Lesson "Functions y = tgx, y = ctgx, their properties and graphs." Lesson "Functions y = tgx, y = ctgx, their properties and graphs" Research of functions with tangent

This video tutorial discusses the properties of functions y =tgx, y = ctgx, shows how to construct their graphs.

The video tutorial begins with a look at the function y =tgx.

The properties of the function are highlighted.

1) Domain of the function y =tgx all real numbers are called, except x =π/2 + 2 πk. Those. there are no points on the graph that belong to the line x =π/2 and x = -π/2, as well as x = 3π/2 and so on (with the same periodicity). So the graph of the function y =tgx will consist of an infinite number of branches that will be located in the spaces between the straight lines x = - 3π/2 and x = -π/2 , x = -π/2 and x = π/2 and so on.

2) Function y =tgx is periodic, where the main period is π. This confirms equality tg(x- π ) = tg x =tg(x+π ) . These equalities were studied earlier, the author invites students to recall them, pointing out that for any valid value t the equalities are valid:

tg(t+ π ) = tg t, and c tg(t+π ) = ctg t. The consequence of these equalities is that if one branch of the graph of the function y = tan x in between the lines X = - π/2 and X= π/2, then the remaining branches can be obtained by shifting this branch along the x axis by π, 2π and so on.

3) Function y =tgx is odd, because . tg (- x) =- tg x.

Next, let's move on to plotting the function y =tgx. As follows from the properties of the function described above, the function y =tgx periodic and odd. Therefore, it is enough to construct part of the graph - one branch in one interval, and then use symmetry for transfer. The author provides a table in which the values ​​are calculated tgx at certain values x for more accurate plotting. These points are marked on the coordinate axis and connected by a smooth line. Because If the graph is symmetrical with respect to the origin of coordinates, then the same branch is constructed, symmetrical with respect to the origin of coordinates. As a result, we get one branch of the graph y =tgx. Next, using a shift along the x axis by π, 2 π, and so on, a graph is obtained y =tgx.

Graph of a function y =tgx is called a tangentoid, and the three branches of the graph shown in the figure are the main branches of the tangentoid.

4) Function y =tgx at each of the intervals (- + ; +) increases.

5) Function graph y =tgx has no restrictions above or below.

6) Function y =tgx does not have the greatest and least value.

7) Function y =tgx continuous on any interval (- - π/2+π;π/2+π). The straight line π/2+π is called the asymptote of the graph of the function y =tgx, because at these points the graph of the function is interrupted.

8) Set of function values y =tgx all real numbers are called.

Further in the video tutorial an example is given: solve the equation with tgx. To solve, we will construct 2 graphs of the function at and find the intersection points of these graphs: this is an infinite set of points whose abscissas differ by πk. The root of this equation will be X= π/6 +πk.

Consider the graph of the function y =ctgx. A function can be graphed in two ways.

The first method involves constructing a graph similar to constructing a graph functions y =tgx. Let's build one branch of the function graph y = ctgx in between the lines X= 0u X= π. Then, using symmetry and periodicity, we will construct other branches of the graph.

The second method is simpler. Graph of a function y = сtgx can be obtained by transforming the tangents using the reduction formula Withtgx = - tg(x +π/2). To do this, let’s shift one branch of the function graph y = tgx along the x-axis by π/2 to the right. The remaining branches are obtained by shifting this branch along the x axis by π, 2π, and so on. Graph of the function y = ctg x is also called a tangentoid, and the branch of the graph in the interval (0;π) is the main branch of the tangentoid.

TEXT DECODING:

We will consider the properties of the function y = tan x (y is equal to tangent x), y = ctg x (y is equal to cotangent x), and construct their graphs. Consider the function y = tgx

Before plotting the function y = tan x, let's write down the properties of this function.

PROPERTY 1. The domain of definition of the function y = tan x is all real numbers, except for numbers of the form x = + πk (x is equal to the sum of pi over two and pi ka).

