a point on the unit circle corresponding to the number t. sin t = y

Cosine (cos) of t

Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x

Tangent (tg) of t

Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t

The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. Point on the circle corresponding to the number t, coincides with the point to which the starting point goes after turning by an angle t radian.

Trigonometric functions of angular and numeric argument

Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except α = 180° k, k ∈ Z (α = π k, k ∈ Z).

We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.

Similarly, we can talk about sine, cosine, tangent and cotangent as functions numeric argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.

Basic functions of trigonometry

Sine, cosine, tangent and cotangent are the basic trigonometric functions.

It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.

Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. Trigonometric definitions sine, cosine, tangent and cotangent are completely consistent with geometric definitions, given using the aspect ratios of a right triangle. Let's show it.

Take a unit circle with center at a rectangular Cartesian system coordinates Let's turn it around starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa from the resulting point A 1 (x, y). In the resulting right triangle, angle A 1 O H equal to angle turn α, the length of the leg O H is equal to the abscissa of point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.

In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side to the hypotenuse.

sin α = A 1 H O A 1 = y 1 = y

This means that determining the sine of an acute angle in a right triangle through the aspect ratio is equivalent to determining the sine of the rotation angle α, with alpha lying in the range from 0 to 90 degrees.

Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.

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In this article we will show how to give definitions of sine, cosine, tangent and cotangent of an angle and number in trigonometry. Here we will talk about notations, give examples of entries, and give graphic illustrations. In conclusion, let us draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.

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Definition of sine, cosine, tangent and cotangent

Let's see how the idea of ​​sine, cosine, tangent and cotangent is formed in school course mathematics. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right triangle is given. And later trigonometry is studied, which talks about sine, cosine, tangent and cotangent of the angle of rotation and number. Let us present all these definitions, give examples and give the necessary comments.

Acute angle in a right triangle

From the geometry course we know the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle. They are given as the ratio of the sides of a right triangle. Let us give their formulations.

Definition.

Sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse.

Definition.

Cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.

Definition.

Tangent of an acute angle in a right triangle– this is the ratio of the opposite side to the adjacent side.

Definition.

Cotangent of an acute angle in a right triangle- this is the ratio of the adjacent side to the opposite side.

The designations for sine, cosine, tangent and cotangent are also introduced there - sin, cos, tg and ctg, respectively.

For example, if ABC is a right triangle with right angle C, then the sine of the acute angle A is equal to the ratio of the opposite side BC to the hypotenuse AB, that is, sin∠A=BC/AB.

These definitions allow you to calculate the values ​​of sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from known values find the lengths of the other sides using sine, cosine, tangent, cotangent and the length of one of the sides. For example, if we knew that in a right triangle the leg AC is equal to 3 and the hypotenuse AB is equal to 7, then we could calculate the value of the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7.

Rotation angle

In trigonometry, they begin to look at the angle more broadly - they introduce the concept of angle of rotation. The magnitude of the rotation angle, unlike an acute angle, is not limited to 0 to 90 degrees; the rotation angle in degrees (and in radians) can be expressed by any real number from −∞ to +∞.

In this light, the definitions of sine, cosine, tangent and cotangent are given not of an acute angle, but of an angle of arbitrary size - the angle of rotation. They are given through the x and y coordinates of the point A 1, to which the so-called starting point A(1, 0) goes after its rotation by an angle α around the point O - the beginning of the rectangular Cartesian coordinate system and the center of the unit circle.

Definition.

Sine of rotation angleα is the ordinate of point A 1, that is, sinα=y.

Definition.

Cosine of the rotation angleα is called the abscissa of point A 1, that is, cosα=x.

Definition.

Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tanα=y/x.

Definition.

Cotangent of the rotation angleα is the ratio of the abscissa of point A 1 to its ordinate, that is, ctgα=x/y.

Sine and cosine are defined for any angle α, since we can always determine the abscissa and ordinate of the point, which is obtained by rotating the starting point by angle α. But tangent and cotangent are not defined for any angle. The tangent is not defined for angles α at which the starting point goes to a point with zero abscissa (0, 1) or (0, −1), and this occurs at angles 90°+180° k, k∈Z (π /2+π·k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for angles α at which the starting point goes to the point with the zero ordinate (1, 0) or (−1, 0), and this occurs for angles 180° k, k ∈Z (π·k rad).

So, sine and cosine are defined for any rotation angles, tangent is defined for all angles except 90°+180°k, k∈Z (π/2+πk rad), and cotangent is defined for all angles except 180° ·k , k∈Z (π·k rad).

