Graph of the function y 2 arcsin x. Inverse trigonometric functions

FUNCTION GRAPHICS

Sine function

- many R all real numbers.

Multiple Function Values— segment [-1; 1], i.e. sine function - limited.

Odd function: sin(−x)=−sin x for all x ∈ R.

The function is periodic

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

sin x = 0 for x = π k , k ∈ Z.

sin x > 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 Domain- many R all real numbers.

Multiple Function Values— segment [-1; 1], i.e. cosine function - limited.

Even function: cos(−x)=cos x for all x ∈ R.

The function is periodic with the smallest positive period 2π:

cos(x+2π k) = cos x, where k ∈ Z for all x ∈ R.

cos x = 0 at
cos x > 0 for everyone
cos x< 0 for everyone
Function increases from −1 to 1 on intervals:
The function is decreasing from −1 to 1 on intervals:
The largest value of the function sin x = 1 at points:
The smallest value of the function sin x = −1 at points:

Tangent function

Multiple Function Values— the entire number line, i.e. tangent - function unlimited.

Odd function: tg(−x)=−tg x
The graph of the function is symmetrical about the OY axis.

The function is periodic with the smallest positive period π, i.e. tg(x+π k) = tan x, k ∈ Z for all x from the domain of definition.

Cotangent function

Multiple Function Values— the entire number line, i.e. cotangent - function unlimited.

Odd function: ctg(−x)=−ctg x for all x from the domain of definition.
The graph of the function is symmetrical about the OY axis.

The function is periodic with the smallest positive period π, i.e. cotg(x+π k)=ctg x, k ∈ Z for all x from the domain of definition.

Arcsine function

Function Domain— segment [-1; 1]

Multiple Function Values- segment -π /2 arcsin x π /2, i.e. arcsine - function limited.

Odd function: arcsin(−x)=−arcsin x for all x ∈ R.
The graph of the function is symmetrical about the origin.

Throughout the entire definition area.

Arc cosine function

Function Domain— segment [-1; 1]

Multiple Function Values— segment 0 arccos x π, i.e. arccosine - function limited.

The function is increasing over the entire definition area.

Arctangent function

Function Domain- many R all real numbers.

Multiple Function Values— segment 0 π, i.e. arctangent - function limited.

Odd function: arctg(−x)=−arctg x for all x ∈ R.
The graph of the function is symmetrical about the origin.

The function is increasing over the entire definition area.

Arc tangent function

Function Domain- many R all real numbers.

Multiple Function Values— segment 0 π, i.e. arccotangent - function limited.

The function is neither even nor odd.
The graph of the function is asymmetrical neither with respect to the origin of coordinates, nor with respect to the Oy axis.

The function is decreasing over the entire definition area.

Problems involving inverse trigonometric functions are often offered in GCSEs and entrance exams in 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 values 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

Lesson No. 3 (2 hours) Topic:

Operations on inverse trigonometric 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 = arctan, b = arctan,

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.

Lesson No. 4 (2 hours) Topic: Operations on inverse trigonometric functions.

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.

1) tg (arctg 2 – arctg) 2) cos( - arctan2) 3) arcsin + arccos	1) cos (arcsin + arcsin) 2) sin (1.5 - arctan 3) 3) arcctg3 – arctg 2

For homework we 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 ))

Lesson No. 5 (2 hours) Topic: Inverse trigonometric operations on trigonometric functions.

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 its graph.

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 with respect to the origin of coordinates 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 (y = arcsin x) is the inverse function of sine (x = siny -1 ≤ x ≤ 1 and the set of values ​​-π /2 ≤ y ≤ π/2.
sin(arcsin x) = x ;
arcsin(sin x) = x .

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

Arc cosine (y = arccos x) is the inverse function of cosine (x = cos y). It has a scope -1 ≤ x ≤ 1 and many meanings 0 ≤ y ≤ π.
cos(arccos x) = x ;
arccos(cos x) = x .

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.

	y= arcsin x	y= arccos x
Scope and continuity	- 1 ≤ x ≤ 1	- 1 ≤ x ≤ 1
Range of values
Ascending, descending	monotonically increases	monotonically decreases
Highs
Minimums
Zeros, y = 0	x = 0	x = 1
Intercept points with the ordinate axis, x = 0	y= 0	y = π/ 2

Table of arcsines and arccosines

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

x	arcsin x		arccos x
	hail	glad.	hail	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: Derivation of formulas for inverse trigonometric functions

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

See also: Deriving formulas

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 .

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

See also: