In a number of problems in mathematics and its applications, it is required to use a known value of a trigonometric function to find the corresponding value of an angle, expressed in degrees or radians. It is known that an infinite number of angles correspond to the same value of the sine, for example, if $\sin α=1/2,$ then the angle $α$ can be equal to $30°$ and $150°,$ or in radian measure $π /6$ and $5π/6,$ and any of the angles that is obtained from these by adding a term of the form $360°⋅k,$ or, respectively, $2πk,$ where $k$ is any integer. This becomes clear from examining the graph of the function $y=\sin x$ on the entire number line (see Fig. $1$): if on the $Oy$ axis we plot a segment of length $1/2$ and draw a straight line parallel to the $Ox axis, $ then it will intersect the sinusoid at an infinite number of points. To avoid possible diversity of answers, inverse trigonometric functions are introduced, otherwise called circular or arc functions (from the Latin word arcus - “arc”).
The main four trigonometric functions $\sin x,$ $\cos x,$ $\mathrm(tg)\,x$ and $\mathrm(ctg)\,x$ correspond to four arc functions $\arcsin x,$ $\arccos x ,$ $\mathrm(arctg)\,x$ and $\mathrm(arcctg)\,x$ (read: arcsine, arccosine, arctangent, arccotangent). Let's consider the functions \arcsin x and \mathrm(arctg)\,x, since the other two are expressed through them using the formulas:
$\arccos x = \frac(π)(2) − \arcsin x,$ $\mathrm(arcctg)\,x = \frac(π)(2) − \mathrm(arctg)\,x.$
The equality $y = \arcsin x$ by definition means the angle $y,$ expressed in radian measure and contained in the range from $−\frac(π)(2)$ to $\frac(π)(2),$ sine which is equal to $x,$ i.e. $\sin y = x.$ The function $\arcsin x$ is the function inverse function$\sin x,$ considered on the interval $\left[−\frac(π)(2),+\frac(π)(2)\right],$ where this function increases monotonically and takes all values from $−1 $ to $+1.$ Obviously, the argument $y$ of the function $\arcsin x$ can only take values from the interval $\left[−1,+1\right].$ So, the function $y=\arcsin x$ is defined on the interval $\left[−1,+1\right],$ is monotonically increasing, and its values fill the interval $\left[−\frac(π)(2),+\frac(π)(2)\ right].$ The graph of the function is shown in Fig. $2.$
Under the condition $−1 ≤ a ≤ 1$, we can represent all solutions of the equation $\sin x = a$ in the form $x=(−1)^n \arcsin a + πn,$ $n=0,±1,± 2, ….$ For example, if
$\sin x = \frac(\sqrt(2))(2)$ then $x = (−1)^n \frac(π)(4)+πn,$ $n = 0, ±1, ±2 ,….$
The relation $y=\mathrm(arcctg)\,x$ is defined for all values of $x$ and by definition means that the angle $y,$ expressed in radian measure, is contained within
$−\frac(π)(2)
and the tangent of this angle is equal to x, i.e. $\mathrm(tg)\,y = x.$ The function $\mathrm(arctg)\,x$ is defined on the entire number line and is the inverse function of the function $\mathrm( tg)\,x$, which is considered only on the interval
$−\frac(π)(2)
The function $y = \mathrm(arctg)\,x$ is monotonically increasing, its graph is shown in Fig. $3.$
All solutions to the equation $\mathrm(tg)\,x = a$ can be written in the form $x=\mathrm(arctg)\,a+πn,$ $n=0,±1,±2,… .$
Note that inverse trigonometric functions are widely used in mathematical analysis. For example, one of the first functions for which a representation by an infinite power series was obtained was the function $\mathrm(arctg)\,x.$ From this series, G. Leibniz, with a fixed value of the argument $x=1$, obtained the famous representation of a number to infinite near
Inverse trigonometric functions- these are arcsine, arccosine, arctangent and arccotangent.
First let's give some definitions.
Arcsine Or, we can say that this is such an angle, belonging to the segment, the sine of which 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, the arithmetic square root of 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
At the same time
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 symbol square root.
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 irrational number This is the exponent 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 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
More about the solution 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 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 into the region limited by lines And
As always when plotting 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 is
| 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 arccosine 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 general view- it is neither even nor odd.
4. The function is strictly decreasing. Highest value, equal to , the function y = arccosx takes at , and the 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 the inverse function 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 the inverse function, that is, the function, has a domain of definition that 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 = 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.
In this lesson we will look at the features inverse functions and repeat inverse trigonometric functions. The properties of all basic inverses will be considered separately. trigonometric functions: arcsine, arccosine, arctangent and arccotangent.
This lesson will help you prepare for one of the types of tasks B7 And C1.
Preparation for the Unified State Exam in mathematics
Experiment
Lesson 9. Inverse trigonometric functions.
Theory
Lesson summary
Let us remember when we encounter such a concept as an inverse function. For example, consider the squaring function. Let us have a square room with sides of 2 meters and we want to calculate its area. To do this, using the square formula, we square two and as a result we get 4 m2. Now imagine the inverse problem: we know the area of a square room and want to find the lengths of its sides. If we know that the area is still the same 4 m2, then we will perform the reverse action to squaring - extracting the arithmetic square root, which will give us the value of 2 m.
Thus, for the function of squaring a number, the inverse function is to take the arithmetic square root.
Specifically, in the above example, we did not have any problems with calculating the side of the room, because we understand that this is a positive number. However, if we take a break from this case and consider the problem in a more general way: “Calculate the number whose square is equal to four,” we are faced with 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 can be solved ambiguously, and the action of determining the number that squared gave the number we know? has two results. It is convenient to show this on a graph:
 |
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 to strictly one thing function value.
In order to introduce precisely the inverse function to squaring, the concept of an arithmetic square root was proposed, which gives only non-negative values. Those. for a function, the inverse function is considered to be .
Similarly, there are functions inverse to trigonometric ones, 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 angles from a known value of the trigonometric function. For example, using a table of values of basic trigonometric functions, you can calculate the sine of which angle is equal to . We find this value in the line of sines and determine which angle it corresponds to. The first thing you want to answer is that this is the angle or, but if you have a table of values at your disposal, you will immediately notice another contender for the answer - this is the angle or. And if we remember the period of the sine, we will understand that there are an infinite number of angles at which the sine is equal. And such a set of angle values corresponding to a given value of the trigonometric function will also be observed for cosines, tangents and cotangents, because they all have periodicity.
Those. we are faced with the same problem that we had for calculating the value of the argument from the value of the function for the squaring action. And in this case, for inverse trigonometric functions, a limitation was introduced on the range of values that they give during calculation. This property of such inverse functions is called narrowing the range of values, and it is necessary in order 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 in 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. Who wants to get acquainted with them in more detail, refer to the chapter “Solving trigonometric equations” in the 10th grade program.
Let's consider the properties of the arcsine function and build its graph.
Definition.Arcsine of the numberx
Basic properties of the arcsine:
1) 
at ,
2) 
at .
Basic properties of the arcsine function:
1) Scope of definition 
;
2) Value range 
;
3) The function is odd. It is advisable to memorize this formula separately, because it is useful for transformations. We also note that the oddity implies the symmetry of the graph of the function relative to the origin;
Let's build a graph of the function:
Please note that none of the sections of the function graph are repeated, which means that the arcsine is not periodic function, unlike sine. The same will apply to all other arc functions.
Let's consider the properties of the arc cosine function and build its graph.
Definition.arc cosine of the numberx is the value of the angle y for which . Moreover, both as restrictions on the values of the sine, and as the selected range of angles.
Basic properties of arc cosine:
1) 
at ,
2) 
at .
Basic properties of the arc cosine function:
1) Scope of definition ;
2) Range of values;
3) The function is neither even nor odd, i.e. general view 
. It is also advisable to remember this formula, it will be useful to us later;
4) The function decreases monotonically.
Let's build a graph of the function:
Let's consider the properties of the arctangent function and build its graph.
Definition.Arctangent of the numberx is the value of the angle y for which . Moreover, because There are no restrictions on the tangent values, but rather on the selected range of angles.
Basic properties of the arctangent:
1) 
at ,
2) 
at .
Basic properties of the arctangent function:
1) Scope of definition;
2) Value range 
;
3) The function is odd 
. This formula is also useful, like others similar to it. As in the case of the arcsine, the oddity implies that the graph of the function is symmetrical about the origin;
4) The function increases monotonically.
Let's build a graph of the function:
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 last 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 [-p/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 takes all numerical 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.
