FUNCTION GRAPHICS

Sine function

- lots of R all real numbers.

Set of function values- segment [-1; 1], ie sine function - limited.

The function is odd: sin (−x) = - sin x for all х ∈ R.

Periodic function

sin (x + 2π k) = sin x, where k ∈ Z for all х ∈ 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 scope- lots of R all real numbers.

Set of function values- segment [-1; 1], ie cosine function - limited.

The function is even: cos (−x) = cos x for all х ∈ R.

Periodic function with the smallest positive period 2π:

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

cos x = 0 at
cos x> 0 for all
cos x< 0 for all
The function is increasing from −1 to 1 at intervals:
The function is decreasing from −1 to 1 at 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

Set of function values- the whole number line, i.e. tangent - function unlimited.

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

Periodic function with the smallest positive period π, i.e. tg (x + π k) = tg x, k ∈ Z for all x from the domain.

Cotangent function

Set of function values- the whole number line, i.e. cotangent - function unlimited.

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

Periodic function with the smallest positive period π, i.e. ctg (x + π k) = ctg x, k ∈ Z for all x from the domain.

Arcsine function

Function scope- segment [-1; 1]

Set of function values- the segment -π / 2 arcsin x π / 2, i.e. arcsine function limited.

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

Over the entire area of ​​definition.

Arc cosine function

Function scope- segment [-1; 1]

Set of function values- segment 0 arccos x π, i.e. inverse cosine - function limited.

The function is ascending over the entire domain of definition.

Arctangent function

Function scope- lots of R all real numbers.

Set of function values- segment 0 π, i.e. arctangent - function limited.

The function is odd: arctan (−x) = - arctan x for all х ∈ R.
The function graph is symmetrical about the origin.

The function is ascending over the entire domain of definition.

Arc cotangent function

Function scope- lots of R all real numbers.

Set of function values- segment 0 π, i.e. arc cotangent - function limited.

The function is neither even nor odd.
The graph of the function is asymmetric neither about the origin nor about the Oy axis.

The function is decreasing over the entire domain of definition.

Inverse trigonometric functions are often offered in high school graduation exams and entrance exams in some universities. A detailed study of this topic can only be achieved in extracurricular activities or at elective courses... The proposed course is designed to develop the abilities of each student as fully as possible, to improve his mathematical training.

The course is designed for 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.

Purpose: full 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 the arcsine and denote it as: y = arcsin x. The graph of the inverse function is symmetric with the graph of the main function relative to the bisector of the 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 is odd: arcsin (-x) = - arcsin x;

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

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

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

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

Example 2. Find .Solution. Reasoning 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: since ... 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 = arctan x, y = arcctg x (similar).

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

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

In this lesson, perform exercises that include finding the domain, domain of values ​​of functions of the type: y = arcsin, y = arccos (x-2), y = arctan (tg x), y = arccos.

It is necessary to build graphs of functions: a) y = arcsin 2x; b) y = 2 arcsin 2x; c) y = arcsin;

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

Example. 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 | ...

Inverse function graphs

Lesson number 3 (2 hours) Topic:

Operations on inverse trigonometric functions.

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

Material for the lesson.

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

Exercises.

a) tg (1,5 + arctan 5) = - ctg (arctan 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 (+ arctan 3). Answer:;

e) tg (- arcctg 4) Answer:.

f) 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] is true arcsin x + arccos x =.

Proof:

arcsin x = - arccos x

sin (arcsin x) = sin (- arccos x)

x = cos (arccos x)

For an independent solution: sin (arccos), cos (arcsin), cos (arcsin ()), sin (arctg (- 3)), tg (arccos), ctg (arccos).

For a homemade 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) arctan 0.5 - arctan 3.

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

Purpose: in this lesson to show the use of ratios in the transformation of more complex expressions.

Material for the lesson.

ORALLY:

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

b) tg (arcсtg 5), ctg (arctan 5);

c) sin (arctg -3), cos (arcсtg ());

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

WRITTEN:

1) cos (arcsin + arcsin + arcsin).

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

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

4)

Independent work will help to identify the level of assimilation of the material

1) tg (arctan 2 - arctg) 2) cos (- arctg2) 3) arcsin + arccos	1) cos (arcsin + arcsin) 2) sin (1.5 - arctan 3) 3) arcctg3 - arctg 2

For homework you can suggest:

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

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

Purpose: to form an idea of ​​students about inverse trigonometric operations on trigonometric functions, focus on increasing the meaningfulness of the theory being studied.

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

Lesson material:

You can start learning new material by examining 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) = sinx, 0<= - x <= .

So,

Having constructed y = arcsin (sin x) on, we continue symmetrically about the origin to [-; 0], taking into account the oddness of this function. Using periodicity, we will continue to the entire number axis.

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

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

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

Definition and notation

Arcsine (y = arcsin x) is the inverse sine function (x = sin y -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:
.

Arcsine function graph

Function graph y = arcsin x

The arcsine plot is obtained from the sine plot by swapping the abscissa and ordinate axes. To eliminate ambiguity, the range of values ​​is limited by the interval over which the function is monotonic. This definition is called the main value of the arcsine.

Arccosine, arccos

Definition and notation

Arc cosine (y = arccos x) is the function inverse to the 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:
.

Arccosine function graph

Function graph y = arccos x

The inverse cosine plot is obtained from the cosine plot by swapping the abscissa and ordinate axes. To eliminate ambiguity, the range of values ​​is limited by the interval over which the function is monotonic. This definition is called the main value of the arccosine.

Parity

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

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

	y = arcsin x	y = arccos x
Domain of definition and continuity	- 1 ≤ x ≤ 1	- 1 ≤ x ≤ 1
Range of values
Increase, decrease	increases monotonically	decreases monotonically
Highs
The minimums
Zeros, y = 0	x = 0	x = 1
Points of intersection with the y-axis, x = 0	y = 0	y = π / 2

Arcsine and arccosine table

This table shows the values ​​of arcsines and arccosines, in degrees and radians, for some 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

Logarithm Expressions, Complex Numbers

See also: Derivation of formulas

Expressions in terms of hyperbolic functions

Derivatives

;
.
See Derivative of inverse and inverse cosine 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 arcsine>>>

Integrals

Substitution x = sin t... We integrate by parts, taking into account that -π / 2 ≤ t ≤ π / 2, cos t ≥ 0:
.

Let us express the inverse cosine in terms of the inverse sine:
.

Series expansion

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

Inverse functions

The inverse of the inverse sine and the inverse cosine are sine and cosine, respectively.

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

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

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Technical Institutions, "Lan", 2009.

See also: