If we construct a unit circle centered at the origin and set an arbitrary value of the argument x0 and count from the axis Ox corner x 0, then this angle on the unit circle corresponds to some point A(Fig. 1) and its projection onto the axis Oh there will be a point M. Cut length OM equal to the absolute value of the abscissa of the point A. given argument value x0 mapped function value y= cos x 0 as the abscissa of a point BUT. Accordingly, the point AT(x 0 ;at 0) belongs to the function graph at= cos X(Fig. 2). If point BUT located to the right of the axis OU, the tocosine will be positive, if to the left it will be negative. But in any case, the point BUT cannot leave the circle. Therefore, the cosine ranges from -1 to 1:

-1 = cos x = 1.

Additional rotation to any angle, multiple of 2 p, returns a 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 that are equal in absolute value but opposite in sign, x and - x, find corresponding points on the circle A x and A-x. As 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).

So, 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 BUT lies on the axis Oh, its abscissa is 1, and therefore cos 0 = 1. With an increase X dot BUT moves around the circle up and to the left, its projection, of course, only to the left, and for x = p/2 cosine becomes 0. Point A at this moment rises to maximum height, and then continues to move to the left, but already decreasing. Its abscissa keeps decreasing until it reaches the smallest value equal to -1 at X= p. Thus, on the segment, the function at= cos X decreases monotonically from 1 to –1 (Fig. 4, 5).

It follows from the parity of the cosine that on the interval [– p, 0], the function increases monotonically from –1 to 1, taking on 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 reached at points X= 2kp, i.e. with step 2 p, and the minima equal to –1 at the points X= p + 2kp.

Function y \u003d sin x.

On the unit circle x 0 corresponds to point BUT(Fig. 7), and its projection onto the axis OU there will be a point N.W function value y 0 = sin x0 defined as the ordinate of a point BUT. Dot AT(corner x 0 ,at 0) belongs to the function graph 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 axle OU located symmetrically about the point O. That's why

sin(- x) = –sin ( x),

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

If the point A rotate about a point O on the corner 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 the cosine, "belated" by p/2, since any cosine value will "repeate" 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). An extremely important property of the sine is expressed by the equality

The geometric meaning of equality can be seen from Fig. 11. Here X - this is half of the arc AB, and sin X - half of the corresponding chord. Obviously, as the points approach BUT and AT the length of the chord is getting closer and closer to the length of the arc. From the same figure, it is easy to extract the inequality

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

The formula (*) is called the wonderful limit by mathematicians. From it, in particular, it follows that sin X» X at small X.

Functions at=tg x, y=ctg X. Two other trigonometric functions - tangent and cotangent are easiest to define as 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 that of sine and cosine. The reason for this is clear: if the sine and cosine both change signs, then their ratio will not change.

Since there is a cosine in the denominator of the tangent, 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 the tangent are the 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 the lines x = kp – its vertical asymptotes. At points x = p/2 +kp the cotangent turns to 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:

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

The periodicity of the 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 axis of tangents.

Basic properties trigonometric functions can be tabulated:

Casting 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 function of the argument a , where 0 a p /2, both the same and additional 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
cosb	sin a	–sin a	-cos a	-cos a	–sin a	sin a	cos a

Therefore, in the tables of trigonometric functions, values ​​\u200b\u200bare given only for acute angles, and it is enough to confine ourselves, for example, to sine and tangent. The table contains only the most commonly used formulas for sine and cosine. From them 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 is an integer, to a function from the argument a :

1) the name of the function 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 the angle a is acute.

For example, when casting ctg (a - p/2) 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.

Multiple angle formulas.

These formulas are derived directly from the addition formulas:

sin 2a \u003d 2 sin a cos a;

cos 2a \u003d cos 2 a - sin 2 a \u003d 2 cos 2 a - 1 \u003d 1 - 2 sin 2 a;

sin 3a \u003d 3 sin a - 4 sin 3 a;

cos 3a \u003d 4 cos 3 a - 3 cos a;

The formula for cos 3a was used by Francois Viet when solving a 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 De 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 from the same argument can be rewritten as a rational expression from a single function tg (a / 2), this is useful when solving some equations:

Formulas for converting sums to products and products to sums.

