Inverse trigonometric functions (Circular functions, arcfunctions) - mathematical functions that are reverse to trigonometric functions.

Arksinus (denotes as arcsin X.; arcsin X. - it's an angle sin. It is equal x.).

Arksinus (y \u003d arcsin x) - reverse trigonometric function to sin. (x \u003d sin y), which has a field of definition and many values . In other words, returns an angle by the meaning of it. sin..

Function y \u003d sin x Continuous and limited on all its numeric straight. Function y \u003d arcsin x - Strictly increases.

Properties of the Arcsin function.

Chart of Arksinus.

Obtaining the ArcSin function.

There is a function y \u003d sin x. On the entire definition area, it is a piecewise monotonous, therefore, the opposite y \u003d arcsin x Not a function. Therefore, we consider the segment on which it only increases and accepts each value of the range of values. Because For function y \u003d sin x On the interval, all values \u200b\u200bof the function turns out at only one value of the argument, it means that there is a reverse function on this segment. y \u003d arcsin xwhich graphics is symmetric graphics function y \u003d sin x on the segment of relatively straight y \u003d X..

Definition and notation

Arksinus is sometimes denoted:
.

Chart of the functions of Arksinus

Schedule function y \u003d arcsin X.

The Arksinus schedule is obtained from the sinus graph, if you change the abscissa and ordinate axis places. To eliminate multi-consciousness, the range of values \u200b\u200blimit the interval on which the monotonna function. Such a definition is called the main value of Arksinus.

Arkkosinus, Arccos.

Definition and notation

Arkkosinus sometimes indicate:
.

Chart of the function of Arkkosinus

Schedule function y \u003d arccos X.

The graph of Arkkosinus is obtained from the cosine graph, if you change the abscissa and ordinate axis places. To eliminate multi-consciousness, the range of values \u200b\u200blimit the interval on which the monotonna function. Such a definition is called the main value of Arkkosinus.

Parity

The Arksinus function is odd:
arcsin (- X) \u003d arcsin (-sin ArcSin X) \u003d arcsin (sin (-arcsin x)) \u003d - Arcsin X.

The function of the ArcCowinus is not even or odd:
arccos (- X) \u003d arcCOS (-COS ArcCOS X) \u003d arcCOS (COS (π-Arccos X)) \u003d π - Arccos X ≠ ± Arccos X

Properties - Extremes, Ascending, Disarm

The functions of the Arksinus and the Arkskosinus are continuous on their field of definition (see proof of continuity). The main properties of the Arksinus and Arkkosinus are presented in the table.

Table of Arksinuses and Arkkosinusov

This table shows the values \u200b\u200bof the arcsinuses and arcsinuses, in degrees and radians, with some values \u200b\u200bof the argument.

≈ 0,7071067811865476
≈ 0,8660254037844386

Formulas

Formulas of the sum and difference

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for

for

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for

Expressions through logarithm, complex numbers

Expressions through hyperbolic functions

Derivatives

;
.
See the derivatives of the Arksinus and Arkkosinus derivatives \u003e\u003e\u003e

Derivatives of higher orders:
,
where is a polynomial degree. It is determined by the formulas:
;
;
.

See the derivatives of the highest orders of Arksinus and Arkkosinus \u003e\u003e\u003e

Integrals

Make the substitution x \u003d sIN T.. We integrate in parts, given that -π / 2 ≤ t ≤ π / 2, cOS T ≥ 0:
.

Express the Arkkosinus through Arksinus:
.

Decomposition in a number

With | x |< 1 The following decomposition takes place:
;
.

Reverse functions

Return to Arksinus and Arkkosinus are sinus and cosine, respectively.

The following formulas are valid throughout the entire field of definition:
sin (arcsin x) \u003d x
cOS (ArcCOS X) \u003d x .

The following formulas are valid only on the set of arcsinus and arcsinus values:
arcsin (SIN X) \u003d X for
arcCOS (COS X) \u003d X at.

References:
I.N. Bronstein, K.A. Semendyaev, a reference book on mathematics for engineers and students of the attendants, "Lan", 2009.

The tasks associated with inverse trigonometric functions are often offered at school final exams and on entrance exams in some universities. A detailed study of this topic can be achieved only at elective classes or in elective courses. The proposed course is designed as fully as possible to develop the ability of each student, to increase its mathematical training.

The course is designed for 10 hours:

1. Functions Arcsin X, ArcCOS X, ArctG X, ArcCTG X (4 hours).

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

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

Lesson 1 (2 hours) Subject: functions y \u003d arcsin x, y \u003d arccos x, y \u003d arctg x, y \u003d arcctg x.

Purpose: full coverage of this issue.

1. Function Y \u003d Arcsin x.

a) For the function y \u003d sin x on the segment there is a reverse (unambiguous) function that the arxinus was called and denoted as follows: y \u003d arcsin x. The reverse function graph is symmetrical with a graph of the main function relative to the bisector I - III coordinate angles.

The properties of the function y \u003d arcsin x.

1) The definition area: segment [-1; one];

2) the area of \u200b\u200bchange: segment;

3) function y \u003d arcsin x is odd: arcsin (-x) \u003d - arcsin x;

4) function y \u003d arcsin x monotonically increasing;

5) The schedule crosses the axis OH, OU at the beginning of the coordinates.

Example 1. Find a \u003d arcsin. This example can be formulated in detail: to find such an argument A lying from the bottom of which is equal to the sinus.

Decision. There are countless arguments, the sinus of which is equal to, for example: etc. But we are only interested in the argument that is on the segment. This argument will be. So, .

Example 2. Find .Decision. Arguing 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 . Does the sense of expression mean:; Arcsin 1.5; ?

c) Place an increase in ascending order: Arcsin, Arcsin (-0.3), Arcsin 0.9.

II. Functions y \u003d arccos x, y \u003d arctg x, y \u003d arcctg x (similarly).

Lesson 2 (2 h) Theme: Inverse trigonometric functions, their graphs.

Objective: At this lesson, it is necessary to work out skills in determining the values \u200b\u200bof trigonometric functions, in the construction of graphs of inverse trigonometric functions using d (y), e (y) and necessary transformations.

At this lesson, perform exercises, including the foundation of the definition area, the values \u200b\u200bof the values \u200b\u200bof the type functions: y \u003d arcsin, y \u003d arccos (x-2), y \u003d arctg (TG x), Y \u003d Arccos.

Function graphs should be built: a) y \u003d arcsin 2x; b) y \u003d 2 arcsin 2x; c) y \u003d arcsin;

d) y \u003d arcsin; e) y \u003d arcsin; e) y \u003d arcsin; g) y \u003d | Arcsin | .

Example.We build a graph Y \u003d Arccos

In the homework, the following exercises can be included: Build graphs of functions: Y \u003d Arccos, Y \u003d 2 ArcCTG X, Y \u003d Arccos | X | .

Reverse Function Charts

Purpose: expanding mathematical knowledge (this is important for applicants in specialty with increased requirements for mathematical preparation) by introducing basic relations for inverse trigonometric functions.

Material for lesson.

Some simple trigonometric operations over inverse trigonometric functions: sin (arcsin x) \u003d x, i xi? one; COS (Ascos X) \u003d X, I XI? one; TG (arctg x) \u003d x, x i r; CTG. (ArcCTG x) \u003d x, x i R.

Exercises.

a) TG (1.5 + Arctg 5) \u003d - CTG (Arctg 5) \u003d .

cTG (arctg x) \u003d; TG (ArcCTG x) \u003d.

b) COS (+ Arcsin 0.6) \u003d - COS (Arcsin 0.6). Let arcsin 0,6 \u003d a, sin a \u003d 0,6;

cos (arcsin x) \u003d; sin (Arccos X) \u003d.

Note: take the "+" sign before the root because A \u003d Arcsin X satisfies.

c) sin (1.5 + arcsin). The answer:;

d) CTG (+ Arctg 3). The answer:;

e) TG (- ArcCTG 4). The answer :.

e) COS (0.5 + ArcCOS). Answer:.

Calculate:

a) sin (2 arctg 5).

Let arctg 5 \u003d a, then Sin 2 A \u003d or sin (2 arctg 5) \u003d ;

b) COS (+ 2 ARCSIN 0.8). The answer: 0.28.

c) arctg + arctg.

Let a \u003d arctg, b \u003d arctg,

then TG (A + B) \u003d .

d) sin (Arcsin + Arcsin).

e) to prove that for all x i [-1; 1] True Arcsin X + Arccos X \u003d.

Evidence:

arcsin X \u003d - ArcCOS X

sIN (Arcsin X) \u003d SIN (- ArcCOS X)

x \u003d COS (ArcCOS X)

For self solutions:sIN (ArcCOS), COS (Arcsin), COS (ArcSin ()), SIN (ArctG (- 3)), TG (ArcCOS), CTG (ArcCOS).

For home solving: 1) SIN (Arcsin 0.6 + Arctg 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.

Purpose: At this lesson, it is to show the use of ratios in converting more complex expressions.

Material for lesson.

ORALLY:

a) sin (Arccos 0.6), COS (Arcsin 0.8);

b) TG (ArcStg 5), CTG (Arctg 5);

c) sin (arctg -3), cos (arcstg ());

d) TG (ArcCOS), CTG (Arccos ()).

Writing:

1) COS (Arcsin + Arcsin + Arcsin).

2) COS (arctg 5-Arccos 0.8) \u003d COS (Arctg 5) COS (Arccos 0.8) + SIN (Arctg 5) SIN (ArcCOS 0.8) \u003d

3) TG (- Arcsin 0,6) \u003d - TG (Arcsin 0,6) \u003d

4)

Independent work will help identify the level of mastering the material

For homework, you can offer:

1) CTG (Arctg + Arctg + Arctg); 2) SIN 2 (Arctg 2 - ArcCTG ()); 3) sin (2 arctg + tg (arcsin)); 4) sin (2 arctg); 5) TG ((Arcsin))

Purpose: to form a presentation of students about inverse trigonometric operations over trigonometric functions, the focus is on the increase in the meaningfulness of the theory under study.

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

Material for lesson:

The study of the new material can be started from the function of the Y \u003d ARCSIN (SIN X) function and building its schedule.

3. Each X i R is put in accordance with Y i, i.e.<= y <= такое, что sin y = sin x.

4. Function is odd: sin (-x) \u003d - SIN X; ARCSIN (SIN (-X)) \u003d - ARCSIN (SIN X).

6. Schedule Y \u003d 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 \u003d SIN (- X) \u003d SINX, 0<= - x <= .

So,

Buing y \u003d arcsin (SIN X) on, will continue symmetrically relative to the start of coordinates on [-; 0], given the accuracy of this function. Using the frequency, we will continue to the entire numeric axis.

Then write some ratios: arcsin (sin a) \u003d a if<= a <= ; arccos (cos A. ) \u003d a if 0<= a <= ; arctg (TG A) \u003d a if< a < ; arcctg (ctg a) = a , если 0 < a < .

And perform the following exercises: a) ArcCOS (SIN 2). RESULT: 2 -; b) Arcsin (COS 0,6). RESULT: - 0.1; c) arctg (TG 2). The answer: 2 -;

d) ArcCTG (TG 0.6). The answer: 0.9; e) ArcCOS (COS (- 2)). Answer: 2 -; e) ARCSIN (SIN (- 0.6)). Answer: - 0,6; g) ARCTG (TG 2) \u003d ArCTG (TG (2 -)). Answer: 2 -; h) arcctg (TG 0.6). Answer: - 0,6; - arctg x; e) Arccos + Arccos

(Circular functions, arcfunctions) - mathematical functions that are reverse to trigonometric functions.

Arkkosinus, reverse function to COS (X \u003d COS Y), y \u003d. Arccos. x. It is determined when and has many values. In other words, returns an angle by the meaning of it. cos..

Arkkosinus (Designation: arccos X.; arccos X. - this is an angle whose cosine is equal x. etc).

Function y \u003d COS X Continuous and limited on all its numeric straight. Function y \u003d Arccos X It is strictly decreasing.

Properties of the Arcsin function.

Obtaining the ArcCOS function.

Dana feature y \u003d COS X. On the entire field of definition, it is a piecewise monotonous, and, it means that inverse y \u003d Arccos X The function is not. Therefore, we consider the segment on which it strictly decreases and takes all its values. On this cut y \u003d COS X strictly monotonically decreases and takes all its values \u200b\u200bonly once, which means there is a reverse function on the segment y \u003d Arccos X, whose graph is symmetrical graphics y \u003d COS X on the segment of relatively straight y \u003d X..

Since trigonometric functions are periodic, then the functions are not unambiguous. So, the equation y \u003d sIN X.When specified, has infinitely many roots. Indeed, due to the periodicity of sinus, if x is such a root, then x + 2πN. (where n is an integer) will also be the root of the equation. In this way, inverse trigonometric functions are meaningful. So that it was easier to work with them, they introduce the concept of their main values. Consider, for example, sinus: y \u003d sIN X.. If you limit the argument x interval, then it is function y \u003d sIN X. Monotonously increases. Therefore, it has an unambiguous reverse function called Arxinus: X \u003d arcsin Y..

If it is not particularly specified, then under the inverse trigonometric functions they mean their main values, which are determined by the following definitions.

Arksinus ( y \u003d. arcsin X.) - this is a function inverse to sinus ( x \u003d. sIN Y.
Arkkosinus ( y \u003d. arccos X.) - this is a function inverse to cosine ( x \u003d. cOS Y.), having a field of definition and many values.
Arctanens ( y \u003d. arctg X.) - this is a function inverse to Tangent ( x \u003d. tG Y.), having a field of definition and many values.
Arkotanens ( y \u003d. arcCTG X.) - this is a function inverse to Kotangent ( x \u003d. cTG Y.), having a field of definition and many values.

Pictures of inverse trigonometric functions

The graphs of inverse trigonometric functions are obtained from the graphs of trigonometric functions with a mirror reflection relative to the direct y \u003d x. See Sine Sections, Kosinus, Tangent, Cotangent.

y \u003d. arcsin X.

y \u003d. arccos X.

y \u003d. arctg X.

y \u003d. arcCTG X.

Basic formulas

Here it is necessary to pay attention to the intervals for which the formulas are valid.

arcsin (SIN X) \u003d X for
sin (arcsin x) \u003d x
arcCOS (COS X) \u003d X for
cOS (ArcCOS X) \u003d x

arctg (TG x) \u003d x for
tG (arctg x) \u003d x
arcCTG (CTG x) \u003d x for
cTG (ArcCTG x) \u003d x

Formulas binding inverse trigonometric functions

Formulas of the sum and difference

at or

at I.

at I.

at or

at I.

at I.

for

for

for

for

for

for

for

for

for

for

References:
I.N. Bronstein, K.A. Semendyaev, a reference book on mathematics for engineers and students of the attendants, "Lan", 2009.