The definitions include the designations already known to us sin, cos, tg and ctg, they are also used to designate sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the designations tan and cotcorresponding to tangent and cotangent). So the sine of a rotation angle of 30 degrees can be written as sin30°, the entries tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α. Recall that when writing the radian measure of an angle, the designation “rad” is often omitted. For example, the cosine of a rotation angle of three pi rad is usually denoted cos3·π.

In conclusion of this point, it is worth noting that when talking about sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase “sine of the rotation angle alpha,” the phrase “sine of the alpha angle” or, even shorter, “sine alpha” is usually used. The same applies to cosine, tangent, and cotangent.

We will also say that the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle are consistent with the definitions just given for sine, cosine, tangent and cotangent of an angle of rotation ranging from 0 to 90 degrees. We will justify this.

Numbers

Definition.

Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the rotation angle in t radians, respectively.

For example, the cosine of the number 8 π by definition is the number equal to cosine angle of 8·π rad. And the cosine of an angle of 8·π rad is equal to one, therefore, the cosine of the number 8·π is equal to 1.

There is another approach to determining the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is associated with a point on the unit circle with the center at the beginning rectangular system coordinates, and sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's look at this in more detail.

Let us show how a correspondence is established between real numbers and points on a circle:

• the number 0 is assigned the starting point A(1, 0);
• the positive number t is associated with a point on the unit circle, which we will get to if we move along the circle from the starting point in a counterclockwise direction and let's walk the path length t;
• negative number t is associated with the point of the unit circle, which we will get to if we move along the circle from the starting point in a clockwise direction and walk a path of length |t| .

Now we move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point on the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1)).

Definition.

Sine of the number t is the ordinate of the point on the unit circle corresponding to the number t, that is, sint=y.

Definition.

Cosine of the number t is called the abscissa of the point of the unit circle corresponding to the number t, that is, cost=x.

Definition.

Tangent of the number t is the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of a number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost.

Definition.

Cotangent of the number t is the ratio of the abscissa to the ordinate of a point on the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is this: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t: ctgt=cost/sint.

Here we note that the definitions just given are consistent with the definition given at the beginning of this paragraph. Indeed, the point on the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point by an angle of t radians.

It is still worth clarifying this point. Let's say we have the entry sin3. How can we understand whether we are talking about the sine of the number 3 or the sine of the rotation angle of 3 radians? This is usually clear from the context, otherwise it is likely not of fundamental importance.

Trigonometric functions of angular and numeric argument

According to the definitions given in the previous paragraph, each rotation angle α corresponds to a very specific value sinα, as well as the value cosα. In addition, all rotation angles other than 90°+180°k, k∈Z (π/2+πk rad) correspond to tgα values, and values ​​other than 180°k, k∈Z (πk rad ) – values ​​of ctgα . Therefore sinα, cosα, tanα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.

We can speak similarly about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a very specific value sint, as well as cost. In addition, all numbers other than π/2+π·k, k∈Z correspond to values ​​tgt, and numbers π·k, k∈Z - values ​​ctgt.

The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.

It is usually clear from the context whether we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can consider the independent variable as a measure of the angle ( angle argument), and a numeric argument.

However, at school they mainly study numeric functions, that is, functions whose arguments, like their corresponding function values, are numbers. Therefore, if we're talking about specifically about functions, it is advisable to consider trigonometric functions as functions of numerical arguments.

Relationship between definitions from geometry and trigonometry

If we consider the rotation angle α ranging from 0 to 90 degrees, then the definitions of sine, cosine, tangent and cotangent of the rotation angle in the context of trigonometry are fully consistent with the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's justify this.

Let us depict the unit circle in the rectangular Cartesian coordinate system Oxy. Let's mark the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get point A 1 (x, y). Let us drop the perpendicular A 1 H from point A 1 to the Ox axis.

It is easy to see that in a right triangle, the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of point A 1, that is, |A 1 H|=y, and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of the acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y. And by definition from trigonometry, the sine of the rotation angle α is equal to the ordinate of point A 1, that is, sinα=y. This shows that determining the sine of an acute angle in a right triangle is equivalent to determining the sine of the rotation angle α when α is from 0 to 90 degrees.

Similarly, it can be shown that the definitions of cosine, tangent and cotangent of an acute angle α are consistent with the definitions of cosine, tangent and cotangent of the rotation angle α.

References.

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