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

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

2 cos a cos b= cos( a-b) + cos( a+b);

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

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

Degree reduction formulas.

From the formulas of a multiple argument, formulas are derived:

With these formulas trigonometric equations can be reduced to equations of lower degrees. In the same way, reduction formulas for higher powers of sine and cosine can be derived.

Every trigonometric function at every 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 into power series. In this case, the functions sin x b cos x appear in rows. convergent for all values x:

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

at | x| p/2;

at 0x| p

(B n are 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 interval [– p, p] orthogonal system of 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 a real argument into the complex plane. Yes, sin z and cos z can be defined using series for sin x and cos x, if instead of x put z:

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

Tangent and cotangent are determined by the formulas:

tg functions z and ctg z are meromorphic functions. Poles tg z and sec z are simple (1st order) and are located at points z=p/2 + pn, ctg poles z and cosec z are also simple and are 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. The formulas are also saved

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

those. the 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 the Euler formulas. Leonhard Euler introduced 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 are hyperbolic sine, cosine and tangent.

Trigonometric functions of complex argument z = x + iy, where x and y- real numbers, can be expressed in terms of 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 ​​greater than 1 in absolute value. For example:

If an unknown angle enters the 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. FROM help various tricks and formulas, trigonometric equations are reduced 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 solution of 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 a function is found that is inverse to it in this section. Such functions are denoted by attributing the prefix arc (arc) to the name of the original function, and are called inverse trigonometric functions or just arc functions.

Inverse trigonometric functions.

For sin X, cos X, tg X and ctg X inverse functions can be defined. They are designated respectively arcsin X(read "arxine x"), arcos x, arctg x and arcctg x. By definition, arcsin X there is such a number y, what

sin at = X.

The same is true for other inverse trigonometric functions. But this definition suffers from some inaccuracy.

If we reflect sin X, cos X, tg X and ctg X with respect 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 an infinite number of angles.

To get rid of the ambiguity, a section of the curve with a width of p, while it is necessary that a one-to-one correspondence be observed between the argument and the value of the function. Areas near the origin are selected. For the sinus as the "interval of one-to-one" is taken 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 in the interval is reflected about the bisector and now you can define inverse trigonometric functions. For example, let the argument value be given x 0 , such that 0 J x 0 Ј 1. Then the value of the function y 0 = arcsin x 0 will be single meaning at 0 , such that - p/2 J at 0 Ј p/2 and x 0 = sin y 0 .

Thus, the arcsine is a function of arcsin a, defined on the interval [–1, 1] and equal for each a 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 there are two points on the circle with an ordinate a, symmetrical about the axis y. One of them is the angle a= arcsin a, and the other is the angle p - a. FROM taking into account the periodicity of the sine, the solution of the equation sin x= a is written as follows:

x =(–1)n arc sin a + 2p n,

where n= 0, ±1, ±2,...

Other simple trigonometric equations are also solved:

cos x = a, –1 =a= 1;

x=±arcos a + 2p n,

where P= 0, ±1, ±2,... (Fig. 16);

tg X = a;

x= arctg 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).

The main properties of inverse trigonometric functions:

arc sin X(Fig. 19): the domain of definition is the segment [–1, 1]; range - [- p/2, p/2], a monotonically increasing function;

arccos X(Fig. 20): the domain of definition is the 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 \u003d 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 are the horizontal asymptotes.

For anyone z = x+iy, where x and y are real numbers, there are inequalities

½| e\ey–e-y| ≤|sin z|≤½( e y +e-y),

½| e y–e-y| ≤|cos z|≤½( e y +e -y),

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

|sin z| » 1/2 e |y| ,

|cos z| » 1/2 e |y| .

