Inverse trigonometric functions and their properties. Let's express it in terms of all inverse trigonometric functions. Basic relations of inverse trigonometric functions
TO inverse trigonometric functions The following 6 functions include: arcsine , arccosine , arctangent , arccotangent , arcsecant And arccosecant .
Since the original trigonometric functions are periodic, then the inverse functions, generally speaking, are polysemantic . To ensure a one-to-one correspondence between two variables, the domains of definition of the original trigonometric functions are limited by considering only them main branches . For example, the function \(y = \sin x\) is considered only in the interval \(x \in \left[ ( - \pi /2,\pi /2) \right]\). On this interval, the inverse arcsine function is uniquely defined.
Arcsine function
The arcsine of the number \(a\) (denoted by \(\arcsin a\)) is the value of the angle \(x\) in the interval \(\left[ ( - \pi /2,\pi /2) \right]\), for which \(\sin x = a\). The inverse function \(y = \arcsin x\) is defined at \(x \in \left[ ( -1,1) \right]\), its range of values is \(y \in \left[ ( - \pi / 2,\pi /2) \right]\).
Arc cosine function
The arccosine of the number \(a\) (denoted \(\arccos a\)) is the value of the angle \(x\) in the interval \(\left[ (0,\pi) \right]\), at which \(\cos x = a\). The inverse function \(y = \arccos x\) is defined at \(x \in \left[ ( -1,1) \right]\), its range of values belongs to the segment \(y \in \left[ (0,\ pi)\right]\).
Arctangent function
Arctangent of the number a(denoted by \(\arctan a\)) is the value of the angle \(x\) in the open interval \(\left((-\pi/2, \pi/2) \right)\), at which \(\tan x = a\). The inverse function \(y = \arctan x\) is defined for all \(x \in \mathbb(R)\), the arctangent range is equal to \(y \in \left((-\pi/2, \pi/2 )\right)\).
Arc tangent function
The arccotangent of the number \(a\) (denoted by \(\text(arccot ) a\)) is the value of the angle \(x\) in the open interval \(\left[ (0,\pi) \right]\), at which \(\cot x = a\). The inverse function \(y = \text(arccot ) x\) is defined for all \(x \in \mathbb(R)\), its range of values is in the interval \(y \in \left[ (0,\pi) \right]\).
Arcsecant function
The arcsecant of the number \(a\) (denoted by \(\text(arcsec ) a\)) is the value of the angle \(x\) at which \(\sec x = a\). The inverse function \(y = \text(arcsec ) x\) is defined at \(x \in \left(( - \infty , - 1) \right] \cup \left[ (1,\infty ) \right)\ ), its range of values belongs to the set \(y \in \left[ (0,\pi /2) \right) \cup \left((\pi /2,\pi ) \right]\).
Arccosecant function
The arccosecant of the number \(a\) (denoted \(\text(arccsc ) a\) or \(\text(arccosec ) a\)) is the value of the angle \(x\) at which \(\csc x = a\ ). The inverse function \(y = \text(arccsc ) x\) is defined at \(x \in \left(( - \infty , - 1) \right] \cup \left[ (1,\infty ) \right)\ ), the range of its values belongs to the set \(y \in \left[ ( - \pi /2,0) \right) \cup \left((0,\pi /2) \right]\).
Principal values of the arcsine and arccosine functions (in degrees)
| \(x\) | \(-1\) | \(-\sqrt 3/2\) | \(-\sqrt 2/2\) | \(-1/2\) | \(0\) | \(1/2\) | \(\sqrt 2/2\) | \(\sqrt 3/2\) | \(1\) |
|---|---|---|---|---|---|---|---|---|---|
| \(\arcsin x\) | \(-90^\circ\) | \(-60^\circ\) | \(-45^\circ\) | \(-30^\circ\) | \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) |
| \(\arccos x\) | \(180^\circ\) | \(150^\circ\) | \(135^\circ\) | \(120^\circ\) | \(90^\circ\) | \(60^\circ\) | \(45^\circ\) | \(30^\circ\) | \(0^\circ\) |
Main values of the arctangent and arccotangent functions (in degrees)
| \(x\) | \(-\sqrt 3\) | \(-1\) | \(-\sqrt 3/3\) | \(0\) | \(\sqrt 3/3\) | \(1\) | \(\sqrt 3\) |
|---|---|---|---|---|---|---|---|
| \(\arctan x\) | \(-60^\circ\) | \(-45^\circ\) | \(-30^\circ\) | \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) |
| \(\text(arccot ) x\) | \(150^\circ\) | \(135^\circ\) | \(120^\circ\) | \(90^\circ\) | \(60^\circ\) | \(45^\circ\) | \(30^\circ\) |
Lessons 32-33. Inverse trigonometric functions
09.07.2015 8495 0Target: consider inverse trigonometric functions and their use for writing solutions to trigonometric equations.
