Arctg cast formulas. Trigonometry. Inverse trigonometric functions. Inverse trigonometric function graphs

Definition and notation

Arcsine (y \u003d arcsin x) is the inverse sine function (x \u003d sin y -1 ≤ x ≤ 1 and the set of values \u200b\u200b-π / 2 ≤ y ≤ π / 2.
sin (arcsin x) \u003d x ;
arcsin (sin x) \u003d x .

Arcsine is sometimes denoted as follows:
.

Arcsine function graph

Function graph y \u003d 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 \u200b\u200bis 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

Arccosine (y \u003d arccos x) is the function inverse to the cosine (x \u003d cos y). It has a scope -1 ≤ x ≤ 1 and many meanings 0 ≤ y ≤ π.
cos (arccos x) \u003d x ;
arccos (cos x) \u003d x .

Arccosine is sometimes denoted as follows:
.

Arccosine function graph

Function graph y \u003d arccos x

The arccosine plot is obtained from the cosine plot by swapping the abscissa and ordinate axes. To eliminate ambiguity, the range of values \u200b\u200bis 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) \u003d arcsin (-sin arcsin x) \u003d arcsin (sin (-arcsin x)) \u003d - arcsin x

The inverse cosine function 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 - 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 \u003d arcsin x	y \u003d arccos x
Scope and continuity	- 1 ≤ x ≤ 1	- 1 ≤ x ≤ 1
Range of values
Increase, decrease	increases monotonically	decreases monotonically
Highs
The minimums
Zeros, y \u003d 0	x \u003d 0	x \u003d 1
Points of intersection with the y-axis, x \u003d 0	y \u003d 0	y \u003d π / 2

Arcsine and Arccosine Table

This table shows the values \u200b\u200bof arcsines and arccosines, in degrees and radians, for some values \u200b\u200bof 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: Deriving formulas

Expressions in terms of hyperbolic functions

Derivatives

;
.
See Derivative Arcsine and Arccosine Derivatives\u003e\u003e\u003e

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\u003e\u003e\u003e

Integrals

Substitution x \u003d 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 arcsine:
.

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) \u003d x
cos (arccos x) \u003d x .

The following formulas are valid only for the set of arcsine and arcsine values:
arcsin (sin x) \u003d x at
arccos (cos x) \u003d x at.

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

See also:

Inverse trigonometric functions are mathematical functions that are inverse trigonometric functions.

Function y \u003d arcsin (x)

The arcsine of a number α is a number α from the interval [-π / 2; π / 2], the sine of which is equal to α.
Function graph
The function y \u003d sin\u2061 (x) on the segment [-π / 2; π / 2] is strictly increasing and continuous; hence, it has an inverse function, strictly increasing and continuous.
The inverse function for the function y \u003d sin\u2061 (x), where x ∈ [-π / 2; π / 2], is called the arcsine and is denoted by y \u003d arcsin (x), where x ∈ [-1; 1].
So, according to the definition of the inverse function, the domain of definition of the arcsine is the segment [-1; 1], and the set of values \u200b\u200bis the segment [-π / 2; π / 2].
Note that the graph of the function y \u003d arcsin (x), where x ∈ [-1; 1]. Is symmetric to the graph of the function y \u003d sin (\u2061x), where x ∈ [-π / 2; π / 2], relative to the bisector of coordinate angles first and third quarters.

Function range y \u003d arcsin (x).

Example # 1.

Find arcsin (1/2)?

Since the range of values \u200b\u200bof the function arcsin (x) belongs to the interval [-π / 2; π / 2], only the value of π / 6 is suitable. Consequently, arcsin (1/2) \u003d π / 6.
Answer: π / 6

Example # 2.
Find arcsin (- (√3) / 2)?

Since the range of values \u200b\u200barcsin (x) х ∈ [-π / 2; π / 2], only the value -π / 3 is suitable. Therefore, arcsin (- (√3) / 2) \u003d - π / 3.

Function y \u003d arccos (x)

The inverse cosine of a number α is a number α from an interval whose cosine is equal to α.