Trigonometric functions arose for the first time 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 ratios were not an independent object of study, so they did not study trigonometric functions as such. They were originally 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 through 30 "with an accuracy of 10 -6. This was the first table of sines. As a ratio sin function a is already found 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-Vef (10th century), who also uses sec a and cosec a. Aryabhata already knew the formula (sin 2 a + cos 2 a) \u003d 1, as well as the sin and cos half-angle formulas, with the help of which he built tables of sines for angles through 3 ° 45 "; based on the known values ​​\u200b\u200bof 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 difference of trigonometric functions of various arguments into a product were derived by Regiomontanus (15th century) and J. Napier in connection with the invention of the latter logarithms (1614). table of sine values ​​​​through 1 ". The expansion of trigonometric functions into power series was obtained by I. Newton (1669). AT modern form the theory of trigonometric functions was introduced by L. Euler (18th century). He owns their definition for the real and complex arguments, the symbolism now accepted, the establishment of a connection with the exponential function and the orthogonality of the system of sines and cosines.

Trigonometric functions are elementary functions whose argument is corner. Trigonometric functions describe the relationships between sides and acute angles in right triangle. The areas of application of trigonometric functions are extremely diverse. So, for example, any periodic processes can be represented as a sum of trigonometric functions (). These functions often appear when solving functional equations.

Trigonometric functions include the following 6 functions: sinus , cosine , tangent , cotangent , secant and cosecant. For each of specified functions exists inverse trigonometric function .

The geometric definition of trigonometric functions is conveniently introduced using unit circle . The figure below shows a circle with radius \(r = 1\). Point \(M\left((x,y) \right)\) is marked on the circle. The angle between the radius vector \(OM\) and the positive direction of the \(Ox\) axis is equal to \(\alpha\).

sinus angle \(\alpha\) is the ratio of the ordinate \(y\) of the point \(M\left((x,y) \right)\) to the radius \(r\):
\(\sin \alpha = y/r\).
Since \(r = 1\), then the sine is equal to the ordinate of the point \(M\left((x,y) \right)\).

cosine the angle \(\alpha\) is the ratio of the abscissa \(x\) of the point \(M\left((x,y) \right)\) to the radius \(r\):
\(\cos \alpha = x/r\)


tangent angle \(\alpha\) is the ratio of the ordinate \(y\) of the point \(M\left((x,y) \right)\) to its abscissa \(x\):
\(\tan \alpha = y/x,\;\;x \ne 0\)


Cotangent the angle \(\alpha\) is the ratio of the abscissa \(x\) of the point \(M\left((x,y) \right)\) to its ordinate \(y\):
\(\cot \alpha = x/y,\;\;y \ne 0\)


Secant angle \(\alpha\) is the ratio of the radius \(r\) to the abscissa \(x\) of the point \(M\left((x,y) \right)\):
\(\sec \alpha = r/x = 1/x,\;\;x \ne 0\)


Cosecant angle \(\alpha\) is the ratio of the radius \(r\) to the ordinate \(y\) of the point \(M\left((x,y) \right)\):
\(\csc \alpha = r/y = 1/y,\;\;y \ne 0\)


In the projection unit circle \(x\), \(y\) the points \(M\left((x,y) \right)\) and the radius \(r\) form a right triangle in which \(x,y \) are legs, and \(r\) is the hypotenuse. Therefore, the above definitions of trigonometric functions as applied to a right triangle are formulated as follows:
sinus the angle \(\alpha\) is the ratio of the opposite leg to the hypotenuse.
cosine angle \(\alpha\) is called the ratio adjacent leg to the hypotenuse.
tangent the angle \(\alpha\) is called the opposite leg to the adjacent one.
Cotangent angle \(\alpha\) is called the adjacent leg to the opposite one.
Secant angle \(\alpha\) is the ratio of the hypotenuse to the adjacent leg.
Cosecant angle \(\alpha\) is the ratio of the hypotenuse to the opposite leg.


sine function graph
\(y = \sin x\), domain: \(x \in \mathbb(R)\), domain: \(-1 \le \sin x \le 1\)




Graph of the cosine function
\(y = \cos x\), domain: \(x \in \mathbb(R)\), domain: \(-1 \le \cos x \le 1\)

Definitions

Definitions of trigonometric functions are given with the help of a trigonometric circle, which is understood as a circle of unit radius centered at the origin.