I. Communicating the topic and purpose of the lessons
II. Learning new material
1. Inverse trigonometric functions
Let's begin our discussion of this topic with the following example.
Example 1
Let's solve the equation: a) sin x = 1/2; b) sin x = a.
a) On the ordinate axis we plot the value 1/2 and construct the angles x 1 and x2, for which sin x = 1/2. In this case x1 + x2 = π, whence x2 = π – x 1 . Using the table of values of trigonometric functions, we find the value x1 = π/6, then
Let's take into account the periodicity of the sine function and write down the solutions to this equation:
where k ∈ Z.
b) Obviously, the algorithm for solving the equation sin x = a is the same as in the previous paragraph. Of course, now the value a is plotted along the ordinate axis. There is a need to somehow designate the angle x1. We agreed to denote this angle with the symbol arcsin A. Then the solutions to this equation can be written in the formThese two formulas can be combined into one: wherein 
The remaining inverse trigonometric functions are introduced in a similar way.
Very often it is necessary to determine the magnitude of an angle from the known value of its trigonometric function. Such a problem is multivalued - there are countless angles whose trigonometric functions are equal to the same value. Therefore, based on the monotonicity of trigonometric functions, the following inverse trigonometric functions are introduced to uniquely determine angles.
Arcsine of the number a (arcsin , whose sine is equal to a, i.e.
Arc cosine of a number a(arccos a) is an angle a from the interval whose cosine is equal to a, i.e.
Arctangent of a number a(arctg a) - such an angle a from the intervalwhose tangent is equal to a, i.e.
tg a = a.
Arccotangent of a number a(arcctg a) is an angle a from the interval (0; π), the cotangent of which is equal to a, i.e. ctg a = a.
Example 2
Let's find:
Taking into account the definitions of inverse trigonometric functions, we obtain:
Example 3
Let's calculate 
Let angle a = arcsin 3/5, then by definition sin a = 3/5 and . Therefore, we need to find cos A. Using the basic trigonometric identity, we get:
It is taken into account that cos a ≥ 0. So, 
Function properties | Function |
|||
y = arcsin x | y = arccos x | y = arctan x | y = arcctg x |
|
Domain | x ∈ [-1; 1] | x ∈ [-1; 1] | x ∈ (-∞; +∞) | 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 = 0) | At x = 0 | At x = 1 | At x = 0 | y ≠ 0 |
Intervals of sign constancy | y > 0 for x ∈ (0; 1], at< 0 при х ∈ [-1; 0) | y > 0 for x ∈ [-1; 1) | y > 0 for x ∈ (0; +∞), at< 0 при х ∈ (-∞; 0) | y > 0 for x ∈ (-∞; +∞) |
Monotone | Increasing | Descending | Increasing | Descending |
Relation to the trigonometric function | sin y = x | cos y = x | tg y = x | ctg y = x |
Schedule | ||||
Let us give a number of more typical examples related to the definitions and basic properties of inverse trigonometric functions.
Example 4
Let's find the domain of definition of the function
In order for the function y to be defined, it is necessary to satisfy the inequality
which is equivalent to the system of inequalities
The solution to the first inequality is the interval x∈
(-∞; +∞), second - This interval and is a solution to the system of inequalities, and therefore the domain of definition of the function
Example 5
Let's find the area of change of the function
Let's consider the behavior of the function z = 2x - x2 (see picture).
