Inverse trigonometric functions and their graphs. Trigonometry

What is arcsine, arccosine? What is arc tangent, arc tangent?

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

To concepts arcsine, arccosine, arctangent, arccotangent the student population is wary. He does not understand these terms and, therefore, does not trust this glorious family.) But in vain. These are very simple concepts. Which, by the way, make life much easier for a knowledgeable person when solving trigonometric equations!

Confused about simplicity? In vain.) Right here and now you will be convinced of this.

Of course, for understanding, it would be nice to know what sine, cosine, tangent and cotangent are. Yes, their table values for some angles ... At least in the most general terms. Then there will be no problems here either.

So, we are surprised, but remember: arcsine, arccosine, arctangent and arctangent are just some angles. No more, no less. There is an angle, say 30°. And there is an angle arcsin0.4. Or arctg(-1.3). There are all kinds of angles.) You can simply write angles in different ways. You can write the angle in degrees or radians. Or you can - through its sine, cosine, tangent and cotangent ...

What does the expression mean

arcsin 0.4?

This is the angle whose sine is 0.4! Yes Yes. This is the meaning of the arcsine. I repeat specifically: arcsin 0.4 is an angle whose sine is 0.4.

And that's it.

To keep this simple thought in my head for a long time, I will even give a breakdown of this terrible term - the arcsine:

arc sin 0,4
corner, whose sine equals 0.4

As it is written, so it is heard.) Almost. Console arc means arc(word arch know?), because ancient people used arcs instead of corners, but this does not change the essence of the matter. Remember this elementary decoding of a mathematical term! Moreover, for the arc cosine, arc tangent and arc tangent, the decoding differs only in the name of the function.

What is arccos 0.8?
This is an angle whose cosine is 0.8.

What is arctan(-1,3) ?
This is an angle whose tangent is -1.3.

What is arcctg 12 ?
This is an angle whose cotangent is 12.

Such an elementary decoding allows, by the way, to avoid epic blunders.) For example, the expression arccos1,8 looks quite solid. Let's start decoding: arccos1,8 is an angle whose cosine is equal to 1.8... Hop-hop!? 1.8!? Cosine cannot be greater than one!

Right. The expression arccos1,8 does not make sense. And writing such an expression in some answer will greatly amuse the verifier.)

Elementary, as you can see.) Each angle has its own personal sine and cosine. And almost everyone has their own tangent and cotangent. Therefore, knowing the trigonometric function, you can write down the angle itself. For this, arcsines, arccosines, arctangents and arccotangents are intended. Further, I will call this whole family a diminutive - arches. to type less.)

Attention! Elementary verbal and conscious deciphering the arches allows you to calmly and confidently solve a variety of tasks. And in unusual tasks only she saves.

Is it possible to switch from arches to ordinary degrees or radians?- I hear a cautious question.)

Why not!? Easily. You can go there and back. Moreover, it is sometimes necessary to do so. Arches are a simple thing, but without them it’s somehow calmer, right?)

For example: what is arcsin 0.5?

Let's look at the decryption: arcsin 0.5 is the angle whose sine is 0.5. Now turn on your head (or Google)) and remember which angle has a sine of 0.5? The sine is 0.5 y angle of 30 degrees. That's all there is to it: arcsin 0.5 is a 30° angle. You can safely write:

arcsin 0.5 = 30°

Or, more solidly, in terms of radians:

That's it, you can forget about the arcsine and work on with the usual degrees or radians.

If you realized what is arcsine, arccosine ... What is arctangent, arccotangent ... Then you can easily deal with, for example, such a monster.)

An ignorant person will recoil in horror, yes ...) And a knowledgeable remember the decryption: the arcsine is the angle whose sine is ... Well, and so on. If a knowledgeable person also knows the table of sines ... The table of cosines. A table of tangents and cotangents, then there are no problems at all!

It is enough to consider that:

I will decipher, i.e. translate the formula into words: angle whose tangent is 1 (arctg1) is a 45° angle. Or, which is the same, Pi/4. Similarly:

and that's all... We replace all the arches with values in radians, everything is reduced, it remains to calculate how much 1 + 1 will be. It will be 2.) Which is the correct answer.