Consider two radii of this circle: fixed (where is the point) and movable (where is the point). Let the movable radius form an angle with the fixed one.

The number equal to the ordinate of the end of the unit radius forming an angle with a fixed radius is called sine of angle : .

The number equal to the abscissa of the end of the unit radius forming an angle with a fixed radius is called cosine of the angle : .

Thus, the point that is the end of the movable radius forming the corner has coordinates.

Tangent of an angle is the ratio of the sine of this angle to its cosine: , .

cotangent of an angle is the ratio of the cosine of this angle to its sine: , .

Geometric meaning of trigonometric functions

The geometric meaning of the sine and cosine on a trigonometric circle is clear from the definition: this is the abscissa and ordinates of the intersection point of the movable radius, which makes an angle with the fixed radius, and the trigonometric circle. That is, .

Consider now the geometric meaning of tangent and cotangent. Triangles are similar in three angles (,), then the relation holds. On the other hand, in, therefore.

Also similar in three corners (,), then the relation holds. On the other hand, in, therefore.

Taking into account the geometric meaning of tangent and cotangent, the concept of the axis of tangents and the axis of cotangents is introduced.

Axes of tangents are called axes, one of which touches the trigonometric circle at a point and is directed upwards, the second touches the circle at a point and is directed downwards.

Cotangent axes are called axes, one of which touches the trigonometric circle at a point and is directed to the right, the second touches the circle at a point and is directed to the left.

Properties of trigonometric functions

Let us consider some basic properties of trigonometric functions. Other properties will be discussed in the section on graphs of trigonometric functions.

Scope and range of values

As mentioned earlier, sine and cosine exist for any angles, i.e. the domain of definition of these functions is the set of real numbers. By definition, the tangent does not exist for angles, but the cotangent for angles, .

Since sine and cosine are the ordinate and abscissa of a point on a trigonometric circle, their values ​​lie in between. The range of tangent and cotangent is the set of real numbers (it is easy to see this by looking at the axes of tangents and cotangents).

Even/Odd

Consider the trigonometric functions of two angles (which corresponds to the movable radius) and (which corresponds to the movable radius). Since, then the point has coordinates. Therefore, i.e. sine - odd function; , i.e. cosine is an even function; , i.e. the tangent is odd; , i.e. the cotangent is also odd.

Constancy intervals

The signs of trigonometric functions for various coordinate quarters follow from the definition of these functions. It should be noted that since tangent and cotangent are ratios of sine and cosine, they are positive when the sine and cosine of an angle have the same sign and negative when they are different.

Periodicity

The periodicity of sine and cosine is based on the fact that angles that differ by an integer number of complete revolutions correspond to the same relative position moving and fixed beams. Accordingly, the coordinates of the point of intersection of the moving beam and the trigonometric circle will be the same for angles that differ by an integer number of full revolutions. So the period of sine and cosine is and where.

Obviously, that is also the period for the tangent and cotangent. But is there a smaller period for these functions? Let's prove that the shortest period for tangent and cotangent is.

Consider two angles and. On the geometric meaning of tangent and cotangent, . Triangles are equal along the side and the angles adjacent to it and, therefore, their sides are also equal, which means and. Similarly, one can prove where. Thus, the period of tangent and cotangent is.

Trigonometric functions of basic angles

Trigonometry formulas

To successfully solve trigonometric problems, it is necessary to master numerous trigonometric formulas. However, there is no need to memorize all the formulas. You need to know by heart only the most basic ones, and you need to be able to deduce the rest of the formulas if necessary.