It is clear that z ∈ (-∞; 1]. Considering that the argument z the arc cotangent function varies within the specified limits, from the table data we obtain that
So the area of change
Example 6
Let us prove that the function y = arctg x odd. Let
Then tg a = -x or x = - tg a = tg (- a), and 
Therefore, - a = arctg x or a = - arctg X. Thus, we see thati.e. y(x) is an odd function.
Example 7
Let us express through all inverse trigonometric functions
Let 
It's obvious that 
Then since
Let's introduce the angle 
Because 
That 
Likewise therefore 
And 
So,
Example 8
Let's build a graph of the function y = cos(arcsin x).
Let us denote a = arcsin x, then 
Let's take into account that x = sin a and y = cos a, i.e. x 2 + y2 = 1, and restrictions on x (x∈
[-1; 1]) and y (y ≥ 0). Then the graph of the function y = cos(arcsin x) is a semicircle.
Example 9
Let's build a graph of the function y = arccos (cos x ).
Since the cos function x changes on the interval [-1; 1], then the function y is defined on the entire numerical axis and varies on the segment . Let's keep in mind that y = arccos(cosx) = x on the segment; the function y is even and periodic with period 2π. Considering that the function has these properties cos x Now it's easy to create a graph.
Let us note some useful equalities:
Example 10
Let's find the smallest and largest values of the function Let's denote 
Then 
Let's get the function 
This function has a minimum at the point z = π/4, and it is equal to 
The greatest value of the function is achieved at the point z = -π/2, and it is equal 
Thus, and
Example 11
Let's solve the equation
Let's take into account that 
Then the equation looks like:
or 
where By definition of arctangent we get:
2. Solving simple trigonometric equations
Similar to example 1, you can obtain solutions to the simplest trigonometric equations.
The equation | Solution |
tgx = a | |
ctg x = a |
Example 12
Let's solve the equation 
Since the sine function is odd, we write the equation in the form
Solutions to this equation:
where do we find it from? 
Example 13
Let's solve the equation 
Using the given formula, we write down the solutions to the equation:
and we'll find 
Note that in special cases (a = 0; ±1) when solving the equations sin x = a and cos x = but it’s easier and more convenient to use not general formulas, and write down solutions based on the unit circle:
for the equation sin x = 1 solution
for the equation sin x = 0 solutions x = π k;
for the equation sin x = -1 solution 
for the cos equation x = 1 solution x = 2π k ;
for the equation cos x = 0 solutions
for the equation cos x = -1 solution 
Example 14
Let's solve the equation 
Since in this example there is a special case of the equation, we will write the solution using the appropriate formula:
where can we find it from? 
III. Control questions (frontal survey)
1. Define and list the main properties of inverse trigonometric functions.
2. Give graphs of inverse trigonometric functions.
3. Solving simple trigonometric equations.
IV. Lesson assignment
§ 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 (g); 16 (b); 18 (c, d); 19 (g); 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 the function:
Answers:
3. Plot a graph of the function:
VII. Summing up the lessons
Inverse trigonometric functions- these are arcsine, arccosine, arctangent and arccotangent.
First let's give some definitions.
Arcsine Or, we can say that this is an angle belonging to a segment whose sine is equal to the number a.
arc cosine number a is called a number such that
Arctangent number a is called a number such that
Arccotangent number a is called a number such that
Let's talk in detail about these four new functions for us - inverse trigonometric ones.
Remember, we have already met.
For example, arithmetic Square root from a number a is a non-negative number whose square is equal to a.
The logarithm of a number b to base a is a number c such that
Wherein
We understand why mathematicians had to “invent” new functions. For example, the solutions to an equation are and We could not write them down without the special arithmetic square root symbol.
The concept of a logarithm turned out to be necessary to write down solutions, for example, to such an equation: The solution to this equation is an irrational number. This is an exponent of the power to which 2 must be raised to get 7.
It's the same with trigonometric equations. For example, we want to solve the equation
It is clear that its solutions correspond to points on the trigonometric circle whose ordinate is equal to And it is clear that this is not the tabular value of the sine. How to write down solutions?
Here we cannot do without a new function, denoting the angle whose sine is equal to a given number a. Yes, everyone has already guessed. This is arcsine.
The angle belonging to the segment whose sine is equal to is the arcsine of one fourth. And this means that the series of solutions to our equation corresponding to the right point on the trigonometric circle is
And the second series of solutions to our equation is
Learn more about solving trigonometric equations -.