This means that on the graph of this function there are no points that belong to the line x = (we get if k = 0 ka is equal to zero) and the line x = (x is equal to minus pi by two) (we get if k = - 1 ka is equal to minus one), and the straight line x = (x is equal to three pi by two) (we obtain if k = 1 is equal to one), etc. This means that the graph of the function y = tan x will consist of an infinite number of branches that will be located in the intervals between straight lines. Namely, in the band between x = and x =-; in the strip x = - and x = ; in the strip x = and x = and so on ad infinitum.

PROPERTY 2. The function y = tan x is periodic with the main period π. (Since the double equality is true

tan(x- π) = tanx = tan (x+π) tangent of x minus pi is equal to tangent of x and equal to tangent of x plus pi). We considered this equality when studying tangent and cotangent. Let's remind him:

For any admissible value of t the equalities are valid:

tg (t + π)= tgt

ctg (t + π) = ctgt

From this equality it follows that, having constructed a branch of the graph of the function y = tan x in the interval from x = - and x =, we obtain the remaining branches by shifting the constructed branch along the X axis by π, 2π, and so on.

PROPERTY 3. The function y = tan x is an odd function, since the equality tg (- x) = - tan x is true.

Let's plot the function y = tan x

Since this function is periodic, consists of an infinite number of branches (in the strip between x = and x =, as well as in the strip between x = and x =, etc.) and odd, we will construct a part of the graph point by point in the interval from zero to pi by two (), then use the symmetry of the origin and periodicity.

Let's build a table of tangent values ​​for plotting.

We find the first point: knowing that at x = 0 tan x = 0 (x is equal to zero, tan x is also equal to zero); next point: at x = tan x = (x equal to pi by six, tangent x is equal to the root of three by three); Let's note the following points: at x = tan x = 1 (x equal to pi by four tan x is equal to one), and at x = tg x = (x equal to pi by three tan x is equal to the square root of three). Mark the resulting points on the coordinate plane and connect them with a smooth line (Fig. 2).

Since the graph of the function is symmetrical with respect to the origin of coordinates, we will construct the same branch symmetrically with respect to the origin of coordinates. (Fig. 3).

And finally, applying periodicity, we obtain a graph of the function y = tan x.

We have constructed a branch of the graph of the function y = tan x in the strip from x = - and x =. We build the remaining branches by shifting the constructed branch along the X axis by π, 2π, and so on.

The plot created is called a tangentoid.

The part of the tangentoid shown in Figure 3 is called the main branch of the tangentoid.

Based on the graph, we will write down some more properties of this function.

PROPERTY 4. The function y = tan x increases on each of the intervals (from minus pi by two plus pi ka to pi by two plus pi ka).

PROPERTY 5. The function y = tan x is not bounded either above or below.

PROPERTY 6. The function y = tan x has neither the largest nor the smallest values.

PROPERTY 7. The function y = tan x is continuous on any interval of the form (from minus pi by two plus pi ka to pi by two plus pi ka).

A straight line of the form x = + πk (x is equal to the sum of pi over two and pi ka) is a vertical asymptote of the graph of the function, since at points of the form x = + πk the function suffers a discontinuity.

PROPERTY 8. The set of values ​​of the function y = tan x are all real numbers, that is (e from eff is equal to the interval from minus infinity to plus infinity).

EXAMPLE 1. Solve the equation tg x = (tangent x is equal to the root of three by three).

Solution. Let us construct graphs of the functions y = tan x in one coordinate system

(the y is equal to the tangent of x) and y = (the y is equal to the root of three divided by three).

We obtained infinitely many intersection points, the abscissa of which differ from each other by πk (pi ka). Since tg x = at x =, then the abscissa of the intersection point on the main branch is equal to (pi by six).

We write all solutions to this equation by the formula x = + πk (x equals pi times six plus pi ka).

Answer: x = + πk.

Let's build a graph of the function y = сtg x.

Let's consider two construction methods.

First way is similar to plotting the function y = tan x.

Since this function is periodic, consists of an infinite number of branches (in the band between x = 0 and x =π, as well as in the band between x =π and x = 2π, etc.) and odd, we will construct a part of the graph point by point on the interval from zero to pi by two (), then we will use symmetry and periodicity.

