Functions graphics. Arksinus, Arkkosinus - Properties, Graphs, Formulas Chart Arcsin x 2

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 values 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

Lesson number 3 (2 h.) Subject:

Operations over inverse trigonometric functions.

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.

Lesson number 4 (2h.) Subject: operations over inverse trigonometric functions.

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

1) TG (Arctg 2 - ArCTG) 2) COS (- Arctg2) 3) Arcsin + Arccos	1) COS (Arcsin + Arcsin) 2) SIN (1.5 - Arctg 3) 3) ArcCTG3 - Arctg 2

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))

Lesson number 5 (2h) Subject: Inverse trigonometric operations on trigonometric functions.

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

Functions graphics

Function sinus

- lots of R.all valid numbers.

Many function values - segment [-1; 1], i.e. sinus function - limited.

Function odd: sin (-x) \u003d - sin x for all x ∈ R..

Periodic function

sin (x + 2π · k) \u003d sin x, where k ∈ Z. for all x ∈ R..

sin x \u003d 0 with x \u003d π · k, k ∈ Z..

sIN X\u003e 0 (positive) for all x ∈ (2π · k, π + 2π · k), k ∈ Z..

sIN X.< 0 (negative) for all x ∈ (π + 2π · k, 2π + 2π · k), k ∈ Z..

Cosine function

Function definition area- lots of R.all valid numbers.

Many function values - segment [-1; 1], i.e. cosine function - limited.

Function even: COS (-X) \u003d COS X for all x ∈ R..

Periodic function With the smallest positive period 2π:

cOS (x + 2π · k.) \u003d COS X, where k. ∈ Z. for all x ∈ R..

cOS X \u003d 0for
cOS X\u003e 0 for all
cOS X.< 0 for all
The function is increasing from -1 to 1 at intervals:
Function decreases from -1 to 1 at intervals:
The greatest value of the function sin x \u003d 1 At points:
The smallest value of the function sin x \u003d -1 At points:

Tangent feature

Many function values - All numeric straight, i.e. Tangent - function unlimited.

Function odd: TG (-X) \u003d - TG X
The graph of the function is symmetrical relative to the OY axis.

Periodic function With the smallest positive period π, i.e. TG (x + π · k.) \u003d TG X, k. ∈ Z. for all x from the definition area.

Cotanence feature

Many function values - All numeric straight, i.e. Kotangent - function unlimited.

Function odd: CTG (-X) \u003d - CTG X for all x from the definition area.
The graph of the function is symmetrical relative to the OY axis.

Periodic function With the smallest positive period π, i.e. CTG (X + π · k.) \u003d CTG x, k. ∈ Z. for all x from the definition area.

Arksinus feature

Function definition area- segment [-1; one]

Many function values - Cut -π / 2 ARCSIN X π / 2, i.e. Arksinus - function limited.

Function odd: Arcsin (-X) \u003d - Arcsin X for all x ∈ R..
The graph of the function is symmetrical on the start of the coordinates.

On the entire definition area.

Arkkosinus function

Function definition area- segment [-1; one]

Many function values - Cut 0 Arccos X π, i.e. Arkkosinus - Function limited.

The function is increasing On the entire definition area.

Function Arctgernes

Function definition area- lots of R.all valid numbers.

Many function values - Cut 0 π, i.e. ARCTANHANCE - FUNCTION limited.

Function odd: arctg (-x) \u003d - arctg x for all x ∈ R..
The graph of the function is symmetrical on the start of the coordinates.

The function is increasing On the entire definition area.

Function Arkkothangence

Function definition area- lots of R.all valid numbers.

Many function values - Cut 0 π, i.e. Arkotangent - function limited.

The function is neither even nor odd.
The graph of the function is asymmetrical or relative to the start of coordinates or relative to the OY axis.

The function is descending On the entire definition area.

Definition and notation

Arksinus (Y \u003d arcsin X.) - this is a function, reverse to sinus (x \u003d sIN Y. -1 ≤ x ≤ 1 and many values \u200b\u200b-π / 2 ≤ y ≤ π / 2.
sin (arcsin x) \u003d x ;
arcsin (SIN X) \u003d X .

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 (Y \u003d arccos X.) is a function inverse to cosine (x \u003d cOS Y.). It has a field of definition -1 ≤ x ≤ 1 and many values 0 ≤ y ≤ π.
cOS (ArcCOS X) \u003d x ;
arcCOS (COS X) \u003d X .

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.

	y \u003d. arcsin X.	y \u003d. arccos X.
Definition and continuity area	- 1 ≤ x ≤ 1	- 1 ≤ x ≤ 1
Region of values
Ascending, descending	Monotonously increase	Monotonously decrease
Maximum
Minima
Zeros, y \u003d 0	x \u003d. 0	x \u003d. 1
Point of intersection with the ordinate axis, x \u003d 0	y \u003d. 0	y \u003d π / 2

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.

X.	arcsin X.		arccos X.
	Grad.	glad.	Grad.	glad.
- 1	- 90 °	-	180 °	π
-	- 60 °	-	150 °
-	- 45 °	-	135 °
-	- 30 °	-	120 °
0	0°	0	90 °
	30 °		60 °
	45 °		45 °
	60 °		30 °
1	90 °		0°	0

≈ 0,7071067811865476
≈ 0,8660254037844386

Formulas

See also: The output of the formulas of 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

Expressions through logarithm, complex numbers

See also: Conclusion of formulas

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.

See also: