Problems related to inverse trigonometric functions are often offered in school final exams and in entrance exams at some universities. A detailed study of this topic can only be achieved in elective classes or elective courses. The proposed course is designed to develop the abilities of each student as fully as possible and improve his mathematical preparation.
The course lasts 10 hours:
1.Functions arcsin x, arccos x, arctg x, arcctg x (4 hours).
2.Operations on inverse trigonometric functions (4 hours).
3. Inverse trigonometric operations on trigonometric functions (2 hours).
Lesson 1 (2 hours) Topic: Functions y = arcsin x, y = arccos x, y = arctan x, y = arcctg x.
Goal: complete coverage of this issue.
1.Function y = arcsin x.
a) For the function y = sin x on the segment there is an inverse (single-valued) function, which we agreed to call arcsine and denote it as follows: y = arcsin x. The graph of the inverse function is symmetrical with the graph of the main function with respect to the bisector of I - III coordinate angles.
Properties of the function y = arcsin x.
1) Domain of definition: segment [-1; 1];
2)Area of change: segment;
3)Function y = arcsin x odd: arcsin (-x) = - arcsin x;
4)The function y = arcsin x is monotonically increasing;
5) The graph intersects the Ox, Oy axes at the origin.
Example 1. Find a = arcsin. This example can be formulated in detail as follows: find an argument a, lying in the range from to, whose sine is equal to.
Solution. There are countless arguments whose sine is equal to , for example: 
etc. But we are only interested in the argument that is on the segment. This would be the argument. So, .
Example 2. Find 
.Solution. Arguing 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: 
, because 
. Do the expressions make sense: ; arcsin 1.5; 
?
c) Arrange in ascending order: arcsin, arcsin (-0.3), arcsin 0.9.
II. Functions y = arccos x, y = arctg x, y = arcctg x (similar).
Lesson 2 (2 hours) Topic: Inverse trigonometric functions, their graphs.
Purpose: in this lesson it is necessary to develop skills in determining values trigonometric functions, in constructing graphs of inverse trigonometric functions using D (y), E (y) and the necessary transformations.
In this lesson, complete exercises that include finding the domain of definition, the domain of value of functions of the type: y = arcsin, y = arccos (x-2), y = arctg (tg x), y = arccos.
You should construct graphs of the functions: a) y = arcsin 2x; b) y = 2 arcsin 2x; c) y = arcsin;
d) y = arcsin; e) y = arcsin; e) y = arcsin; g) y = | arcsin | .
Example. Let's plot y = arccos
You can include the following exercises in your homework: build graphs of functions: y = arccos, y = 2 arcctg x, y = arccos | x | .
Graphs of Inverse Functions
Lesson No. 3 (2 hours) Topic:
Operations on inverse trigonometric functions.Goal: to expand mathematical knowledge (this is important for those entering specialties with increased requirements for mathematical training) by introducing basic relations for inverse trigonometric functions.
Material for the lesson.
Some simple trigonometric operations on inverse trigonometric functions: sin (arcsin x) = x , i xi ? 1; cos (arсcos x) = x, i xi? 1; tg (arctg x)= x , x I R; ctg (arcctg x) = x , x I R.
Exercises.
a) tg (1.5 + arctg 5) = - ctg (arctg 5) = 
.
ctg (arctg x) = ; tg (arcctg x) = .
b) cos ( + arcsin 0.6) = - cos (arcsin 0.6). Let arcsin 0.6 = a, sin a = 0.6;
cos (arcsin x) = ; sin (arccos x) = .
Note: we take the “+” sign in front of the root because a = arcsin x satisfies .
c) sin (1.5 + arcsin). Answer: ;
d) ctg ( + arctg 3). Answer: ;
e) tg ( – arcctg 4). Answer: .
e) cos (0.5 + arccos). Answer: .
Calculate:
a) sin (2 arctan 5) .
Let arctan 5 = a, then sin 2 a = 
or sin (2 arctan 5) = 
;
b) cos ( + 2 arcsin 0.8). Answer: 0.28.
c) arctg + arctg.
Let a = arctg, b = arctg,
then tg(a + b) = 
.
d) sin (arcsin + arcsin).
e) Prove that for all x I [-1; 1] true arcsin x + arccos x = .
Proof:
arcsin x = – arccos x
sin (arcsin x) = sin ( – arccos x)
x = cos (arccos x)
To solve it yourself: sin (arccos), cos (arcsin), cos (arcsin ()), sin (arctg (- 3)), tg (arccos), ctg (arccos).
For a home 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) arctg 0.5 – arctg 3.
Lesson No. 4 (2 hours) Topic: Operations on inverse trigonometric functions.
Goal: In this lesson, demonstrate the use of ratios in transforming more complex expressions.
Material for the lesson.
ORALLY:
a) sin (arccos 0.6), cos (arcsin 0.8);
b) tg (arcсtg 5), ctg (arctg 5);
c) sin (arctg -3), cos (arcсtg());
d) tg (arccos), ctg (arccos()).
IN WRITTEN:
1) cos (arcsin + arcsin + arcsin).
