Tangent and arctangent are mutually inverse functions. Inverse trigonometric functions

sin functions, cos, tg and ctg are always accompanied by arcsine, arccosine, arctangent and arccotangent. One is a consequence of the other, and pairs of functions are equally important for working with trigonometric expressions.

Let's look at the drawing unit circle, which graphically displays the values trigonometric functions.

If we calculate arcs OA, arcos OC, arctg DE and arcctg MK, then they will all be equal to the value of angle α. The formulas below reflect the relationship between the basic trigonometric functions and their corresponding arcs.

To understand more about the properties of the arcsine, it is necessary to consider its function. Schedule has the form of an asymmetric curve passing through the coordinate center.

Properties of arcsine:

If we compare the graphs sin And arcsin, two trigonometric functions can have common principles.

arc cosine

Arccos of a number is the value of the angle α, the cosine of which is equal to a.

Curve y = arcos x mirrors arcsin graph x, with the only difference that it passes through the point π/2 on the OY axis.

Let's look at the arc cosine function in more detail:

1. The function is defined on the interval [-1; 1].
2. ODZ for arccos - .
3. The graph is entirely located in the first and second quarters, and the function itself is neither even nor odd.
4. Y = 0 at x = 1.
5. The curve decreases along its entire length. Some properties of the arc cosine coincide with the cosine function.

Some properties of the arc cosine coincide with the cosine function.

Perhaps schoolchildren will find such a “detailed” study of “arches” unnecessary. However, otherwise, some basic typical Unified State Exam assignments may lead students into confusion.

Task 1. Indicate the functions shown in the figure.

Answer: rice. 1 – 4, Fig. 2 – 1.

IN in this example the emphasis is on the little things. Typically, students are very inattentive to the construction of graphs and the appearance of functions. Indeed, why remember the type of curve if it can always be plotted using calculated points. Do not forget that under test conditions the time spent on drawing for simple task, will be required to solve more complex tasks.

Arctangent

Arctg the numbers a are the value of the angle α such that its tangent is equal to a.

If we consider the arctangent graph, we can highlight the following properties:

1. The graph is infinite and defined on the interval (- ∞; + ∞).
2. Arctangent odd function, therefore, arctan (- x) = - arctan x.
3. Y = 0 at x = 0.
4. The curve increases throughout the entire definition range.

Here's a short comparative analysis tg x and arctg x in table form.

Arccotangent

Arcctg of a number - takes a value α from the interval (0; π) such that its cotangent is equal to a.

Properties of the arc cotangent function:

1. The function definition interval is infinity.
2. Region acceptable values– interval (0; π).
3. F(x) is neither even nor odd.
4. Throughout its entire length, the graph of the function decreases.

It is very simple to compare ctg x and arctg x; you just need to make two drawings and describe the behavior of the curves.

Task 2. Match the graph and the notation form of the function.

If we think logically, it is clear from the graphs that both functions are increasing. Therefore, both figures reflect a certain function arctan. From the properties of the arctangent it is known that y=0 at x = 0,

Answer: rice. 1 – 1, fig. 2 – 4.

Trigonometric identities arcsin, arcos, arctg and arcctg

Previously, we have already identified the relationship between arches and the basic functions of trigonometry. This dependence can be expressed by a number of formulas that allow one to express, for example, the sine of an argument through its arcsine, arccosine, or vice versa. Knowledge of such identities can be useful when solving specific examples.

There are also relationships for arctg and arcctg:

Another useful pair of formulas sets the value for the sum of arcsin and arcos, as well as arcctg and arcctg of the same angle.

Examples of problem solving

Trigonometry tasks can be divided into four groups: calculate the numerical value of a specific expression, construct a graph of a given function, find its domain of definition or ODZ and perform analytical transformations to solve the example.

When solving the first type of problem, you must adhere to next plan actions:

When working with function graphs, the main thing is knowledge of their properties and appearance crooked. To solve trigonometric equations and inequalities, identity tables are needed. How more formulas remembers the student, the easier it is to find the answer to the task.

Let’s say in the Unified State Examination you need to find the answer for an equation like:

If we correctly transform the expression and lead to the right type, then solving it is very simple and quick. First, let's move arcsin x to right side equality.

