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; therefore, 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 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 the 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) x ∈ [-π / 2; π / 2], then 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 the 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 х ∈ [-1; 1], is symmetric to the graph of the function y \u003d cos (\u2061x), where х ∈, relative to the bisector of the coordinate angles of the first and third quarters.
The domain of the function y \u003d arccos (x).
Example # 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) x ∈ (-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 equal to 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 will be R, and the set values \u200b\u200b- 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
Inverse trigonometric tasks are often offered in school final exams and on entrance exams in some universities. A detailed study of this topic can be achieved only in extracurricular activities or in elective courses... The offered 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 \u003d arcsin x, y \u003d arccos x, y \u003d arctan x, y \u003d arcctg x.
Purpose: full coverage of this issue.
1. Function y \u003d arcsin x.
a) For the function y \u003d sin x on the segment there is an inverse (single-valued) function, which we agreed to call the arcsine and denote it as: y \u003d 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 \u003d arcsin x.
1) Domain of definition: segment [-1; 1];
2) Area of \u200b\u200bchange: segment;
3) Function y \u003d arcsin x is odd: arcsin (-x) \u003d - arcsin x;
4) The function y \u003d arcsin x is monotonically increasing;
5) The graph crosses the Ox, Oy axes at the origin.
Example 1. Find a \u003d arcsin. This example in detail can be formulated as follows: find an argument a, lying in the range from to, whose sine is equal to.
Decision. 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 
.Decision. 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 \u003d arccos x, y \u003d arctan x, y \u003d arcctg x (similar).
Lesson 2 (2 hours) Topic: Inverse trigonometric functions, their graphs.
Purpose: on this lesson you need to practice skills in determining values trigonometric functions, in the construction of graphs of inverse trigonometric functions using D (y), E (y) and the necessary transformations.
In this lesson, perform exercises that include finding the domain of definition, the domain of values \u200b\u200bof functions of the type: y \u003d arcsin, y \u003d arccos (x-2), y \u003d arctan (tg x), y \u003d arccos.
It is necessary to build graphs of functions: a) y \u003d arcsin 2x; b) y \u003d 2 arcsin 2x; c) y \u003d arcsin;
d) y \u003d arcsin; e) y \u003d arcsin; f) y \u003d arcsin; g) y \u003d | arcsin | ...
Example.Plot y \u003d arccos
You can include the following exercises in your homework: build graphs of functions: y \u003d arccos, y \u003d 2 arcctg x, y \u003d 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 to specialties with increased requirements for mathematical training) by introducing basic relations for inverse trigonometric functions.
Lesson material.
Some of the simplest trigonometric operations on inverse trigonometric functions: sin (arcsin x) \u003d x, i xi? 1; cos (arсcos x) \u003d x, i xi? 1; tg (arctan x) \u003d x, x I R; ctg (arcctg x) \u003d x, x I R.
Exercises.
a) tg (1,5 + arctan 5) \u003d - ctg (arctan 5) \u003d 
.
ctg (arctg x) \u003d; tg (arcctg x) \u003d.
b) cos (+ arcsin 0.6) \u003d - cos (arcsin 0.6). Let arcsin 0.6 \u003d a, sin a \u003d 0.6;
cos (arcsin x) \u003d; sin (arccos x) \u003d.
Note: we take the plus sign in front of the root because a \u003d 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 \u003d a, then sin 2 a \u003d 
or sin (2 arctan 5) \u003d 
;
b) cos (+ 2 arcsin 0.8). Answer: 0.28.
c) arctg + arctg.
Let a \u003d arctan, b \u003d arctan,
then tg (a + b) \u003d 
.
d) sin (arcsin + arcsin).
e) Prove that for all x I [-1; 1] arcsin x + arccos x \u003d is true.
Evidence:
arcsin x \u003d - arccos x
sin (arcsin x) \u003d sin (- arccos x)
x \u003d 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.
Lesson material.
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) \u003d cos (arctan 5) cos (arccos 0.8) + sin (arctan 5) sin (arccos 0.8) \u003d
3) tg (- arcsin 0.6) \u003d - tg (arcsin 0.6) \u003d 
4) 
Independent work will help to identify the level of assimilation of the material
| 1) tg (arctan 2 - arctg) 2) cos (- arctg2) 3) arcsin + arccos |
1) cos (arcsin + arcsin) 2) sin (1.5 - arctan 3) 3) arcctg3 - arctg 2 |
For homework you can suggest:
1) ctg (arctg + arctg + arctg); 2) sin 2 (arctan 2 - arcctg ()); 3) sin (2 arctan + tg (arcsin)); 4) sin (2 arctg); 5) tg ((arcsin))
Lesson № 5 (2 hours) Topic: Inverse trigonometric operations on trigonometric functions.