2) cos (arctg 5–arccos 0.8) = cos (arctg 5) cos (arccos 0.8) + sin (arctg 5) sin (arccos 0.8) =
3) tg ( - arcsin 0.6) = - tg (arcsin 0.6) = 
4) 
Independent work will help to identify the level of mastery of the material.
| 1) tg (arctg 2 – arctg) 2) cos( - arctan2) 3) arcsin + arccos |
1) cos (arcsin + arcsin) 2) sin (1.5 - arctan 3) 3) arcctg3 – arctg 2 |
For homework we can suggest:
1) ctg (arctg + arctg + arctg); 2) sin 2 (arctg 2 – arcctg ()); 3) sin (2 arctg + tan ( arcsin )); 4) sin(2 arctg); 5) tg ( (arcsin ))
Lesson No. 5 (2 hours) Topic: Inverse trigonometric operations on trigonometric functions.
Goal: to form students’ understanding of inverse trigonometric operations on trigonometric functions, focusing on increasing the comprehension of the theory being studied.
When studying this topic, it is assumed that the volume of theoretical material to be memorized is limited.
Lesson material:
You can start learning new material by studying the function y = arcsin (sin x) and plotting its graph.
3. Each x I R is associated with y I, i.e.<= y <= такое, что sin y = sin x.
4. The function is odd: sin(-x) = - sin x; arcsin(sin(-x)) = - arcsin(sin x).
6. Graph y = arcsin (sin x) on:
a) 0<= x <= имеем y = arcsin(sin x) = x, ибо sin y = sin x и <= y <= .
b)<= x <= получим y = arcsin (sin x) = arcsin ( - x) = - x, ибо
sin y = sin ( – x) = sin x , 0<= - x <= .
So, 
Having constructed y = arcsin (sin x) on , we continue symmetrically about the origin on [- ; 0], given the oddness of this function. Using periodicity, we continue along the entire number line.
Then write down some relationships: arcsin (sin a) = a if<= a <= ; arccos (cos a ) = a if 0<= a <= ; arctg (tg a) = a if< a < ; arcctg (ctg a) = a , если 0 < a < .
And do the following exercises:a) arccos(sin 2).Answer: 2 - ; b) arcsin (cos 0.6). Answer: - 0.1; c) arctg (tg 2). Answer: 2 - ;
d) arcctg(tg 0.6).Answer: 0.9; e) arccos (cos ( - 2)). Answer: 2 - ; e) arcsin (sin ( - 0.6)). Answer: - 0.6; g) arctg (tg 2) = arctg (tg (2 - )). Answer: 2 - ; h) аrcctg (tg 0.6). Answer: - 0.6; - arctan x; e) arccos + arccos
Inverse trigonometric functions 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, 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
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 this 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 are mutually inverse 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 an even nor an 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 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
Inverse trigonometric functions(circular functions, arc functions) - mathematical functions that are inverse to trigonometric functions.
These usually include 6 functions:
- arcsine(designation: arcsin x; arcsin x- this is the angle sin which is equal to x),
- arc cosine(designation: arccos x; arccos x is the angle whose cosine is equal to x and so on),
- arctangent(designation: arctan x or arctan x),
- arccotangent(designation: arcctg x or arccot x or arccotan x),
- arcsecant(designation: arcsec x),
- arccosecant(designation: arccosec x or arccsc x).
arcsine (y = arcsin x) - inverse function to sin (x = sin y 
. In other words, returns the angle by its value sin.
arc cosine (y = arccos x) - inverse function to cos (x = cos y cos.
Arctangent (y = arctan x) - inverse function to tg (x = tan y), which has a domain and a set of values 
. In other words, returns the angle by its value tg.
Arccotangent (y = arcctg x) - inverse function to ctg (x = cotg y), which has a domain of definition and a set of values. In other words, returns the angle by its value ctg.
arcsec- arcsecant, returns the angle according to the value of its secant.
arccosec- arccosecant, returns an angle based on the value of its cosecant.
When the inverse trigonometric function is not defined at a specified point, then its value will not appear in the final table. Functions arcsec And arccosec are not determined on the segment (-1,1), but arcsin And arccos are determined only on the interval [-1,1].
The name of the inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix “arc-” (from Lat. arc us- arc). This is due to the fact that geometrically, the value of the inverse trigonometric function is associated with the length of the arc of the unit circle (or the angle that subtends this arc), which corresponds to one or another segment.
Sometimes in foreign literature, as well as in scientific/engineering calculators, they use notations like sin−1, cos −1 for arcsine, arccosine and the like, this is considered not completely accurate, because there is likely to be confusion with raising a function to a power −1 (« −1 » (minus the first power) defines the function x = f -1 (y), the inverse of the function y = f(x)).
Basic relations of inverse trigonometric functions.
Here it is important to pay attention to the intervals for which the formulas are valid.
Formulas relating inverse trigonometric functions.