If you remember the formula arcsin (sin α) = α, then we can reduce the search for answers to solving a system of two equations:

The restriction on the model x arose, again from the properties of arcsin: ODZ for x [-1; 1]. When a ≠0, part of the system is quadratic equation with roots x1 = 1 and x2 = - 1/a. When a = 0, x will be equal to 1.

Inverse trigonometric functions are mathematical functions, which are inverses of trigonometric functions.

Function y=arcsin(x)

The arcsine of a number α is a number α from the interval [-π/2;π/2] whose sine is equal to α.
Graph of a function
The function у= sin⁡(x) on the interval [-π/2;π/2], is strictly increasing and continuous; therefore, it has an inverse function, strictly increasing and continuous.
The inverse function for the function y= sin⁡(x), where x ∈[-π/2;π/2], is called the arcsine and is denoted y=arcsin(x), where x∈[-1;1].
So, according to the definition of the inverse function, the domain of definition of the arcsine is the segment [-1;1], and the set of values ​​is the segment [-π/2;π/2].
Note that the graph of the function y=arcsin(x), where x ∈[-1;1], is symmetrical to the graph of the function y= sin(⁡x), where x∈[-π/2;π/2], with respect to the bisector coordinate angles first and third quarters.

Function range y=arcsin(x).

Example No. 1.

Find arcsin(1/2)?

Since the range of values ​​of the function arcsin(x) belongs to the interval [-π/2;π/2], then only the value π/6 is suitable. Therefore, arcsin(1/2) =π/6.
Answer:π/6

Example No. 2.
Find arcsin(-(√3)/2)?

Since the range of values ​​arcsin(x) x ∈[-π/2;π/2], then only the value -π/3 is suitable. Therefore, arcsin(-(√3)/2) =- π/3.

Function y=arccos(x)

The arc cosine of a number α is a number α from the interval whose cosine is equal to α.

Graph of a function

The function y= cos(⁡x) on the segment is strictly decreasing and continuous; therefore, it has an inverse function, strictly decreasing and continuous.
The inverse function for the function y= cos⁡x, where x ∈, is called arc cosine and is denoted by y=arccos(x),where x ∈[-1;1].
So, according to the definition of the inverse function, the domain of definition of the arc cosine is the segment [-1;1], and the set of values ​​is the segment.
Note that the graph of the function y=arccos(x), where x ∈[-1;1] is symmetrical to the graph of the function y= cos(⁡x), where x ∈, with respect to the bisector of the coordinate angles of the first and third quarters.

Function range y=arccos(x).

Example No. 3.

Find arccos(1/2)?

Since the range of values ​​of the function arccos(x) belongs to the interval, then only the value 3π/4 is suitable. Therefore, arccos(-(√2)/2) = 3π/4.

Answer: 3π/4

Function y=arctg(x)

The arctangent of a number α is a number α from the interval [-π/2;π/2] whose tangent is equal to α.

Graph of a function

The tangent function is continuous and strictly increasing on the interval (-π/2;π/2); therefore, it has an inverse function that is continuous and strictly increasing.
The inverse function for the function y= tan⁡(x), where x∈(-π/2;π/2); is called the arctangent and is denoted by y=arctg(x), where x∈R.
So, according to the definition of the inverse function, the domain of definition of the arctangent is the interval (-∞;+∞), and the set of values ​​is the interval
(-π/2;π/2).
Note that the graph of the function y=arctg(x), where x∈R, is symmetrical to the graph of the function y= tan⁡x, where x ∈ (-π/2;π/2), relative to the bisector of the coordinate angles of the first and third quarters.

Function range y=arctg(x).

Example No. 5?

Find arctan((√3)/3).

Since the range of values ​​arctg(x) x ∈(-π/2;π/2), then only the value π/6 is suitable. Therefore, arctg((√3)/3) =π/6.
Example No. 6.
Find arctg(-1)?

Since the range of values ​​arctg(x) x ∈(-π/2;π/2), then only the value -π/4 is suitable. Therefore, arctg(-1) = - π/4.

Function y=arcctg(x)

Graph of a function

On the interval (0;π), the cotangent function strictly decreases; in addition, it is continuous at every point of this interval; therefore, on the interval (0;π), this function has an inverse function, which is strictly decreasing and continuous.
The inverse function for the function y=ctg(x), where x ∈(0;π), is called arccotangent and is denoted y=arcctg(x), where x∈R.
So, according to the definition of the inverse function, the domain of definition of the arc cotangent will be R, and by a set values ​​– interval (0;π).The graph of the function y=arcctg(x), where x∈R is symmetrical to the graph of the function y=ctg(x) x∈(0;π),relative to the bisector of the coordinate angles of the first and third quarters.

Function range y=arcctg(x).

Example No. 8.
Find arcctg(-(√3)/3)?

Since the range of values ​​arcctg(x) x∈(0;π), then only the value 2π/3 is suitable. Therefore, arccos(-(√3)/3) = 2π/3.

Editors: Ageeva Lyubov Aleksandrovna, Gavrilina Anna Viktorovna

Definitions of inverse trigonometric functions and their graphs are given. As well as formulas connecting inverse trigonometric functions, formulas for sums and differences.

Definition of inverse trigonometric functions

Since trigonometric functions are periodic, their inverse functions are not unique. So, the equation y = sin x, for a given , has infinitely many roots. Indeed, due to the periodicity of the sine, if x is such a root, then so is x + 2πn(where n is an integer) will also be the root of the equation. Thus, inverse trigonometric functions are multivalued. To make it easier to work with them, the concept of their main meanings is introduced. Consider, for example, sine: y = sin x. If we limit the argument x to the interval , then on it the function y = sin x increases monotonically. Therefore, it has a unique inverse function, which is called the arcsine: x = arcsin y.

Unless otherwise stated, by inverse trigonometric functions we mean their main values, which are determined by the following definitions.

Arcsine ( y = arcsin x) is the inverse function of sine ( x = siny

Arc cosine ( y = arccos x) is the inverse function of cosine ( x = cos y), having a domain of definition and a set of values.

Arctangent ( y = arctan x) is the inverse function of tangent ( x = tg y), having a domain of definition and a set of values.

arccotangent ( y = arcctg x) is the inverse function of cotangent ( x = ctg y), having a domain of definition and a set of values.

Graphs of inverse trigonometric functions

Graphs of inverse trigonometric functions are obtained from graphs of trigonometric functions mirror image relative to the straight line y = x. See sections Sine, cosine, Tangent, cotangent.

y = arcsin x

y = arccos x

y = arctan x

y = arcctg x

Basic formulas

Here you should pay special attention to the intervals for which the formulas are valid.

arcsin(sin x) = x at
sin(arcsin x) = x
arccos(cos x) = x at
cos(arccos x) = x

arctan(tg x) = x at
tg(arctg x) = x
arcctg(ctg x) = x at
ctg(arcctg x) = x

Formulas relating inverse trigonometric functions

Sum and difference formulas

at or

at and

at and

at or

at and

at and

at

at

at

at

Inverse cosine function

The range of values ​​of the function y=cos x (see Fig. 2) is a segment. On the segment the function is continuous and monotonically decreasing.

Rice. 2

This means that the function inverse to the function y=cos x is defined on the segment. This inverse function is called arc cosine and is denoted y=arccos x.

Definition

The arccosine of a number a, if |a|1, is the angle whose cosine belongs to the segment; it is denoted by arccos a.

Thus, arccos a is an angle that satisfies the following two conditions: сos (arccos a)=a, |a|1; 0? arccos a ?р.

For example, arccos, since cos and; arccos, since cos and.

The function y = arccos x (Fig. 3) is defined on a segment; its range of values ​​is the segment. On the segment, the function y=arccos x is continuous and monotonically decreases from p to 0 (since y=cos x is a continuous and monotonically decreasing function on the segment); at the ends of the segment it reaches its extreme values: arccos(-1)= p, arccos 1= 0. Note that arccos 0 = . The graph of the function y = arccos x (see Fig. 3) is symmetrical to the graph of the function y = cos x relative to the straight line y=x.