This is how you can (and should) move from arcsines, arccosines, arctangents and arctangents to ordinary degrees and radians. This greatly simplifies scary examples!

Often, in such examples, inside the arches are negative values. Like, arctg(-1.3), or, for example, arccos(-0.8)... That's not a problem. Here are some simple formulas for going from negative to positive:

You need, say, to determine the value of an expression:

You can solve this using a trigonometric circle, but you don't want to draw it. Well, okay. Going from negative values inside the arc cosine to positive according to the second formula:

Inside the arccosine on the right already positive meaning. What

you just have to know. It remains to substitute the radians instead of the arc cosine and calculate the answer:

That's all.

Restrictions on arcsine, arccosine, arctangent, arccotangent.

Is there a problem with examples 7 - 9? Well, yes, there is some trick there.)

All of these examples, from 1st to 9th, are carefully sorted out on the shelves in Section 555. What, how and why. With all the secret traps and tricks. Plus ways to dramatically simplify the solution. By the way, this section contains a lot of useful information and practical advice trigonometry in general. And not only in trigonometry. Helps a lot.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Tasks related to inverse trigonometric functions are often offered at school final exams and at entrance exams in some universities. A detailed study of this topic can only be achieved in extracurricular classes or at elective courses. The proposed 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 of 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 = arctg x, y = 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 as follows: y \u003d 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.

Function properties y = arcsin x .

1)Scope of definition: segment [-1; one];

2) Area of change: cut ;

3) Function y = arcsin x odd: arcsin (-x) = - arcsin x;

4) The function y = arcsin x is monotonically increasing;

5) The graph crosses the Ox, Oy axes at the origin.

Example 1. Find a = arcsin . This example can be formulated in detail as follows: find such an argument a , lying in the range from to , whose sine is equal to .

Solution. There are countless arguments whose sine is , for example: etc. But we are only interested in the argument that is on the interval . This argument will be . 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 (similarly).

Lesson 2 (2 hours) Topic: Inverse trigonometric functions, their graphs.

Target: on this lesson it is necessary to develop skills in determining the values trigonometric functions, in plotting 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 scope of functions of the type: y = arcsin , y = arccos (x-2), y = arctg (tg x), y = arccos .

It is necessary to build graphs of functions: a) y = arcsin 2x; b) y = 2 arcsin 2x; c) y \u003d arcsin;

d) y \u003d arcsin; e) y = arcsin; f) 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 #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 preparation) by introducing the basic relationships for inverse trigonometric functions.

Lesson material.

Some simple trigonometric operations on inverse trigonometric functions: sin (arcsin x) \u003d x, i xi? one; cos (arсcos x) = x, i xi? one; 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 (arctgx) = ; tg (arctg x) = .

b) cos (+ arcsin 0.6) = - cos (arcsin 0.6). Let arcsin 0.6 \u003d a, sin a \u003d 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: .

f) cos (0.5 + arccos) . Answer: .

Calculate:

a) sin (2 arctan 5) .

Let arctg 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 tan(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)

For a standalone solution: sin (arccos ), cos (arcsin ) , cos (arcsin ()), sin (arctg (- 3)), tg (arccos ) , ctg (arccos ).

For a home solution: 1) sin (arcsin 0.6 + arctg 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.

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 (arctg 5), ctg (arctg 5);

c) sin (arctg -3), cos (arctg ());

d) tg (arccos ), ctg (arccos()).

WRITTEN:

1) cos (arcsin + arcsin + arcsin).

2) cos (arctg 5 - arccos 0.8) = cos (arctg 5) cos (arctg 0.8) + sin (arctg 5) sin (arccos 0.8) =

3) tg (- arcsin 0.6) = - tg (arcsin 0.6) =

Independent work will help to determine the level of assimilation of the material

1) tg ( arctg 2 - arctg )

2) cos( - arctg2)

3) arcsin + arccos

1) cos (arcsin + arcsin)

2) sin (1.5 - arctg 3)

3) arcctg3 - arctg 2

For homework can offer:

1) ctg (arctg + arctg + arctg); 2) sin 2 (arctg 2 - arcctg ()); 3) sin (2 arctg + tg ( arcsin )); 4) sin (2 arctan); 5) tg ( (arcsin ))

Lesson No. 5 (2h) Topic: Inverse trigonometric operations on trigonometric functions.