Basic trigonometric identity and consequences from it

All trigonometric functions of an arbitrary angle are interconnected, i.e. knowing one function, you can always find the rest. This connection is given by the formulas considered in this section.

Theorem 1 (Basic trigonometric identity). For any, the identity

The proof consists in applying the Pythagorean theorem for a right triangle with legs, and a hypotenuse.

A more general theorem is also true.

Theorem 2. In order for two numbers to be taken as the cosine and sine of the same real angle, it is necessary and sufficient that the sum of their squares be equal to one:

Consider the consequences of the main trigonometric identity.

Let's express sine in terms of cosine and cosine in terms of sine:

In these formulas, the plus or minus sign in front of the root is chosen depending on the quarter in which the angle lies.

Substituting the formulas obtained above into the formulas that determine the tangent and cotangent, we obtain:

Dividing the basic trigonometric identity term by term by or we get, respectively:

These ratios can be rewritten as:

The following formulas give the relationship between tangent and cotangent. Since when, and when, then the equality takes place:

Cast formulas

With the help of reduction formulas, one can express the values ​​of trigonometric functions of arbitrary angles in terms of the values ​​of functions of an acute angle. All reduction formulas can be generalized using the following rule.

Any trigonometric function of an angle, in absolute value, is equal to the same function of the angle, if the number is even, and the co-function of the angle, if the number is odd. Moreover, if the function of the angle is positive, when is an acute positive angle, then the signs of both functions are the same, if negative, then they are different.

Formulas for sum and difference of angles

Theorem 3 . For any real and the following formulas are true:

The proof of the remaining formulas is based on the formulas for reduction and even/odd for trigonometric functions.

Q.E.D.

Theorem 4. For any real and such that

1. , the following formulas are valid

2. , the following formulas are valid

Proof. By definition of tangent

The last transformation is obtained by dividing the numerator and denominator of this fraction.

Similarly for the cotangent (the numerator and denominator in this case are divided by):

Q.E.D.

Attention should be paid to the fact that the right and left parts of the last equalities have different ranges of admissible values. Therefore, the use of these formulas without restrictions on the possible values ​​of the angles can lead to incorrect results.

Double and half angle formulas

Formulas double angle allow us to express the trigonometric functions of an arbitrary angle in terms of functions of an angle half the original. These formulas are consequences of the formulas for the sum of two angles, if we put the angles in them equal to each other.

The last formula can be transformed using the basic trigonometric identity:

Thus, for the cosine of a double angle, there are three formulas:

It should be noted that this formula is valid only for

The last formula is valid for, .

Similarly to double angle functions, triple angle functions can be obtained. Here these formulas are given without proof:

The half-angle formulas are consequences of the double-angle formulas and allow you to express the trigonometric functions of a certain angle in terms of functions of an angle twice the original one.

In this article, we will show how definitions of sine, cosine, tangent and cotangent of angle and number in trigonometry. Here we will talk about notation, give examples of records, give graphic illustrations. In conclusion, we 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 follow how the concept of sine, cosine, tangent and cotangent is formed in the school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in right triangle. And later trigonometry is studied, which refers to the sine, cosine, tangent and cotangent of the angle of rotation and the number. We give all these definitions, give examples and give the necessary comments.

Acute angle in a right triangle

From the course of geometry, the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle are known. They are given as the ratio of the sides of a right triangle. We present their formulations.

Definition.

Sine of an acute angle in a right triangle is the ratio of the opposite leg 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 is the ratio of the opposite leg to the adjacent leg.

Definition.

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

The notation of sine, cosine, tangent and cotangent is also introduced there - sin, cos, tg and ctg, respectively.

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

These definitions make it possible to calculate the values ​​of the 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 sine, cosine, tangent, cotangent and the length of one of the sides to find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is 3 and the hypotenuse AB is 7 , then we could calculate the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7 .