Function graph

The function y \u003d cos (\u2061x) on a segment is strictly decreasing and continuous; hence, it has an inverse function, strictly decreasing and continuous.
The inverse function for the function y \u003d cos\u2061x, where x ∈, is called inverse cosine and is denoted by y \u003d arccos (x), where х ∈ [-1; 1].
So, according to the definition of the inverse function, the domain of definition of the arccosine is the segment [-1; 1], and the set of values \u200b\u200bis the segment.
Note that the graph of the function y \u003d arccos (x), where x ∈ [-1; 1], is symmetric to the graph of the function y \u003d cos (\u2061x), where x ∈, relative to the bisector of the coordinate angles of the first and third quarters.

The domain of the function y \u003d arccos (x).

Example No. 3.

Find arccos (1/2)?

Since the range of values \u200b\u200bis arccos (x) х∈, only the value π / 3 is suitable; therefore, arccos (1/2) \u003d π / 3.
Example No. 4.
Find arccos (- (√2) / 2)?

Since the range of values \u200b\u200bof the function arccos (x) belongs to the interval, only the value 3π / 4 is suitable; therefore, arccos (- (√2) / 2) \u003d 3π / 4.

Answer: 3π / 4

Function y \u003d arctan (x)

The arctangent of a number α is a number α from the interval [-π / 2; π / 2], the tangent of which is equal to α.

Function graph

The tangent function is continuous and strictly increasing on the interval (-π / 2; π / 2); hence, it has an inverse function, which is continuous and strictly increasing.
The inverse function for the function y \u003d tg\u2061 (x), where х∈ (-π / 2; π / 2); is called the arctangent and is denoted by y \u003d arctan (x), where х∈R.
So, according to the definition of the inverse function, the domain of definition of the arctangent is the interval (-∞; + ∞), and the set of values \u200b\u200bis the interval
(-π / 2; π / 2).
Note that the graph of the function y \u003d arctan (x), where х∈R, is symmetric to the graph of the function y \u003d tg\u2061x, where х ∈ (-π / 2; π / 2), relative to the bisector of the coordinate angles of the first and third quarters.

Function range y \u003d arctan (x).

Example # 5?

Find arctan ((√3) / 3).

Since the range of values \u200b\u200barctan (x) х ∈ (-π / 2; π / 2), only the value π / 6 is suitable. Therefore, arctg ((√3) / 3) \u003d π / 6.
Example # 6.
Find arctg (-1)?

Since the range of values \u200b\u200barctan (x) х ∈ (-π / 2; π / 2), only the value -π / 4 is suitable. Therefore, arctg (-1) \u003d - π / 4.

Function y \u003d arcctg (x)

The arccotangent of a number α is a number α from the interval (0; π), the cotangent of which is α.

Function graph

On the interval (0; π), the cotangent function is strictly decreasing; moreover, it is continuous at every point of this interval; therefore, on the interval (0; π), this function has an inverse function, which is strictly decreasing and continuous.
The inverse function for the function y \u003d ctg (x), where х ∈ (0; π), is called the arc cotangent and is denoted by y \u003d arcctg (x), where х∈R.
So, according to the definition of the inverse function, the domain of definition of the arc cotangent is R, and the set of values \u200b\u200bis the interval (0; π). The graph of the function y \u003d arcctg (x), where х∈R is symmetric to the graph of the function y \u003d ctg (x) х∈ (0 ; π), relative to the bisector of the coordinate angles of the first and third quarters.

Function range y \u003d arcctg (x).

Example # 7.
Find arcctg ((√3) / 3)?

Since the range of values \u200b\u200bis arcctg (x) х ∈ (0; π), then only π / 3 is suitable; therefore, arccos ((√3) / 3) \u003d π / 3.

Example # 8.
Find arcctg (- (√3) / 3)?

Since the range of values \u200b\u200bis arcctg (x) х∈ (0; π), only the value 2π / 3 is suitable; therefore, arccos (- (√3) / 3) \u003d 2π / 3.

Editors: Ageeva Lyubov Alexandrovna, Gavrilina Anna Viktorovna

In this lesson we will look at the features inverse functions and repeat inverse trigonometric functions... The properties of all the main inverse trigonometric functions will be considered separately: arcsine, arccosine, arctangent and arccotangent.

This lesson will help you prepare for one of the types of assignments AT 7 and C1.