Purpose: to form students' understanding of 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.

Material for the lesson:

You can start learning new material by examining the function y = 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)) = - 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 \u003d sin ( - x) \u003d sinx, 0<= - x <= .

So,

Having built y = arcsin (sin x) on , we continue symmetrically about the origin on [- ; 0], taking into account the oddness of this function. Using periodicity, we continue to the entire numerical axis.

Then write down some ratios: 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 -; f) arcsin (sin (- 0.6)). Answer: - 0.6; g) arctg (tg 2) = arctg (tg (2 - )). Answer: 2 - ; h) arcctg (tg 0.6). Answer: - 0.6; - arctanx; e) arccos + arccos

Function inverse to cosine

The range of the function y=cos x (see Fig. 2) is a segment. On the interval, the function is continuous and monotonically decreasing.

Rice. 2

This means that a function is defined on the interval that is inverse to the function y=cos x. This inverse function is called the arccosine and is denoted y=arccos x .

Definition

The arccosine of the number a, if |a|1, is the angle whose cosine belongs to the segment; it is designated arccos a.

Thus, arccos a is an angle that satisfies the following two conditions: cos (arccos a)=a, |a|1; 0? arccos a ?r.

For example, arccos, since cos and; arccos, since cos.

The function y = arccos x (Fig. 3) is defined on a segment, its range is a segment. On the segment, the function y=arccos x is continuous and decreases monotonically 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 \u003d arccos x (see Fig. 3) is symmetrical to the graph of the function y \u003d cos x with respect to the straight line y \u003d x.

Rice. 3

Let us show that the equality arccos(-x) = p-arccos x takes place.

Indeed, 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 monotonically decreases on a segment, there cannot be two different angles on it that have equal cosines. Find the cosines of the angles arccos(-x) and p-arccos x. By definition cos (arccos x) = - x, by 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 that the angles themselves are equal.

Function inverse to sine

Consider the function y=sin x (Fig. 6), which on the segment [-p/2; p/2] is increasing, continuous and takes values from the segment [-1; one]. Hence, on the segment [- p / 2; p/2] a function is defined that is inverse to the function y=sin x.

Rice. 6

This inverse function is called the arcsine and denoted y=arcsin x. We introduce the definition of the arcsine of the number a.

The arcsine of the number a, if they 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 designated arcsin a.

Thus, arcsin a is an angle that satisfies the following conditions: sin (arcsin a)=a, |a| ?one; -r/2 ? arcsin huh? p/2. For example, since sin and [- p/2; p/2]; arcsin since sin = and [-p/2; p/2].

The function y=arcsin x (Fig. 7) is defined on the segment [- 1; 1], its range is the segment [-р/2;р/2]. On the segment [- 1; 1] the function y=arcsin x is continuous and monotonically increasing 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 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 -r / 2. At x \u003d 0, the function is zero: arcsin 0 \u003d 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: - р/2 ? arcsin x ? ? p/2. So the angles are arcsin(-x) and - arcsin x belong to the same segment [ - p/2; p/2].

Find the sines of these angles: sin (arcsin (-x)) = - 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; p/2], are equal, which means that the angles themselves are equal, i.e. arcsin (-x) = - arcsin x. Hence, the function y=arcsin x is odd. The graph of the function y=arcsin x is symmetrical with respect to the origin.

Let us show that arcsin (sin x) = x for any x [-p/2; p/2].

Indeed, by definition -p/2 ? arcsin (sin x) ? р/2, and according to the condition -р/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=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 have sin x, for the angle arcsin (sin x) we have sin (arcsin (sin x)) = sin x. We got 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 modulo transformations from the graph y=arcsin (sin x) (depicted by the dashed line in Fig. 8). The desired graph y=arcsin (sin |x-/4|) is obtained from it by a shift of /4 to the right along the x-axis (depicted by a solid line in Fig. 8)

Function inverse to tangent

The function y=tg x on the interval takes all numeric values: E (tg x)=. On this interval, it is continuous and monotonically increasing. Hence, a function is defined on the interval that is inverse to the function y = tg x. This inverse function is called the arc tangent and denoted y = arctg x.