It remains to be found out - why does the definition of arcsine indicate that this is an angle belonging to the segment?
The fact is that there are infinitely many angles whose sine is equal to, for example, . We need to choose one of them. We choose the one that lies on the segment .
Take a look at the trigonometric circle. You will see that on the segment each angle corresponds to a certain sine value, and only one. And vice versa, any value of the sine from the segment corresponds to a single value of the angle on the segment. This means that on a segment you can define a function taking values from to
Let's repeat the definition again:
The arcsine of a number is the number , such that
Designation: The arcsine definition area is a segment. The range of values is a segment.
You can remember the phrase “arcsines live on the right.” Just don’t forget that it’s not just on the right, but also on the segment.
We are ready to graph the function
As usual, we plot the x values on the horizontal axis and the y values on the vertical axis.
Because , therefore, x lies in the range from -1 to 1.
This means that the domain of definition of the function y = arcsin x is the segment
We said that y belongs to the segment . This means that the range of values of the function y = arcsin x is the segment.
Note that the graph of the function y=arcsinx fits entirely within the area bounded by the lines and
As always when plotting a graph of an unfamiliar function, let's start with a table.
By definition, the arcsine of zero is a number from the segment whose sine is equal to zero. What is this number? - It is clear that this is zero.
Similarly, the arcsine of one is a number from the segment whose sine is equal to one. Obviously this
We continue: - this is a number from the segment whose sine is equal to . Yes it
| 0 | |||||
| 0 |
Building a graph of a function
Function properties
1. Scope of definition
2. Range of values
3., that is, this function is odd. Its graph is symmetrical about the origin.
4. The function increases monotonically. Its minimum value, equal to - , is achieved at , and its greatest value, equal to , at
5. What do the graphs of functions and ? Don't you think that they are "made according to the same pattern" - just like the right branch of a function and the graph of a function, or like the graphs of exponential and logarithmic functions?
Imagine that we cut out a small fragment from to to from an ordinary sine wave, and then turned it vertically - and we will get an arcsine graph.
What for a function on this interval are the values of the argument, then for the arcsine there will be the values of the function. That's how it should be! After all, sine and arcsine - reciprocal functions. Other examples of pairs of mutually inverse functions are at and , as well as exponential and logarithmic functions.
Recall that the graphs of mutually inverse functions are symmetrical with respect to the straight line
Similarly, we define the function. We only need a segment on which each angle value corresponds to its own cosine value, and knowing the cosine, we can uniquely find the angle. A segment will suit us
The arc cosine of a number is the number , such that
It’s easy to remember: “arc cosines live from above,” and not just from above, but on the segment
Designation: The arc cosine definition area is a segment. The range of values is a segment.
Obviously, the segment was chosen because on it each cosine value is taken only once. In other words, each cosine value, from -1 to 1, corresponds to a single angle value from the interval
Arc cosine is neither even nor odd function. But we can use the following obvious relationship:
Let's plot the function
We need a section of the function where it is monotonic, that is, it takes each value exactly once.
Let's choose a segment. On this segment the function decreases monotonically, that is, the correspondence between sets is one-to-one. Each x value has a corresponding y value. On this segment there is a function inverse to cosine, that is, the function y = arccosx.
Let's fill in the table using the definition of arc cosine.
The arc cosine of a number x belonging to the interval will be a number y belonging to the interval such that
This means, since ;
Because ;
Because ,
Because ,
| 0 | |||||
| 0 |
Here is the arc cosine graph:
Function properties
1. Scope of definition
2. Range of values
This function is of a general form - it is neither even nor odd.
4. The function is strictly decreasing. The function y = arccosx takes its greatest value, equal to , at , and its smallest value, equal to zero, takes at
5. The functions and are mutually inverse.
The next ones are arctangent and arccotangent.
The arctangent of a number is the number , such that
Designation: . The area of definition of the arctangent is the interval. The area of values is the interval.
Why are the ends of the interval - points - excluded in the definition of arctangent? Of course, because the tangent at these points is not defined. There is no number a equal to the tangent of any of these angles.
Let's build a graph of the arctangent. According to the definition, the arctangent of a number x is a number y belonging to the interval such that
How to build a graph is already clear. Since arctangent is a function reciprocal of tangent, we proceed as follows:
We select a section of the graph of the function where the correspondence between x and y is one-to-one. This is the interval C. In this section the function takes values from to
Then have inverse function, that is, the function , domain, definition will be the entire number line, from to and the range of values will be the interval
Means,
Means,
Means,
But what happens for infinitely large values of x? In other words, how does this function behave as x tends to plus infinity?
We can ask ourselves the question: for which number in the interval does the tangent value tend to infinity? - Obviously this
This means that for infinitely large values of x, the arctangent graph approaches the horizontal asymptote
Similarly, if x approaches minus infinity, the arctangent graph approaches the horizontal asymptote
The figure shows a graph of the function
Function properties
1. Scope of definition
2. Range of values
3. The function is odd.
4. The function is strictly increasing.
6. Functions and are mutually inverse - of course, when the function is considered on the interval
Similarly, we define the inverse tangent function and plot its graph.
The arccotangent of a number is the number , such that
Function graph:
Function properties
1. Scope of definition
2. Range of values
3. The function is of general form, that is, neither even nor odd.
4. The function is strictly decreasing.
5. Direct and - horizontal asymptotes of this function.
6. The functions and are mutually inverse if considered on the interval
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 functionsSum 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 formulasExpressions 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 = sin t. 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 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 .
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
Inverse cosine function
The range of values of the function y=cos x (see Fig. 2) is a segment. On the segment the function is continuous and monotonically decreasing.
Rice. 2
This means that the function inverse to the function y=cos x is defined on the segment. This inverse function is called arc cosine and is denoted y=arccos x.
Definition
The arccosine of a number a, if |a|1, is the angle whose cosine belongs to the segment; it is denoted by arccos a.
Thus, arccos a is an angle that satisfies the following two conditions: сos (arccos a)=a, |a|1; 0? arccos a ?р.
For example, arccos, since cos and; arccos, since cos and.
The function y = arccos x (Fig. 3) is defined on a segment; its range of values is the segment. On the segment, the function y=arccos x is continuous and monotonically decreases from p to 0 (since y=cos x is a continuous and monotonically decreasing function on the segment); at the ends of the segment it reaches its extreme values: arccos(-1)= p, arccos 1= 0. Note that arccos 0 = . The graph of the function y = arccos x (see Fig. 3) is symmetrical to the graph of the function y = cos x relative to the straight line y=x.
Rice. 3
Let us show that the equality arccos(-x) = p-arccos x holds.
In fact, by definition 0? arccos x? R. Multiplying by (-1) all parts of the latter double inequality, we get - p? arccos x? 0. Adding p to all parts of the last inequality, we find that 0? p-arccos x? R.
Thus, the values of the angles arccos(-x) and p - arccos x belong to the same segment. Since the cosine decreases monotonically on a segment, there cannot be two different angles on it that have equal cosines. Let's find the cosines of the angles arccos(-x) and p-arccos x. By definition, cos (arccos x) = - x, according to the reduction formulas and by definition we have: cos (p - - arccos x) = - cos (arccos x) = - x. So, the cosines of the angles are equal, which means the angles themselves are equal.
Inverse sine function
Let's consider the function y=sin x (Fig. 6), which on the segment [-р/2;р/2] is increasing, continuous and takes values from the segment [-1; 1]. This means that on the segment [- p/2; p/2] the inverse function of the function y=sin x is defined.
Rice. 6
This inverse function is called the arcsine and is denoted y=arcsin x. Let us introduce the definition of the arcsine of a number.
The arcsine of a number is an angle (or arc) whose sine is equal to the number a and which belongs to the segment [-р/2; p/2]; it is denoted by arcsin a.
Thus, arcsin a is an angle satisfying the following conditions: sin (arcsin a)=a, |a| ?1; -r/2 ? arcsin huh? r/2. For example, since sin and [- p/2; p/2]; arcsin, since sin = u [- p/2; p/2].