Angle of rotation

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

In this light, the definitions of the sine, cosine, tangent and cotangent are no longer an acute angle, but an angle of arbitrary magnitude - the angle of rotation. They are given through the x and y coordinates of the point A 1 , into which the so-called initial point A(1, 0) passes after it rotates through an angle α around the point O - the beginning of a rectangular Cartesian coordinate system and the center of the unit circle.

Definition.

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

Definition.

cosine of the angle of rotationα is called the abscissa of the 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, tgα=y/x .

Definition.

The cotangent of the angle of rotationα is the ratio of the abscissa of the point A 1 to its ordinate, that is, ctgα=x/y .

The sine and cosine are defined for any angle α , since we can always determine the abscissa and ordinate of a point, which is obtained by rotating the starting point through the angle α . And tangent and cotangent are not defined for any angle. The tangent is not defined for such angles α at which the initial point goes to a point with zero abscissa (0, 1) or (0, −1) , and this takes place 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 such angles α at which the starting point goes to to a point with zero ordinate (1, 0) or (−1, 0) , and this is the case for angles 180° k , k ∈Z (π k rad).

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

The notations already known to us appear in the definitions sin, cos, tg and ctg, they are also used to denote the sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the notation tan and cotcorresponding to tangent and cotangent). So the sine of the rotation angle of 30 degrees can be written as sin30°, the records 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 notation "rad" is often omitted. For example, the cosine of a rotation angle of three pi rads is usually denoted cos3 π .

In conclusion of this paragraph, it is worth noting that in talking about the 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 angle of rotation alpha", the phrase "sine of the angle of alpha" is usually used, or even shorter - "sine of alpha". The same applies to cosine, and tangent, and cotangent.

Let's also say that the definitions of the sine, cosine, tangent, and cotangent of an acute angle in a right triangle are consistent with the definitions just given for the sine, cosine, tangent, and cotangent of a rotation angle ranging from 0 to 90 degrees. We will substantiate 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 angle of rotation in t radians, respectively.

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

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is assigned a point of the unit circle centered at the origin of the rectangular coordinate system, and the sine, cosine, tangent and cotangent are determined in terms of the coordinates of this point. Let's dwell on this in more detail.

Let us show how the correspondence between real numbers and points of the circle is established:

• the number 0 is assigned the starting point A(1, 0) ;
• a positive number t is associated with a point on the unit circle, which we will get to if we move around the circle from the starting point in a counterclockwise direction and go through a path of length t;
• negative number t corresponds to a point on the unit circle, which we will reach if we move around the circle from the starting point in a clockwise direction and go through a path of length |t| .

Now let's 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 of the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).

Definition.

The sine of a number t is the ordinate of the unit circle point corresponding to the number t , that is, sint=y .

Definition.

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

Definition.

Cotangent of a number t is the ratio of the abscissa to the ordinate of the point of the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is as follows: 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 agree with the definition given at the beginning of this subsection. Indeed, the point of the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point through an angle of t radians.

It is also worth clarifying this point. Let's say we have a sin3 entry. How to understand whether the sine of the number 3 or the sine of the rotation angle of 3 radians is in question? This is usually clear from the context, otherwise it probably doesn't matter.

Trigonometric functions of angular and numerical argument

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

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

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

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

However, the school mainly studies numeric functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking about functions, then it is advisable to consider trigonometric functions as functions of numerical arguments.

Connection of definitions from geometry and trigonometry

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

Draw in a rectangle Cartesian system Oxy coordinates unit circle. Note the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get the point A 1 (x, y) . Let's drop the perpendicular A 1 H from the point A 1 to the Ox axis.

It is easy to see that in a right triangle the angle A 1 OH equal to the angle turn α , the length of the leg OH adjacent to this corner is equal to the abscissa of the point A 1 , that is, |OH|=x , the length of the leg opposite to the corner A 1 H is equal to the ordinate of the 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 an 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 angle of rotation α is equal to the ordinate of the point A 1, that is, sinα=y. This shows that the definition of the sine of an acute angle in a right triangle is equivalent to the definition of the sine of the angle of rotation α for α from 0 to 90 degrees.

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

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