Single-digit numbers are added using an addition table. Addition table, and ...
Inverse trigonometric functions(circular functions, arc functions) - mathematical functions that are inverse to trigonometric functions.
Arcsine(denoted as arcsin x; arcsin x Is the angle sin it equals x).
Arcsine (y = arcsin x) is the inverse trigonometric function to sin (x = sin y), which has a domain and a set of values 
... In other words, it returns the angle by its value sin.
Function y = sin x continuous and bounded on 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... It is piecewise monotone throughout its domain of definition, 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 the 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, in which the graph is symmetric to the graph of the function y = sin x on a segment relative to a straight line y = x.
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 functionsSum 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 formulasExpressions in terms of hyperbolic functions
Derivatives
;
.
See Derivative 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 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
Inverse to arcsine and arccosine 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 on 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.
Inverse trigonometric tasks are often offered in high school graduation exams and 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, 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 follows: 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 the values of trigonometric functions, in plotting inverse trigonometric functions using D (y), E (y) and the necessary transformations.
In this lesson, perform exercises that include finding the domain, the 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 (arctg 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 offer:
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 (tan 0.6). Answer: - 0.6; - arctg x; e) arccos + arccos
Since trigonometric functions are periodic, their inverse functions are not single-valued. So, the equation y = sin x, for a given, has infinitely many roots. Indeed, due to the periodicity of the sine, if x is such a root, then x + 2πn(where n is an integer) will also be the root of the equation. Thus, inverse trigonometric functions are multivalued... To make it easier to work with them, they introduce the concept of their main meanings. Consider, for example, sine: y = sin x... If we restrict the argument x by an interval, then on it the function y = sin x increases monotonically. Therefore, it has a single-valued inverse function, which is called the arcsine: x = arcsin y.
Unless otherwise stated, inverse trigonometric functions mean their main meanings, which are determined by the following definitions.
Arcsine ( y = arcsin x) is the inverse sine function ( x = sin y
Arccosine ( y = arccos x) is the inverse function of cosine ( x = cos y), which has a domain and many values.
Arc tangent ( y = arctg x) is the inverse function of the tangent ( x = tg y), which has a domain and many values.
Arccotangent ( y = arcctg x) is the inverse function of the cotangent ( x = ctg y), which has a domain and many values.
Inverse trigonometric function graphs
The graphs of inverse trigonometric functions are obtained from the graphs of trigonometric functions by mirroring them relative to the straight line y = x. See sections Sine, Cosine, Tangent, Cotangent.
y = arcsin x
y = arccos x
y = arctg x
y = arcctg x
Basic formulas
Here you should pay special attention to the intervals for which the formulas are valid.
arcsin (sin x) = x at
sin (arcsin x) = x
arccos (cos x) = x at
cos (arccos x) = x
arctan (tg x) = x at
tg (arctan x) = x
arcctg (ctg x) = x at
ctg (arcctg x) = x
Formulas Relating Inverse Trigonometric Functions
See also: Derivation of formulas for inverse trigonometric functionsSum and Difference Formulas
at or
at and
at and
at or
at and
at and
at
at
at
at
at
at
at
at
at
at
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Technical Institutions, "Lan", 2009.