Rice. 3

Let us show that the equality arccos(-x) = p-arccos x holds.

In fact, by definition 0? arccos x? r. Multiplying by (-1) all parts of the last double inequality, we get - p? arccos x? 0. Adding p to all parts of the last inequality, we find that 0? p-arccos x? r.

Thus, the values ​​of the angles arccos(-x) and p - arccos x belong to the same segment. Since the cosine decreases monotonically on a segment, there cannot be two different angles on it that have equal cosines. Let's find the cosines of the angles arccos(-x) and p-arccos x. By definition, cos (arccos x) = - x, according to the reduction formulas and by definition we have: cos (p - - arccos x) = - cos (arccos x) = - x. So, the cosines of the angles are equal, which means the angles themselves are equal.

Inverse sine function

Let's consider the function y=sin x (Fig. 6), which on the segment [-р/2;р/2] is increasing, continuous and takes values ​​from the segment [-1; 1]. This means that on the segment [- p/2; р/2] the inverse function of the function y=sin x is defined.

Rice. 6

This inverse function is called the arcsine and is denoted y=arcsin x. Let us introduce the definition of the arcsine of a number.

The arcsine of a number is an angle (or arc) whose sine is equal to the number a and which belongs to the segment [-р/2; p/2]; it is denoted by arcsin a.

Thus, arcsin a is the angle satisfying following conditions: sin (arcsin a)=a, |a| ?1; -r/2 ? arcsin huh? r/2. For example, since sin and [- p/2; p/2]; arcsin, since sin = u [- p/2; p/2].

The function y=arcsin x (Fig. 7) is defined on the segment [- 1; 1], the range of its values ​​is the segment [-р/2;р/2]. On the segment [- 1; 1] the function y=arcsin x is continuous and increases monotonically from -p/2 to p/2 (this follows from the fact that the function y=sin x on the segment [-p/2; p/2] is continuous and increases monotonically). It takes the greatest value at x = 1: arcsin 1 = p/2, and the smallest at x = -1: arcsin (-1) = -p/2. At x = 0 the function is zero: arcsin 0 = 0.

Let us show that the function y = arcsin x is odd, i.e. arcsin(-x) = - arcsin x for any x [ - 1; 1].

Indeed, by definition, if |x| ?1, we have: - p/2 ? arcsin x ? ? r/2. Thus, the angles arcsin(-x) and - arcsin x belong to the same segment [ - p/2; p/2].

Let's find the sines of these angles: sin (arcsin(-x)) = - x (by definition); since the function y=sin x is odd, then sin (-arcsin x)= - sin (arcsin x)= - x. So, the sines of angles belonging to the same interval [-р/2; p/2], are equal, which means the angles themselves are equal, i.e. arcsin (-x)= - arcsin x. This means that the function y=arcsin x is odd. The graph of the function y=arcsin x is symmetrical about the origin.

Let us show that arcsin (sin x) = x for any x [-р/2; p/2].

Indeed, by definition -p/2? arcsin (sin x) ? p/2, and by condition -p/2? x? r/2. This means that the angles x and arcsin (sin x) belong to the same interval of monotonicity of the function y=sin x. If the sines of such angles are equal, then the angles themselves are equal. Let's find the sines of these angles: for angle x we ​​have sin x, for angle arcsin (sin x) we have sin (arcsin(sin x)) = sin x. We found that the sines of the angles are equal, therefore, the angles are equal, i.e. arcsin(sin x) = x. .

Rice. 7

Rice. 8

The graph of the function arcsin (sin|x|) is obtained ordinary transformations, associated with the module, from the graph y=arcsin (sin x) (shown as a dashed line in Fig. 8). The desired graph y=arcsin (sin |x-/4|) is obtained from it by shifting by /4 to the right along the x-axis (shown as a solid line in Fig. 8)

Inverse function of tangent

The function y=tg x on the interval accepts everything numeric values: E (tg x)=. Over this interval it is continuous and increases monotonically. This means that a function inverse to the function y = tan x is defined on the interval. This inverse function is called the arctangent and is denoted y = arctan x.