Preparation for the exam in mathematics

Experiment

Lesson 9. Inverse trigonometric functions.

Theory

Lesson summary

Let's remember when we come across such a concept as an inverse function. For example, consider the squaring function. Suppose we have a square room with sides of 2 meters and we want to calculate its area. To do this, using the formula for the square of the square, we raise the two to a square and as a result we get 4 m 2. Now let's imagine the inverse problem: we know the area of \u200b\u200ba square room and we want to find the lengths of its sides. If we know that the area is still the same 4 m 2, then we will perform the reverse action to squaring - extracting the arithmetic square root, which will give us a value of 2 m.

Thus, for the function of squaring a number, the inverse function is to extract the arithmetic square root.

Specifically, in the above example, we had no problems calculating the side of the room, since we understand that this is a positive number. However, if we break away from this case and consider the problem in a more general way: “Calculate a number whose square is four”, we will face a problem - there are two such numbers. These are 2 and -2 because is also equal to four. It turns out that the inverse problem in the general case is solved ambiguously, and the action of determining the number that squared gave us the number we know? has two results. It is convenient to show it on the chart:

And this means that we cannot call such a law of correspondence of numbers a function, since for a function one value of the argument corresponds strictly one function value.

In order to introduce precisely the inverse function to squaring, the concept of arithmetic square root was proposed, which gives only non-negative values. Those. for a function, the inverse function is considered.

Similarly, there are functions inverse to trigonometric functions, they are called inverse trigonometric functions... Each of the functions we have considered has its own inverse, they are called: arcsine, arccosine, arctangent and arccotangent.

These functions solve the problem of calculating the angles from the known value of the trigonometric function. For example, using a table of values \u200b\u200bof basic trigonometric functions, you can calculate the sine of which angle is. We find this value in the line of sines and determine which angle it corresponds to. The first thing I want to answer is that this is an angle or, but if you have a table of values \u200b\u200bbefore, you will immediately notice another contender for an answer - this is an angle or. And if we remember the period of the sine, then we understand that the angles at which the sine is equal are infinite. And such a set of angle values \u200b\u200bcorresponding to a given value of the trigonometric function will be observed for cosines, tangents and cotangents, since they all have periodicity.

Those. we are facing the same problem we had for calculating the argument value from the function value for the square action. And in this case, for inverse trigonometric functions, a restriction on the range of values \u200b\u200bthat they give when calculating was introduced. This property of such inverse functions is called narrowing the range, and it is necessary for them to be called functions.

For each of the inverse trigonometric functions, the range of angles that it returns is different, and we will consider them separately. For example, arcsine returns angle values \u200b\u200bin the range from to.

The ability to work with inverse trigonometric functions will be useful to us when solving trigonometric equations.

We will now indicate the basic properties of each of the inverse trigonometric functions. If you want to learn more about them, refer to the chapter "Solving trigonometric equations" in the 10th grade program.

Consider the properties of the arcsine function and build its graph.

Definition.Arcsine of a numberx

The main properties of the arcsine:

1) at,

2) at.

Basic properties of the arcsine function:

1) Scope ;

2) Range of values ;

3) The function is odd. It is desirable to remember this formula separately, since it is useful for transformations. Also note that the oddness implies the symmetry of the function graph relative to the origin;

Let's plot the function:

Note that none of the sections of the function graph is repeated, which means that the arcsine is not a periodic function, in contrast to the sine. The same will apply to all other arc functions.

Consider the properties of the inverse cosine function and build its graph.

Definition.Arccosine numberx is called the value of the angle y for which. Moreover, as restrictions on the values \u200b\u200bof the sine, but as the selected range of angles.

The main properties of the arccosine:

1) at,

2) at.

Basic properties of the inverse cosine function:

1) Scope ;

2) Range of values;

3) The function is neither even nor odd, i.e. general view ... It is also desirable to remember this formula, it will be useful to us later;

4) The function decreases monotonically.

Let's plot the function:

Consider the properties of the arctangent function and build its graph.

Definition.Arctangent of the numberx called the value of the angle y for which Moreover, since there are no restrictions on the tangent values, but as the selected range of angles.

The main properties of the arctangent:

1) at,

2) at.