Purpose: to form an idea of \u200b\u200bstudents 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 \u003d 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) \u003d - sin x; arcsin (sin (-x)) \u003d - arcsin (sin x).
6. Graph y \u003d 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 \u003d sin (- x) \u003d sinx, 0<= - x <= .
So, 
Having constructed y \u003d arcsin (sin x) on, we continue symmetrically about the origin to [-; 0], considering the oddness of this function. Using periodicity, let's continue to the entire number axis.
Then write down some ratios: arcsin (sin a) \u003d a if<= a <= ; arccos (cos a ) \u003d a if 0<= a <= ; arctan (tg a) \u003d 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) \u003d arctan (tg (2 -)). Answer: 2 -; h) arcctg (tan 0.6). Answer: - 0.6; - arctg x; e) arccos + arccos
Inverse cosine function
The range of values \u200b\u200bof the function y \u003d cos x (see Fig. 2) is a segment. On a segment, the function is continuous and decreases monotonically.
Figure: 2
This means that a function is defined on the segment inverse to the function y \u003d cos x. This inverse function is called the inverse cosine and is denoted by y \u003d arccos x.
Definition
The arcosine of the 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 satisfying the following two conditions: cos (arccos a) \u003d a, | a | 1; 0? arccos a? p.
For example, arccos, since cos and; arccos since cosi.
The function y \u003d arccos x (Fig. 3) is defined on a segment, the range of its values \u200b\u200bis a segment. On the segment, the function y \u003d arccos x is continuous and monotonically decreases from p to 0 (since y \u003d 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) \u003d p, arccos 1 \u003d 0. Note that arccos 0 \u003d. The graph of the function y \u003d arccos x (see Fig. 3) is symmetric to the graph of the function y \u003d cos x relative to the straight line y \u003d x.
Figure: 3
Let us show that the equality arccos (-x) \u003d р-arccos x holds.
Indeed, by definition, 0? arcсos x? R. Multiplying by (-1) all parts of the last double inequality, we get - p? arcсos x? 0. Adding p to all parts of the last inequality, we find that 0? p-arccos x? R.
Thus, the values \u200b\u200bof 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 with equal cosines. Find the cosines of the angles arccos (-x) and p-arccos x. By definition, cos (arccos x) \u003d - x, by the reduction formulas and by definition, we have: cos (p - - arccos x) \u003d - cos (arccos x) \u003d - x. So, the cosines of the angles are equal, which means that the angles themselves are also equal.
Inverse sine function
Consider the function y \u003d sin x (Fig. 6), which is increasing, continuous on the segment [-p / 2; p / 2] and takes values \u200b\u200bfrom the segment [-1; 1]. Hence, on the segment [- p / 2; р / 2] a function is defined that is inverse to the function y \u003d sin x.
Figure: 6
This inverse function is called the arcsine and is denoted y \u003d arcsin x. Let us introduce the definition of the arcsine of the number a.
The arcsine of the number a, if you call the angle (or arc), the sine of which 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) \u003d a, | a | ?1; -p / 2? arcsin huh? p / 2. For example, since sin and [- p / 2; p / 2]; arcsin, since sin \u003d and [- p / 2; p / 2].
The function y \u003d arcsin х (Fig. 7) is defined on the segment [- 1; 1], the range of its values \u200b\u200bis the segment [-p / 2; p / 2]. On the segment [- 1; 1] the function y \u003d arcsin x is continuous and monotonically increases from -p / 2 to p / 2 (this follows from the fact that the function y \u003d sin x on the interval [-p / 2; p / 2] is continuous and monotonically increasing). It takes the greatest value at x \u003d 1: arcsin 1 \u003d p / 2, and the smallest at x \u003d -1: arcsin (-1) \u003d -p / 2. For x \u003d 0, the function is zero: arcsin 0 \u003d 0.
Let us show that the function y \u003d arcsin x is odd, i.e. arcsin (-x) \u003d - arcsin x for any x [ - 1; 1].
Indeed, by definition, if | x | ? 1, we have: - р / 2? arcsin x? ? p / 2. Thus, the angles arcsin (-x) and - arcsin x belong to the same segment [ - p / 2; p / 2].
Find the sinuses of theseangles: sin (arcsin (-x)) \u003d - x (by definition); since the function y \u003d sin x is odd, then sin (-arcsin x) \u003d - sin (arcsin x) \u003d - x. So, the sines of the angles belonging to the same interval [-p / 2; р / 2], are equal, which means that the angles themselves are also equal, that is, arcsin (-x) \u003d - arcsin x. Hence, the function y \u003d arcsin x is odd. The graph of the function y \u003d arcsin x is symmetrical about the origin.
