Arcsin graph and properties. Inverse trigonometric functions, their graphs and formulas

Inverse trigonometric functions(circular functions, arc functions) - mathematical functions that are inverse to trigonometric functions.

arcsine(denoted as arcsin x; arcsin x- this is the angle sin his equals x).

arcsine (y = arcsin x) - inverse trigonometric function to sin (x = sin y), which has a domain of definition and a set of values . In other words, returns the angle by its value sin.

Function y=sin x is continuous and bounded along its entire number line. Function y=arcsin x- strictly increases.

Properties of the arcsin function.

Arcsine plot.

Getting the arcsin function.

There is a function y = sin x. Throughout its entire domain of definition it is piecewise monotonic, thus the inverse correspondence y = arcsin x is not a function. Therefore, we consider the segment on which it only increases and takes each value of the range of values ​​- . Because for function y = sin x on the interval, all values ​​of the function are obtained with only one value of the argument, which means that on this interval there is an inverse function y = arcsin x, whose graph is symmetrical to the graph of the function y = sin x on a segment relatively straight y = x.

Problems related to inverse trigonometric functions are often offered in school final exams and in entrance exams at some universities. A detailed study of this topic can only be achieved in elective classes or elective courses. The proposed course is designed to develop the abilities of each student as fully as possible and improve his mathematical preparation.

The course lasts 10 hours:

1.Functions arcsin x, arccos x, arctg x, arcctg x (4 hours).

2.Operations on inverse trigonometric functions (4 hours).

3. Inverse trigonometric operations on trigonometric functions (2 hours).

Lesson 1 (2 hours) Topic: Functions y = arcsin x, y = arccos x, y = arctan x, y = arcctg x.

Goal: complete coverage of this issue.

1.Function y = arcsin x.

a) For the function y = sin x on the segment there is an inverse (single-valued) function, which we agreed to call arcsine and denote it as follows: y = arcsin x. The graph of the inverse function is symmetrical with the graph of the main function with respect to the bisector of I - III coordinate angles.

Properties of the function y = arcsin x.

1) Domain of definition: segment [-1; 1];

2)Area of ​​change: segment;

3)Function y = arcsin x odd: arcsin (-x) = - arcsin x;

4)The function y = arcsin x is monotonically increasing;

5) The graph intersects the Ox, Oy axes at the origin.

Example 1. Find a = arcsin. This example can be formulated in detail as follows: find an argument a, lying in the range from to, whose sine is equal to.

Solution. There are countless arguments whose sine is equal to , for example: etc. But we are only interested in the argument that is on the segment. This would be the argument. So, .

Example 2. Find .Solution. Arguing in the same way as in Example 1, we get .

b) oral exercises. Find: arcsin 1, arcsin (-1), arcsin, arcsin(), arcsin, arcsin(), arcsin, arcsin(), arcsin 0. Sample answer: , because . Do the expressions make sense: ; arcsin 1.5; ?

c) Arrange in ascending order: arcsin, arcsin (-0.3), arcsin 0.9.

II. Functions y = arccos x, y = arctg x, y = arcctg x (similar).

Lesson 2 (2 hours) Topic: Inverse trigonometric functions, their graphs.

Purpose: in this lesson it is necessary to develop skills in determining the values ​​of trigonometric functions, in constructing graphs of inverse trigonometric functions using D (y), E (y) and the necessary transformations.

In this lesson, complete exercises that include finding the domain of definition, the domain of value of functions of the type: y = arcsin, y = arccos (x-2), y = arctg (tg x), y = arccos.

You should construct graphs of the functions: a) y = arcsin 2x; b) y = 2 arcsin 2x; c) y = arcsin;

d) y = arcsin; e) y = arcsin; e) y = arcsin; g) y = | arcsin | .

Example. Let's plot y = arccos

You can include the following exercises in your homework: build graphs of functions: y = arccos, y = 2 arcctg x, y = arccos | x | .

Graphs of Inverse Functions

Goal: to expand mathematical knowledge (this is important for those entering specialties with increased requirements for mathematical training) by introducing basic relations for inverse trigonometric functions.

Material for the lesson.

Some simple trigonometric operations on inverse trigonometric functions: sin (arcsin x) = x , i xi ? 1; cos (arсcos x) = x, i xi? 1; tg (arctg x)= x , x I R; ctg (arcctg x) = x , x I R.

Exercises.

a) tg (1.5 + arctg 5) = - ctg (arctg 5) = .

ctg (arctg x) = ; tg (arcctg x) = .

b) cos ( + arcsin 0.6) = - cos (arcsin 0.6). Let arcsin 0.6 = a, sin a = 0.6;

cos (arcsin x) = ; sin (arccos x) = .

Note: we take the “+” sign in front of the root because a = arcsin x satisfies .

c) sin (1.5 + arcsin). Answer: ;

d) ctg ( + arctg 3). Answer: ;

e) tg ( – arcctg 4). Answer: .

e) cos (0.5 + arccos). Answer: .

Calculate:

a) sin (2 arctan 5) .

Let arctan 5 = a, then sin 2 a = or sin (2 arctan 5) = ;

b) cos ( + 2 arcsin 0.8). Answer: 0.28.

c) arctg + arctg.

Let a = arctg, b = arctg,

then tg(a + b) = .

d) sin(arcsin + arcsin).

e) Prove that for all x I [-1; 1] true arcsin x + arccos x = .

Proof:

arcsin x = – arccos x

sin (arcsin x) = sin ( – arccos x)

x = cos (arccos x)

To solve it yourself: sin (arccos), cos (arcsin), cos (arcsin ()), sin (arctg (- 3)), tg (arccos), ctg (arccos).