The arctangent of a is an angle from an interval whose tangent is equal to a. Thus, arctg a is an angle that satisfies the following conditions: tg (arctg a) = a and 0? arctg a ? r.

So, any number x always corresponds to a single value of the function y = arctan x (Fig. 9).

It is obvious that D (arctg x) = , E (arctg x) = .

The function y = arctan x is increasing because the function y = tan x is increasing on the interval. It is not difficult to prove that arctg(-x) = - arctgx, i.e. that arctangent is an odd function.

Rice. 9

The graph of the function y = arctan x is symmetrical to the graph of the function y = tan x relative to the straight line y = x, the graph y = arctan x passes through the origin of coordinates (since arctan 0 = 0) and is symmetrical relative to the origin (like the graph of an odd function).

It can be proven that arctan (tan x) = x if x.

Cotangent inverse function

The function y = ctg x on an interval takes all numeric values ​​from the interval. The range of its values ​​coincides with the set of all real numbers. In the interval, the function y = cot x is continuous and increases monotonically. This means that on this interval a function is defined that is inverse to the function y = cot x. The inverse function of cotangent is called arccotangent and is denoted y = arcctg x.

The arccotangent of the number a is the angle intervening, whose cotangent is equal to a.

Thus, аrcctg a is an angle satisfying the following conditions: ctg (arcctg a)=a and 0? arcctg a ? r.

From the definition of the inverse function and the definition of arctangent it follows that D (arcctg x) = , E (arcctg x) = . The arc cotangent is a decreasing function because the function y = ctg x decreases in the interval.

The graph of the function y = arcctg x does not intersect the Ox axis, since y > 0 R. For x = 0 y = arcctg 0 =.

The graph of the function y = arcctg x is shown in Figure 11.

Rice. 11

Note that for everyone real values x the identity is true: arcctg(-x) = р-arcctg x.

Lessons 32-33. Inverse trigonometric functions

Target: consider inverse trigonometric functions and their use for writing solutions to trigonometric equations.

I. Communicating the topic and purpose of the lessons

II. Learning new material

1. Inverse trigonometric functions

Let's begin our discussion of this topic with the following example.

Example 1

Let's solve the equation: a) sin x = 1/2; b) sin x = a.

a) On the ordinate axis we plot the value 1/2 and construct the angles x 1 and x2, for which sin x = 1/2. In this case x1 + x2 = π, whence x2 = π – x 1 . Using the table of values ​​of trigonometric functions, we find the value x1 = π/6, thenLet's take into account the periodicity of the sine function and write down the solutions given equation: where k ∈ Z.

b) Obviously, the algorithm for solving the equation sin x = a is the same as in the previous paragraph. Of course, now the value a is plotted along the ordinate axis. There is a need to somehow designate the angle x1. We agreed to denote this angle with the symbol arcsin A. Then the solutions to this equation can be written in the formThese two formulas can be combined into one: at the same time

The remaining inverse trigonometric functions are introduced in a similar way.

Very often it is necessary to determine the magnitude of the angle by known value its trigonometric function. Such a problem is multivalued - there are countless angles whose trigonometric functions are equal to the same value. Therefore, based on the monotonicity of trigonometric functions, for unambiguous definition angles introduce the following inverse trigonometric functions.

Arcsine of the number a (arcsin , whose sine is equal to a, i.e.

Arc cosine of a number a(arccos a) is an angle a from the interval whose cosine is equal to a, i.e.

Arctangent of a number a(arctg a) - such an angle a from the intervalwhose tangent is equal to a, i.e.tg a = a.

Arccotangent of a number a(arcctg a) is an angle a from the interval (0; π), the cotangent of which is equal to a, i.e. ctg a = a.

Example 2

Let's find:

Taking into account the definitions of inverse trigonometric functions, we obtain:

Example 3

Let's calculate

Let angle a = arcsin 3/5, then by definition sin a = 3/5 and . Therefore, we need to find cos A. Using basic trigonometric identity, we get:It is taken into account that cos a ≥ 0. So,

Let's give a few more typical examples related to the definitions and basic properties of inverse trigonometric functions.

Example 4

Let's find the domain of definition of the function

In order for the function y to be defined, it is necessary to satisfy the inequalitywhich is equivalent to the system of inequalitiesThe solution to the first inequality is the interval x∈ (-∞; +∞), second - This interval and is a solution to the system of inequalities, and therefore the domain of definition of the function

Example 5

Let's find the area of ​​change of the function

Let's consider the behavior of the function z = 2x - x2 (see picture).

It is clear that z ∈ (-∞; 1]. Considering that the argument z arccotangent function changes in within specified limits, from the table data we obtain thatSo the area of ​​change

Example 6

Let us prove that the function y = arctg x odd. LetThen tg a = -x or x = - tg a = tg (- a), and Therefore, - a = arctg x or a = - arctg X. Thus, we see thati.e. y(x) is an odd function.

Example 7

Let us express through all inverse trigonometric functions

Let It's obvious that Then since

Let's introduce the angle Because That

Likewise therefore And

So,

Example 8

Let's build a graph of the function y = cos(arcsin x).

Let us denote a = arcsin x, then Let's take into account that x = sin a and y = cos a, i.e. x 2 + y2 = 1, and restrictions on x (x∈ [-1; 1]) and y (y ≥ 0). Then the graph of the function y = cos(arcsin x) is a semicircle.

Example 9

Let's build a graph of the function y = arccos (cos x ).

Since the cos function x changes on the interval [-1; 1], then the function y is defined on the entire number axis and changes on the segment . Let's keep in mind that y = arccos(cosx) = x on the segment; the function y is even and periodic with period 2π. Considering that the function has these properties cos x Now it's easy to create a graph.

Let us note some useful equalities:

Example 10

Let's find the smallest and highest value functions Let's denote Then Let's get the function This function has a minimum at the point z = π/4, and it is equal to The greatest value of the function is achieved at the point z = -π/2, and it is equal Thus, and

Example 11

Let's solve the equation

Let's take into account that Then the equation looks like:or where By definition of arctangent we get:

2. Solving simple trigonometric equations

Similar to example 1, you can obtain solutions to the simplest trigonometric equations.

Example 12

Let's solve the equation

Since the sine function is odd, we write the equation in the formSolutions to this equation:where do we find it from?

Example 13

Let's solve the equation

Using the given formula, we write down the solutions to the equation:and we'll find

Note that in special cases (a = 0; ±1) when solving the equations sin x = a and cos x = but it’s easier and more convenient to use not general formulas, and write down solutions based on the unit circle:

for the equation sin x = 1 solution

for the equation sin x = 0 solutions x = π k;

for the equation sin x = -1 solution

for the cos equation x = 1 solutions x = 2π k ;

for the equation cos x = 0 solutions

for the equation cos x = -1 solution

Example 14

Let's solve the equation

Since in this example there is special case equations, then by corresponding formula Let's write down the solution:where can we find it from?

III. Security questions(frontal survey)

1. Define and list the main properties of inverse trigonometric functions.

2. Give graphs of inverse trigonometric functions.

3. Solving simple trigonometric equations.

IV. Lesson assignment

§ 15, No. 3 (a, b); 4 (c, d); 7(a); 8(a); 12 (b); 13(a); 15 (c); 16(a); 18 (a, b); 19 (c); 21;

§ 16, No. 4 (a, b); 7(a); 8 (b); 16 (a, b); 18(a); 19 (c, d);

§ 17, No. 3 (a, b); 4 (c, d); 5 (a, b); 7 (c, d); 9 (b); 10 (a, c).

V. Homework

§ 15, No. 3 (c, d); 4 (a, b); 7 (c); 8 (b); 12(a); 13(b); 15 (g); 16 (b); 18 (c, d); 19 (g); 22;

§ 16, No. 4 (c, d); 7(b); 8(a); 16 (c, d); 18 (b); 19 (a, b);

§ 17, No. 3 (c, d); 4 (a, b); 5 (c, d); 7 (a, b); 9 (d); 10 (b, d).

VI. Creative tasks

1. Find the domain of the function:

Answers:

2. Find the range of the function:

Answers:

3. Plot a graph of the function:

VII. Summing up the lessons