Let's use the table of cotangent values ​​to build a graph.

Mark the resulting points on the coordinate plane and connect them with a smooth line.

Since the graph of the function is relatively symmetrical, we will construct the same branch symmetrically.

Let us apply periodicity and obtain a graph of the function y = сtg x.

We have constructed a branch of the graph of the function y = сtg x in the strip from x = 0 and x =π. We construct the remaining branches by shifting the constructed branch along the x axis by π, - π, 2π, - 2π and so on.

Second way plotting the function y =сtg x.

The easiest way to obtain a graph of the function y =сtg x is to transform the tangent, using the reduction formula (cotangent x is equal to minus the tangent of the sum of x and pi by two).

In this case, first, we shift the branch of the graph of the function y =tg x along the abscissa axis to the right, we get

y = tg (x+), and then we perform the symmetry of the resulting graph relative to the abscissa axis. The result will be a branch of the graph of the function y =сtg x (Fig. 4). Knowing one branch, we can build the entire graph using the periodicity of the function. We construct the remaining branches by shifting the constructed branch along the x axis by π, 2π, and so on.

The graph of the function y =сtg x is also called a tangentoid, just like the graph of the function y =tg x. The branch that lies in the interval from zero to pi is called the main branch of the graph of the function y = сtg x.

Centered at point A.
α is the angle expressed in radians.

Tangent ( tan α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC|

to the length of the adjacent leg |AB| . Cotangent () is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB|

to the length of the opposite leg |BC| .

Tangent Where n

- whole.
.
;
;
.

In Western literature, tangent is denoted as follows:

Graph of the tangent function, y = tan x

Tangent Where n

Cotangent
.
In Western literature, cotangent is denoted as follows:
;
;
.

The following notations are also accepted:

Graph of the cotangent function, y = ctg x

Properties of tangent and cotangent

Periodicity Functions y = tg x and y = ctg x

are periodic with period π.

Parity

The tangent and cotangent functions are odd.

Areas of definition and values, increasing, decreasing Where The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table (

Intercept points with the ordinate axis, x =

Formulas

; ;
; ;
;

Expressions using sine and cosine

Formulas for tangent and cotangent from sum and difference

The remaining formulas are easy to obtain, for example

Product of tangents

Formula for the sum and difference of tangents

This table presents the values ​​of tangents and cotangents for certain values ​​of the argument.

Expressions using complex numbers

;
;

Expressions through hyperbolic functions

; .

.
Derivatives
.
Derivative of the nth order with respect to the variable x of the function:

Deriving formulas for tangent > > > ; for cotangent > > >

Integrals

Series expansions To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in a power series for the functions sin x And cos x

and divide these polynomials by each other, .

This produces the following formulas.
At . at . Where
;
;
Bn
- Bernoulli numbers. They are determined either from the recurrence relation:

Where .

Or according to Laplace's formula:

Inverse functions

The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively. Where n

Arctangent, arctg

The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively. Where n

, Where
Arccotangent, arcctg
References:

I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.

, [−5π/2; −3π/2],. . . - in a word, on all segments [−π/2 + 2πk; π/2 + 2πk], where k Z, and decreases on all segments

[π/2 + 2πn; 3π/2 + 2πn], where n Z.

§ 12. Graphs of tangent and cotangent

Let's plot the function y = tan x. First, let's construct it for numbers x belonging to the interval (−π/2; π/2).

If x = 0, then tan x = 0; when x increases from 0 to π/2, tan x also increases - this can be seen if you look at the tangent axis (Fig. 12.1 a). As x approaches π/2, remaining smaller

Rice. 12.2. y = tan x.

π/2, the value of tan x increases (point M in Fig. 12.1 a runs higher and higher) and can, obviously, become an arbitrarily large positive number. Likewise, as x decreases from 0 to −π/2, tan x becomes a negative number whose absolute value increases as x approaches −π/2. For x = π/2 or −π/2, the function tan x is undefined. Therefore, the graph y = tan x for x (−π/2; π/2) looks approximately like in Fig. 12.1 b.