The function y=arcsin x (Fig. 7) is defined on the segment [- 1; 1], the range of its values is the segment [-р/2;р/2]. On the segment [- 1; 1] the function y=arcsin x is continuous and increases monotonically from -p/2 to p/2 (this follows from the fact that the function y=sin x on the segment [-p/2; p/2] is continuous and increases monotonically). It takes the greatest value at x = 1: arcsin 1 = p/2, and the smallest at x = -1: arcsin (-1) = -p/2. At x = 0 the function is zero: arcsin 0 = 0.
Let us show that the function y = arcsin x is odd, i.e. arcsin(-x) = - arcsin x for any x [ - 1; 1].
Indeed, by definition, if |x| ?1, we have: - p/2 ? arcsin x ? ? r/2. Thus, the angles arcsin(-x) and - arcsin x belong to the same segment [ - p/2; p/2].
Let's find the sines of these angles: sin (arcsin(-x)) = - x (by definition); since the function y=sin x is odd, then sin (-arcsin x)= - sin (arcsin x)= - x. So, the sines of angles belonging to the same interval [-р/2; p/2], are equal, which means the angles themselves are equal, i.e. arcsin (-x)= - arcsin x. This means that the function y=arcsin x is odd. The graph of the function y=arcsin x is symmetrical about the origin.
Let us show that arcsin (sin x) = x for any x [-р/2; p/2].
Indeed, by definition -p/2? arcsin (sin x) ? p/2, and by condition -p/2? x? r/2. This means that the angles x and arcsin (sin x) belong to the same interval of monotonicity of the function y=sin x. If the sines of such angles are equal, then the angles themselves are equal. Let's find the sines of these angles: for angle x we have sin x, for angle arcsin (sin x) we have sin (arcsin(sin x)) = sin x. We found that the sines of the angles are equal, therefore, the angles are equal, i.e. arcsin(sin x) = x. .
Rice. 7
Rice. 8
The graph of the function arcsin (sin|x|) is obtained by the usual transformations associated with the modulus from the graph y=arcsin (sin x) (shown by the dashed line in Fig. 8). The desired graph y=arcsin (sin |x-/4|) is obtained from it by shifting by /4 to the right along the x-axis (shown as a solid line in Fig. 8)
Inverse function of tangent
The function y=tg x on the interval accepts everything numeric values: E (tg x)=. Over this interval it is continuous and increases monotonically. This means that a function inverse to the function y = tan x is defined on the interval. This inverse function is called the arctangent and is denoted y = arctan x.
The arctangent of a is an angle from an interval whose tangent is equal to a. Thus, arctg a is an angle that satisfies the following conditions: tg (arctg a) = a and 0? arctg a ? R.
So, any number x always corresponds to a single value of the function y = arctan x (Fig. 9).
It is obvious that D (arctg x) = , E (arctg x) = .
The function y = arctan x is increasing because the function y = tan x is increasing on the interval. It is not difficult to prove that arctg(-x) = - arctgx, i.e. that arctangent is an odd function.
Rice. 9
The graph of the function y = arctan x is symmetrical to the graph of the function y = tan x relative to the straight line y = x, the graph y = arctan x passes through the origin of coordinates (since arctan 0 = 0) and is symmetrical relative to the origin (like the graph of an odd function).
It can be proven that arctan (tan x) = x if x.
Cotangent inverse function
The function y = ctg x on an interval takes all numeric values from the interval. The range of its values coincides with the set of all real numbers. In the interval, the function y = cot x is continuous and increases monotonically. This means that on this interval a function is defined that is inverse to the function y = cot x. The inverse function of cotangent is called arccotangent and is denoted y = arcctg x.
The arc cotangent of a is an angle belonging to an interval whose cotangent is equal to a.
Thus, аrcctg a is an angle satisfying the following conditions: ctg (arcctg a)=a and 0? arcctg a ? R.
From the definition of the inverse function and the definition of arctangent it follows that D (arcctg x) = , E (arcctg x) = . The arc cotangent is a decreasing function because the function y = ctg x decreases in the interval.
The graph of the function y = arcctg x does not intersect the Ox axis, since y > 0 R. For x = 0 y = arcctg 0 =.
The graph of the function y = arcctg x is shown in Figure 11.
Rice. 11
Note that for all real values of x the identity is true: arcctg(-x) = p-arcctg x.
Loading...