Let us show that arcsin (sin x) \u003d x for any x [-p / 2; p / 2].
Indeed, by definition -p / 2? arcsin (sin x)? p / 2, and by condition -p / 2? x? p / 2. This means that the angles x and arcsin (sin x) belong to the same interval of monotonicity of the function y \u003d 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 the angle x we \u200b\u200bhave sin x, for the angle arcsin (sin x) we have sin (arcsin (sin x)) \u003d sin x. We got that the sines of the angles are equal, therefore, the angles are equal, i.e. arcsin (sin x) \u003d x. ...
Figure: 7
Figure: 8
The graph of the function arcsin (sin | x |) is obtained by the usual transformations associated with the modulus from the graph y \u003d arcsin (sin x) (shown by the dashed line in Fig. 8). The desired graph y \u003d arcsin (sin | x- / 4 |) is obtained from it by shifting / 4 to the right along the abscissa axis (shown by the solid line in Fig. 8)
Inverse of the tangent function
The function y \u003d tg x on the interval takes all numerical values: E (tg x) \u003d. On this interval, it is continuous and increases monotonically. Hence, on the interval, a function is defined that is inverse to the function y \u003d tg x. This inverse function is called the arctangent and is denoted by y \u003d arctan x.
The arctangent of the number a is the angle from the interval, the tangent of which is equal to a. Thus, arctan a is an angle satisfying the following conditions: tg (arctan a) \u003d a and 0? arctg a? R.
So, any number x always corresponds to a single value of the function y \u003d arctan x (Fig. 9).
Obviously, D (arctan x) \u003d, E (arctan x) \u003d.
The function y \u003d arctan x is increasing because the function y \u003d tg x is increasing over the interval. It is not hard to prove that arctg (-x) \u003d - arctgx, i.e. that the arctangent is an odd function.
Figure: 9
The graph of the function y \u003d arctan x is symmetric to the graph of the function y \u003d tg x relative to the straight line y \u003d x, the graph of y \u003d arctan x passes through the origin (because arctan 0 \u003d 0) and is symmetric about the origin (like the graph of an odd function).
It can be proved that arctan (tg x) \u003d x if x.
Inverse cotangent function
The function y \u003d ctg x on the interval takes all numerical values \u200b\u200bfrom the interval. Its range of values \u200b\u200bcoincides with the set of all real numbers. In the interval, the function y \u003d ctg x is continuous and monotonically increasing. Hence, on this interval the function inverse to the function y \u003d ctg x is defined. The inverse function of the cotangent is called the arc cotangent and is denoted y \u003d arcctg x.
The arc cotangent of the number a is the angle belonging to the interval whose cotangent is equal to a.
Thus, arcctg a is an angle satisfying the following conditions: ctg (arcctg a) \u003d a and 0? arcctg a? R.
From the definition of the inverse function and the definition of the arctangent it follows that D (arcctg x) \u003d, E (arcctg x) \u003d. The arc cotangent is a decreasing function, since the function y \u003d ctg x decreases in the interval.
The graph of the function y \u003d arcctg x does not intersect the Ox axis, since y\u003e 0 R. At x \u003d 0 y \u003d arcctg 0 \u003d.
The graph of the function y \u003d arcctg x is shown in Figure 11.
Figure: 11
Note that for all real values \u200b\u200bof x the identity is true: arcctg (-x) \u003d p-arcctg x.
Inverse trigonometric functions are arcsine, arccosine, arctangent and arccotangent.
First, let's give definitions.
Arcsine Or, we can say that this is an angle belonging to a segment whose sine is equal to the number a.
Arccosine number a is a number such that
Arctangent number a is a number such that
Arccotangent number a is a number such that
Let's talk in detail about these four new functions for us - inverse trigonometric.
Remember, we've already met with.
For example, the arithmetic square root of a is a non-negative number whose square is a.
The logarithm of b to base a is a number c such that
Wherein
We understand why mathematicians had to “invent” new functions. For example, solutions to an equation are and We could not have written them without the special symbol of the arithmetic 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 an irrational number This is an exponent to which 2 must be raised to get 7.
So it is with trigonometric equations. For example, we want to solve the equation
It is clear that his solutions correspond to points on the trigonometric circle, the ordinate of which is equal to AND, it is clear that this is not a tabular value of the sine. How do you write down decisions?
Here we cannot do without a new function that denotes the angle whose sine is equal to the given number a. Yes, everyone already guessed. This is the arcsine.
An angle belonging to a segment whose sine is equal to is the arcsine of one fourth. So, 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
Read more about solving trigonometric equations -.
It remains to find out - why is it indicated in the definition of the arcsine that this is an angle belonging to a segment?
The fact is that there are infinitely many angles whose sine is equal, 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 corner corresponds to a certain sine value, and only one. Conversely, any sine value from a segment corresponds to a single angle value on the segment. This means that on the segment, you can specify a function that takes values \u200b\u200bfrom to
Let's repeat the definition one more time:
The arcsine of a number a is the number , such that
Designation: Area of \u200b\u200bdefinition of arcsine - segment Area of \u200b\u200bvalues \u200b\u200b- segment.
You can remember the phrase "arcsines live on the right." Do not forget that not just to the right, but also on the segment.
We are ready to plot the function
As usual, plot the x values \u200b\u200balong the horizontal axis and the y values \u200b\u200balong the vertical axis.
Since, therefore, x lies in the range from -1 to 1.
Hence, the domain of the function y \u003d arcsin x is the segment
We said that y belongs to the segment. This means that the range of values \u200b\u200bof the function y \u003d arcsin x is a segment.
Note that the graph of the function y \u003d arcsinx is all placed in the area bounded by the 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 a segment whose sine is equal to zero. What is this number? - It is clear that this is zero.
Likewise, the arcsine of one is a number from a segment whose sine is equal to one. Obviously this
We continue: - this is such a number from the segment, the sine of which is equal to. Yes it
| 0 | |||||
| 0 |
Plotting a function
Function properties
1. Scope
2. Range of values
3., that is, this function is odd. Its graph is symmetrical about the origin.
4. The function increases monotonically. Its smallest value, equal to -, is achieved at, and the largest value, equal to, at
5. What do the graphs of functions and have in common? Don't you think that they are "made according to the same template" - just like the right branch of a function and a graph of a function, or like graphs of exponential and logarithmic functions?
Imagine that we cut out a small fragment from to from an ordinary sinusoid, and then unfold it vertically - and we will get a graph of the arcsine.
The fact that for the function in this interval are the values \u200b\u200bof the argument, then for the arcsine there will be the values \u200b\u200bof the function. It should be so! After all, sine and arcsine are mutually inverse functions. Other examples of pairs of mutually inverse functions are for and, as well as exponential and logarithmic functions.
Recall that the graphs of mutually inverse functions are symmetric with respect to the straight line
Similarly, we define the function Only a segment we need is one on which each value of the angle corresponds to its own value of the cosine, and knowing the cosine, you can uniquely find the angle. The segment is suitable for us
The inverse cosine of a number a is the number , such that
It is easy to remember: "arc cosines live on top", and not just on top, but on a segment
Designation: Area of \u200b\u200bdefinition of inverse cosine - segment Range of values \u200b\u200b- 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 portion of the function where it is monotonic, that is, it takes each of its values \u200b\u200bexactly once.
Let's choose a segment. On this interval, the function decreases monotonically, that is, the correspondence between the sets and is one-to-one. Each value of x corresponds to its own value of y. On this segment, there is a function inverse to the cosine, that is, the function y \u003d arccosx.
Let's fill in the table using the definition of the arccosine.
The inverse cosine of a number x belonging to an interval is a number y belonging to an interval such that
Hence, since;
Because ;
Because ,
Because ,
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Here is the arccosine plot:
Function properties
1. Scope
2. Range of values
This function is general - it is neither even nor odd.
4. The function is strictly decreasing. The largest value, equal to, the function y \u003d arccosx takes at, and the smallest value, equal to zero, takes at
5. Functions and are mutually inverse.
The next ones are arc tangent and arc cotangent.
The arctangent of a number a is the number , such that
Designation:. Arctangent definition area - interval Value area - interval.
Why are the ends of the interval - points - excluded in the definition of the 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. By definition, the arctangent of a number x is a number y belonging to an interval such that
How to build a graph is already clear. Since the arctangent is the inverse function of the tangent, we proceed as follows:
We choose such a plot of the function graph, where the correspondence between x and y is one-to-one. This is the interval Ts.In this section, the function takes values \u200b\u200bfrom to
Then the inverse function, that is, the function, domain, definition will have the entire number line, from to and the range of values \u200b\u200bwill be the interval
Hence,
Hence,
Hence,
And what will happen for infinitely large values \u200b\u200bof x? In other words, how does this function behave if x tends to plus infinity?