The main properties of the arctangent function:

1) Scope of definition;

2) Range of values ;

3) The function is odd ... This formula is useful as well as similar ones. As in the case with the arcsine, the oddness implies the symmetry of the function graph relative to the origin;

4) The function increases monotonically.

Let's plot the function:

Lessons 32-33. Inverse trigonometric functions

09.07.2015 8936 0

Purpose: consider inverse trigonometric functions, their use to write solutions of trigonometric equations.

I. Communication of the topic and purpose of the lessons

II. Learning new material

1. Inverse trigonometric functions

Let's start our discussion of this topic with the following example.

Example 1

Let's solve the equation:a) sin x \u003d 1/2; b) sin x \u003d a.

a) On the ordinate, we postpone the value 1/2 and plot the anglesx 1 and x2, for whichsin x \u003d 1/2. Moreover, x1 + x2 \u003d π, whence x2 \u003d π -x 1 ... According to the table of values \u200b\u200bof trigonometric functions, we find the value x1 \u003d π / 6, thenLet us take into account the periodicity of the sine function and write down the solutions of this equation:where k ∈ Z.

b) It is obvious that the algorithm for solving the equationsin x \u003d a is the same as in the previous paragraph. Of course, now the value a is plotted along the ordinate. It becomes necessary to somehow designate the angle x1. We agreed to denote such an angle by the symbolarcsin and. Then the solutions of this equation can be written in the formThese two formulas can be combined into one:wherein

The rest of the inverse trigonometric functions are introduced in a similar way.

It is very often necessary to determine the value of an angle from the known value of its trigonometric function. This problem is multivalued - there are countless angles, the trigonometric functions of which are equal to the same value. Therefore, proceeding from the monotonicity of trigonometric functions, the following inverse trigonometric functions are introduced to uniquely determine the angles.

Arcsine of number a (arcsin , whose sine is equal to a, i.e.

Arccosine numbera (arccos a) is such an angle a from an interval whose cosine is equal to a, i.e.

Arc tangent of a numbera (arctg a) - such an angle a from the intervalwhose tangent is equal to a, i.e.tg a \u003d a.

Arccotangent of numbera (arcctg a) is an angle a from the interval (0; π), the cotangent of which is equal to a, i.e.ctg a \u003d a.

Example 2

Let's find:

Taking into account the definitions of inverse trigonometric functions, we get:

Example 3

We calculate

Let the angle a \u003d arcsin 3/5, then by definitionsin a \u003d 3/5 and ... Therefore, it is necessary to findcos and. Using the main trigonometric identity, we get:It was taken into account that cos a ≥ 0. So,

Function properties	Function
	y \u003d arcsin x	y \u003d arccos x	y \u003d arctan x	y \u003d arcctg x
Domain	x ∈ [-1; 1]	x ∈ [-1; 1]	х ∈ (-∞; + ∞)	x ∈ (-∞ + ∞)
Range of values	y ∈ [-π / 2; π / 2]	y ∈	y ∈ (-π / 2; π / 2)	y ∈ (0; π)
Parity	Odd	Neither even nor odd	Odd	Neither even nor odd
Function zeros (y \u003d 0)	For x \u003d 0	For x \u003d 1	For x \u003d 0	y ≠ 0
Intervals of constancy	y\u003e 0 for x ∈ (0; 1], at< 0 при х ∈ [-1; 0)	y\u003e 0 for x ∈ [-1; 1)	y\u003e 0 for х ∈ (0; + ∞), at< 0 при х ∈ (-∞; 0)	y\u003e 0 for x ∈ (-∞; + ∞)
Monotone	Increasing	Decreases	Increasing	Decreases
Relationship with trigonometric function	sin y \u003d x	cos y \u003d x	tg y \u003d x	ctg y \u003d x
Schedule

Here are some more typical examples related to the definitions and basic properties of inverse trigonometric functions.

Example 4

Find the domain of the function

For the function y to be defined, the inequalitywhich is equivalent to the system of inequalitiesThe solution to the first inequality is the interval x∈ (-∞; + ∞), the second -This gap and is a solution to the system of inequalities, and hence the domain of definition of the function

Example 5

Find the area of \u200b\u200bchange of the function

Consider the behavior of the functionz \u003d 2x - x2 (see figure).