Let us denote any of the values of inverse trigonometric functions by Arcsin x, Arccos x, Arctan x, Arccot x and keep the notation: arcsin x, arcos x, arctan x, arccot x for their main values, then the connection between them is expressed by such relationships.
Lessons 32-33. Inverse trigonometric functions
09.07.2015 8936 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 a 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 = and it’s easier and more convenient to use not general formulas, but to 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 solutions 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. Graph the function:
VII. Summing up the lessons
Inverse trigonometric functions are mathematical functions that are the inverse of trigonometric functions.
Function y=arcsin(x)
The arcsine of a number α is a number α from the interval [-π/2;π/2] whose sine is equal to α.
Graph of a function
The function у= sin(x) on the interval [-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= sin(x), where x ∈[-π/2;π/2], is called the arcsine and is denoted y=arcsin(x), where x∈[-1;1].
So, according to the definition of the inverse function, the domain of definition of the arcsine is the segment [-1;1], and the set of values is the segment [-π/2;π/2].
Note that the graph of the function y=arcsin(x), where x ∈[-1;1], is symmetrical to the graph of the function y= sin(x), where x∈[-π/2;π/2], with respect to the bisector of the coordinate angles first and third quarters.
Function range y=arcsin(x).
Example No. 1.
Find arcsin(1/2)?
Since the range of values of the function arcsin(x) belongs to the interval [-π/2;π/2], then only the value π/6 is suitable. Therefore, arcsin(1/2) =π/6.
Answer:π/6
Example No. 2.
Find arcsin(-(√3)/2)?
Since the range of values arcsin(x) x ∈[-π/2;π/2], then only the value -π/3 is suitable. Therefore, arcsin(-(√3)/2) =- π/3.
Function y=arccos(x)
The arc cosine of a number α is a number α from the interval whose cosine is equal to α.
Graph of a function
The function y= cos(x) on the segment is strictly decreasing and continuous; therefore, it has an inverse function, strictly decreasing and continuous.
The inverse function for the function y= cosx, where x ∈, is called arc cosine and is denoted by y=arccos(x),where x ∈[-1;1].
So, according to the definition of the inverse function, the domain of definition of the arc cosine is the segment [-1;1], and the set of values is the segment.
Note that the graph of the function y=arccos(x), where x ∈[-1;1] is symmetrical to the graph of the function y= cos(x), where x ∈, with respect to the bisector of the coordinate angles of the first and third quarters.
Function range y=arccos(x).
Example No. 3.
Find arccos(1/2)?
Since the range of values is arccos(x) x∈, then only the value π/3 is suitable. Therefore, arccos(1/2) =π/3.
Example No. 4.
Find arccos(-(√2)/2)?
Since the range of values of the function arccos(x) belongs to the interval, then only the value 3π/4 is suitable. Therefore, arccos(-(√2)/2) = 3π/4.
Answer: 3π/4
Function y=arctg(x)
The arctangent of a number α is a number α from the interval [-π/2;π/2] whose tangent is equal to α.
Graph of a function 
The tangent function is continuous and strictly increasing on the interval (-π/2;π/2); therefore, it has an inverse function that is continuous and strictly increasing.
The inverse function for the function y= tan(x), where x∈(-π/2;π/2); is called the arctangent and is denoted by y=arctg(x), where x∈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 is the interval
(-π/2;π/2).
Note that the graph of the function y=arctg(x), where x∈R, is symmetrical to the graph of the function y= tanx, where x ∈ (-π/2;π/2), relative to the bisector of the coordinate angles of the first and third quarters.
The range of the function y=arctg(x).
Example No. 5?
Find arctan((√3)/3).
Since the range of values arctg(x) x ∈(-π/2;π/2), then only the value π/6 is suitable. Therefore, arctg((√3)/3) =π/6.
Example No. 6.
Find arctg(-1)?
Since the range of values arctg(x) x ∈(-π/2;π/2), then only the value -π/4 is suitable. Therefore, arctg(-1) = - π/4.
Function y=arcctg(x)
The arc cotangent of a number α is a number α from the interval (0;π) whose cotangent is equal to α.
Graph of a function
On the interval (0;π), the cotangent function strictly decreases; in addition, 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=ctg(x), where x ∈(0;π), is called arccotangent and is denoted y=arcctg(x), where x∈R.
So, according to the definition of the inverse function, the domain of definition of the arccotangent will be R, and the set of values will be the interval (0;π). The graph of the function y=arcctg(x), where x∈R is symmetrical to the graph of the function y=ctg(x) x∈(0 ;π), relative to the bisector of the coordinate angles of the first and third quarters.
Function range y=arcctg(x).
Example No. 7.
Find arcctg((√3)/3)?
Since the range of values arcctg(x) x ∈(0;π), then only the value π/3 is suitable. Therefore arccos((√3)/3) =π/3.
Example No. 8.
Find arcctg(-(√3)/3)?
Since the range of values is arcctg(x) x∈(0;π), then only the value 2π/3 is suitable. Therefore, arccos(-(√3)/3) = 2π/3.
Editors: Ageeva Lyubov Aleksandrovna, Gavrilina Anna Viktorovna