For a home solution: 1) sin (arcsin 0.6 + arctan 0); 2) arcsin + arcsin ; 3) ctg ( – arccos 0.6); 4) cos (2 arcctg 5) ; 5) sin (1.5 – arcsin 0.8); 6) arctg 0.5 – arctg 3.

Goal: In this lesson, demonstrate the use of ratios in transforming more complex expressions.

Material for the lesson.

ORALLY:

a) sin (arccos 0.6), cos (arcsin 0.8);

b) tg (arcсtg 5), ctg (arctg 5);

c) sin (arctg -3), cos (arcсtg());

d) tg (arccos), ctg (arccos()).

IN WRITTEN:

1) cos (arcsin + arcsin + arcsin).

2) cos (arctg 5–arccos 0.8) = cos (arctg 5) cos (arccos 0.8) + sin (arctg 5) sin (arccos 0.8) =

3) tg ( - arcsin 0.6) = - tg (arcsin 0.6) =

4)

Independent work will help to identify the level of mastery of the material.

For homework you can suggest:

1) ctg (arctg + arctg + arctg); 2) sin 2 (arctg 2 – arcctg ()); 3) sin (2 arctg + tan ( arcsin )); 4) sin(2 arctan); 5) tg ( (arcsin ))

Goal: to form students’ understanding of inverse trigonometric operations on trigonometric functions, focusing on increasing the comprehension of the theory being studied.

When studying this topic, it is assumed that the volume of theoretical material to be memorized is limited.

Lesson material:

You can start learning new material by studying the function y = arcsin (sin x) and plotting it.

3. Each x I R is associated with y I, i.e.<= y <= такое, что sin y = sin x.

4. The function is odd: sin(-x) = - sin x; arcsin(sin(-x)) = - arcsin(sin x).

6. Graph y = arcsin (sin x) on:

a) 0<= x <= имеем y = arcsin(sin x) = x, ибо sin y = sin x и <= y <= .

b)<= x <= получим y = arcsin (sin x) = arcsin ( - x) = - x, ибо

sin y = sin ( – x) = sin x , 0<= - x <= .

So,

Having constructed y = arcsin (sin x) on , we continue symmetrically about the origin on [- ; 0], given the oddness of this function. Using periodicity, we continue along the entire number line.

Then write down some relationships: arcsin (sin a) = a if<= a <= ; arccos (cos a ) = a if 0<= a <= ; arctg (tg a) = a if< a < ; arcctg (ctg a) = a , если 0 < a < .

And do the following exercises:a) arccos(sin 2).Answer: 2 - ; b) arcsin (cos 0.6). Answer: - 0.1; c) arctg (tg 2). Answer: 2 - ;

d) arcctg(tg 0.6).Answer: 0.9; e) arccos (cos ( - 2)). Answer: 2 - ; e) arcsin (sin ( - 0.6)). Answer: - 0.6; g) arctg (tg 2) = arctg (tg (2 - )). Answer: 2 - ; h) аrcctg (tg 0.6). Answer: - 0.6; - arctan x; e) arccos + arccos

Definition and notation

Arcsine is sometimes denoted as follows:
.

Graph of arcsine function

Graph of the function y = arcsin x

The arcsine graph is obtained from the sine graph if the abscissa and ordinate axes are swapped. To eliminate ambiguity, the range of values ​​is limited to the interval over which the function is monotonic. This definition is called the principal value of the arcsine.

Arccosine, arccos

Definition and notation

Arccosine is sometimes denoted as follows:
.

Graph of arc cosine function

Graph of the function y = arccos x

The arc cosine graph is obtained from the cosine graph if the abscissa and ordinate axes are swapped. To eliminate ambiguity, the range of values ​​is limited to the interval over which the function is monotonic. This definition is called the principal value of the arc cosine.

Parity

The arcsine function is odd:
arcsin(- x) = arcsin(-sin arcsin x) = arcsin(sin(-arcsin x)) = - arcsin x

The arc cosine function is not even or odd:
arccos(- x) = arccos(-cos arccos x) = arccos(cos(π-arccos x)) = π - arccos x ≠ ± arccos x

Properties - extrema, increase, decrease

The functions arcsine and arccosine are continuous in their domain of definition (see proof of continuity). The main properties of arcsine and arccosine are presented in the table.

Table of arcsines and arccosines

This table presents the values ​​of arcsines and arccosines, in degrees and radians, for certain values ​​of the argument.

≈ 0,7071067811865476
≈ 0,8660254037844386

Formulas

Sum and difference formulas

at or

at and

at and

at or

at and

at and

at

at

at

at

Expressions through logarithms, complex numbers

Expressions through hyperbolic functions

Derivatives

;
.
See Derivation of arcsine and arccosine derivatives > > >

Higher order derivatives:
,
where is a polynomial of degree . It is determined by the formulas:
;
;
.

See Derivation of higher order derivatives of arcsine and arccosine > > >

Integrals

We make the substitution x = sint. We integrate by parts, taking into account that -π/ 2 ≤ t ≤ π/2, cos t ≥ 0:
.

Let's express arc cosine through arc sine:
.

Series expansion

When |x|< 1 the following decomposition takes place:
;
.

Inverse functions

The inverses of arcsine and arccosine are sine and cosine, respectively.

The following formulas are valid throughout the entire domain of definition:
sin(arcsin x) = x
cos(arccos x) = x .

The following formulas are valid only on the set of arcsine and arccosine values:
arcsin(sin x) = x at
arccos(cos x) = x at .

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