It is seen that z ∈ (-∞; 1]. Considering that the argumentz the arc cotangent function varies within the indicated limits, from the data in the table we obtain thatSo the area of \u200b\u200bchange

Example 6

Let us prove that the function y \u003darctg x is odd. Let beThen tan a \u003d -x or x \u003d - tan a \u003d tan (- a), and Therefore, - a \u003d arctan x or a \u003d - arctan x. Thus, we see thatthat is, y (x) is an odd function.

Example 7

Let us express in terms of all inverse trigonometric functions

Let be It's obvious that Then Since

Let's introduce an angle As then

Similarly, therefore and

So,

Example 8

Let us plot the function y \u003dcos (arcsin x).

We denote a \u003d arcsin x, then We take into account that x \u003d sin a and y \u003d cos a, i.e. x 2 + y2 \u003d 1, and restrictions on x (x∈ [-1; 1]) and y (y ≥ 0). Then the graph of the function y \u003dcos (arcsin x) is a semicircle.

Example 9

Let us plot the function y \u003darccos (cos x).

Since the function cos x changes on the segment [-1; 1], then the function y is defined on the entire numerical axis and changes on the segment. We will keep in mind that y \u003darccos (cos x) \u003d x on the segment; the function y is even and periodic with a period of 2π. Taking into account that these properties are possessed by the functioncos x, now it's easy to plot.

Here are some useful equalities:

Example 10

Find the smallest and largest values \u200b\u200bof the functionWe denote then We get the function This function has a minimum at the pointz \u003d π / 4, and it is equal to The greatest value of the function is attained at the pointz \u003d -π / 2, and it is equal to Thus, and

Example 11

Let's solve the equation

Let's take into account that Then the equation has the form: or from where By the definition of the arctangent, we get:

2. Solution of the simplest trigonometric equations

Similarly to example 1, you can get solutions to the simplest trigonometric equations.

The equation	Decision
tgx \u003d a
ctg x \u003d a

Example 12

Let's solve the equation

Since the sine function is odd, we write the equation in the formSolutions to this equation:where do we find

Example 13

Let's solve the equation

Using the above formula, we write down the solutions to the equation:and find

Note that in particular cases (a \u003d 0; ± 1), when solving the equationssin x \u003d a and cos x \u003d and it is easier and more convenient to use not general formulas, but to write down solutions based on the unit circle:

for the equation sin х \u003d 1 solutions

for the equation sin х \u003d 0 solutions х \u003d π k;

for the equation sin x \u003d -1 solutions

for the equation cos x \u003d 1 solutions x \u003d 2πk;

for the equation cos x \u003d 0 solutions

for the equation cos x \u003d -1 solutions

Example 14

Let's solve the equation

Since in this example there is a special case of the equation, then using the corresponding formula we write the solution:where will we find

III. Test questions (frontal survey)

1. Give a definition and list the main properties of inverse trigonometric functions.

2. Give the graphs of inverse trigonometric functions.

3. Solution of the simplest trigonometric equations.

IV. Assignment in the classroom

§ 15, No. 3 (a, b); 4 (c, d); 7 (a); 8 (a); 12 (b); 13 (a); 15 (c); 16 (a); 18 (a, b); 19 (c); 21;

§ 16, No. 4 (a, b); 7 (a); 8 (b); 16 (a, b); 18 (a); 19 (c, d);

§ 17, No. 3 (a, b); 4 (c, d); 5 (a, b); 7 (c, d); 9 (b); 10 (a, c).

V. Homework

§ 15, No. 3 (c, d); 4 (a, b); 7 (c); 8 (b); 12 (a); 13 (b); 15 (d); 16 (b); 18 (c, d); 19 (d); 22;

§ 16, No. 4 (c, d); 7 (b); 8 (a); 16 (c, d); 18 (b); 19 (a, b);

§ 17, No. 3 (c, d); 4 (a, b); 5 (c, d); 7 (a, b); 9 (d); 10 (b, d).

Vi. Creative tasks

1. Find the domain of the function:

Answers:

2. Find the range of values \u200b\u200bof the function:

Answers:

3. Plot the function:

Vii. Summing up the lessons