We can ask ourselves the question: for what number from the interval does the value of the tangent tend to infinity? - Obviously this
This means that for infinitely large values \u200b\u200bof x, the arctangent graph approaches the horizontal asymptote
Likewise, if x tends to minus infinity, the arctangent graph approaches the horizontal asymptote
The figure shows the graph of the function
Function properties
1. Scope
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 function of arc cotangent and plot its graph.
The arc cotangent of a number a is the number , such that
Function graph:
Function properties
1. Scope
2. Range of values
3. The function is of a general type, that is, neither even nor odd.
4. The function is strictly decreasing.
5. Direct and - horizontal asymptotes of this function.
6. Functions and are mutually inverse if considered on the interval
In a number of problems of mathematics and its applications, it is required to find the corresponding value of the angle, expressed in degree or in radian measure, from the known value of the trigonometric function. It is known that an infinite set of angles corresponds to the same sine value, for example, if $ \\ sin α \u003d 1/2, $ then the angle $ α $ can be equal to both $ 30 ° $ and $ 150 °, $ or in radian measure $ π / 6 $ and $ 5π / 6, $ and any of the angles obtained from these by adding a term of the form $ 360 ° ⋅k, $ or, respectively, $ 2πk, $ where $ k $ is any integer. This becomes clear from considering the graph of the function $ y \u003d \\ sin x $ on the whole number line (see Fig. $ 1 $): if on the $ Oy $ axis we put off a segment of length $ 1/2 $ and draw a straight line parallel to the $ Ox axis, $ then it will intersect a sinusoid at an infinite number of points. To avoid the possible variety of answers, inverse trigonometric functions are introduced, otherwise called circular, or arc functions (from the Latin word arcus - "arc").
The four basic 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). Consider the functions \\ arcsin x and \\ mathrm (arctg) \\, x, since the other two are expressed in terms of them by the formulas:
$ \\ arccos x \u003d \\ frac (π) (2) - \\ arcsin x, $ $ \\ mathrm (arcctg) \\, x \u003d \\ frac (π) (2) - \\ mathrm (arctg) \\, x. $
Equality $ y \u003d \\ arcsin x $ by definition means such an angle $ y, $ expressed in radian measure and enclosed in the range from $ - \\ frac (π) (2) $ to $ \\ frac (π) (2), $ sine which is equal to $ x, $ that is, $ \\ sin y \u003d x. $ The function $ \\ arcsin x $ is the inverse function of the function $ \\ sin x $ considered on the segment $ \\ left [- \\ frac (π) (2 ), + \\ frac (π) (2) \\ right], $ where this function increases monotonically and takes all values \u200b\u200bfrom $ −1 $ to $ + 1. $ Obviously, the argument $ y $ of the function $ \\ arcsin x $ can take values \u200b\u200bonly from the segment $ \\ left [−1, + 1 \\ right]. $ So, the function $ y \u003d \\ arcsin x $ is defined on the segment $ \\ left [−1, + 1 \\ right], $ is monotonically increasing, and its values \u200b\u200bfill the segment $ \\ left [- \\ frac (π) (2), + \\ frac (π) (2) \\ right]. $ The graph of the function is shown in fig. $ 2. $
Under the condition $ −1 ≤ a ≤ 1 $ all solutions of the equation $ \\ sin x \u003d a $ can be represented as $ x \u003d (- 1) ^ n \\ arcsin a + πn, $ $ n \u003d 0, ± 1, ± 2, …. $ For example, if
$ \\ sin x \u003d \\ frac (\\ sqrt (2)) (2) $ then $ x \u003d (−1) ^ n \\ frac (π) (4) + πn, $ $ n \u003d 0, ± 1, ± 2 ,…. $
The relation $ y \u003d \\ mathrm (arcctg) \\, x $ is defined for all values \u200b\u200bof $ x $ and by definition means that the angle $ y, $ expressed in radian measure, is within the limits
$ - \\ frac (π) (2)
and the tangent of this angle is x, that is, $ \\ mathrm (tg) \\, y \u003d x. $ The function $ \\ mathrm (arctg) \\, x $ is defined on the whole number line, is a function inverse to the function $ \\ mathrm ( tg) \\, x $, which is considered only on the interval
$ - \\ frac (π) (2)
The function $ y \u003d \\ mathrm (arctg) \\, x $ is monotonically increasing, its graph is shown in Fig. $ 3. $
All solutions of the equation $ \\ mathrm (tg) \\, x \u003d a $ can be written as $ x \u003d \\ mathrm (arctg) \\, a + πn, $ $ n \u003d 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 of an infinite power series was obtained was the function $ \\ mathrm (arctg) \\, x. $ From this series, G. Leibniz, for a fixed value of the argument $ x \u003d 1 $, obtained the famous nearby