Near the origin of coordinates, our curve is close to the straight line y = x x: after all, for small acute angles the approximate equality tg x ≈ x is true. We can say that the line y = x touches the graph of the function y = tan x at the origin. In addition, the curve in Fig. 12.1 b is symmetrical about the origin. This is explained by the fact that the function y = tan x is odd, that is, the identity tg(−x) = − tan x holds.

To plot the function y = tan x for all x, recall that tan x is a periodic function with period π. Therefore, in order to obtain a complete graph of the function y = tan x, it is necessary to repeat the curve in Fig. infinitely many times. 12.1 b, moving it along the abscissa to distances πn, where n is an integer. The final view of the graph of the function y = tan x is in Fig. 12.2.

According to the graph, we once again see that the function y = tan x

Rice. 12.3. y = cotg x.

is not defined for x = π/2 + πn, n Z, that is, for those x for which cos x = 0. Vertical lines with equations x = π/2, 3π/2,. . . , to which the branches of the graph approach are called asymptotes of the graph.

In the same fig. 12.2 we depicted solutions to the equation tg x = a.

Let's plot the function y = cot x. The easiest way is to use the reduction formula ctg x = tan(π/2 − x) to obtain this graph from the graph of the function y = tan x using transformations similar to those we described in the previous paragraph. The result is shown in Fig. 12.3

Problem 12.1. The graph of the function y = ctg x is obtained from the graph of the function y = tan x using symmetry about a certain line. Which one? Are there other lines with this property?

Problem 12.2. What does the equation of a straight line tangent to the graph of the function y = cot x look like at a point with coordinates (π/2; 0)?

Problem 12.3. Compare the numbers: a) tg(13π/11) and tg 3.3π; b) tan 9.6π and cot(−11.3π).

Problem 12.4. Arrange the numbers in ascending order: tg 1, tg 2, tg 3, tg 4, tg 5.

Problem 12.5. Graph the functions:

Problem 12.7. Plot the function y = arctan x + arctan(1/x).

§ 13. What is sin x + cos x equal?

In this section we will try to solve the following problem: what is the largest value that the expression sin x+cos x can take?

If you counted correctly, you should have found that of all the x included in this table, the largest value is sin x + cos x

is obtained for x close to 45◦, or, in radian measure, to π/4.

If x = π/4, the exact value of sin x+cos x is 2. It turns out that our result obtained experimentally, and in

is actually true: for all x the inequality sin x + cos x 6 is true

2, so 2 is the largest value accepted by this expression.

We do not yet have enough means to prove this inequality in the most natural way. For now, we will show how to reduce it to a planimetry problem.

If 0< x < π/2, то sin x и cos x - катеты прямоугольного треугольника с гипотенузой 1 и острым углом x (рис. 13.1 ).

Therefore, our task is reformulated as follows: to prove that the sum of the lengths of the legs of a right triangle with hypotenuse 1 will be maximum if this triangle is isosceles.

Problem 13.1. Prove this statement.

Since an isosceles right triangle with hy-

Potenuse 1, the sum of the lengths of the legs is equal to 2√, the result of this problem implies the inequality sin x + cos x 6 2 for all x lying in the interval (0; π/2). From here it is not difficult to conclude that this inequality holds for all x in general.

The result of Problem 13.1 is not only true for right triangles.

Problem 13.2. Prove that among all triangles with given values ​​of side AC and angle B, the greatest sum AB + BC will be for an isosceles triangle with base AC.

Let's return to trigonometry.

Problem 13.3. Using the table of sines from § 3, construct a point-by-point graph of the function y = sin x + cos x.

Note. Remember that x must be expressed in radians; For x values ​​outside the interval, use the reduction formulas.

If you did everything correctly, you should have a curve that looks like a sine wave. Later we will see that this curve is not just similar, but is a sinusoid. We will also learn to find the largest values ​​of expressions such as 3 sin x + 4 cos x (by the way, the graph of the function y = 3 sin x + 4 cos x is also a sinusoid!).