The arc tangent of the number a is the angle from the interval, the tangent of which 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 the only value of the function y \u003d arctg x (Fig. 9).

Obviously, D (arctg x) = , E (arctg x) = .

The function y = arctg x is increasing because the function y = tg x is increasing on the interval. It is easy to prove that arctg(-x) = - arctgx, i.e. that the arc tangent is an odd function.

Rice. 9

The graph of the function y = arctg x is symmetrical to the graph of the function y = tg x with respect to the straight line y = x, the graph y = arctg x passes through the origin (because arctan 0 = 0) and is symmetrical with respect to the origin (as the graph of an odd function).

It can be proved that arctg (tg x) = x if x.

Cotangent inverse function

The function y = ctg x on the interval takes all the numeric values from the interval. Its range of values coincides with the set of all real numbers. In the interval, the function y = ctg x is continuous and monotonically increasing. Hence, a function is defined on this interval that is inverse to the function y = ctg x. The inverse function of the cotangent is called the arc cotangent and is denoted y = arcctg x.

The arc tangent of the number a is the angle belonging to the interval, the cotangent of which is equal to a.

Thus, arcctg a is an angle that satisfies the following conditions: ctg (arcctg a)=a and 0 ? arcctg a ? R.

It follows from the definition of the inverse function and the definition of the arc tangent that D (arcctg x) = , E (arcctg x) = . The arc tangent is a decreasing function because the function y = ctg x decreases in the interval.

The graph of the function y \u003d arcctg x does not cross the Ox axis, since y\u003e 0 R. At x \u003d 0 y \u003d arcctg 0 \u003d.

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.

The sin, cos, tg, and ctg functions are always accompanied by an arcsine, arccosine, arctangent, and arccotangent. One is a consequence of the other, and pairs of functions are equally important for working with trigonometric expressions.

Consider a drawing of a unit circle, which graphically displays the values of trigonometric functions.

If you calculate arcs OA, arcos OC, arctg DE and arcctg MK, then they will all be equal to the value of the angle α. The formulas below reflect the relationship between the main trigonometric functions and their corresponding arcs.

To understand more about the properties of the arcsine, it is necessary to consider its function. Schedule has the form of an asymmetric curve passing through the center of coordinates.

Arcsine properties:

If we compare graphs sin and arc sin, two trigonometric functions can find common patterns.

Arc cosine

Arccos of the number a is the value of the angle α, the cosine of which is equal to a.

Curve y = arcos x mirrors the plot of arcsin x, with the only difference being that it passes through the point π/2 on the OY axis.

Consider the arccosine function in more detail:

The function is defined on the segment [-1; one].
ODZ for arccos - .
The graph is entirely located in the I and II quarters, and the function itself is neither even nor odd.
Y = 0 for x = 1.
The curve decreases along its entire length. Some properties of the arc cosine are the same as the cosine function.

Some properties of the arc cosine are the same as the cosine function.

It is possible that such a “detailed” study of the “arches” will seem superfluous to schoolchildren. However, otherwise, some elementary typical USE tasks can lead students to a dead end.

Exercise 1. Specify the functions shown in the figure.

Answer: rice. 1 - 4, fig. 2 - 1.

In this example, the emphasis is on the little things. Usually, students are very inattentive to the construction of graphs and the appearance of functions. Indeed, why memorize the form of the curve, if it can always be built from calculated points. Do not forget that in the conditions of the test, the time spent on drawing for a simple task will be required to solve more complex tasks.

Arctangent

Arctg the number a is such a value of the angle α that its tangent is equal to a.

If we consider the plot of the arc tangent, we can distinguish the following properties:

The graph is infinite and defined on the interval (- ∞; + ∞).
Arctangent is an odd function, therefore, arctan (- x) = - arctan x.
Y = 0 for x = 0.
The curve increases over the entire domain of definition.

Let's give a brief comparative analysis of tg x and arctg x in the form of a table.

Arc tangent

Arcctg of the number a - takes such a value of α from the interval (0; π) that its cotangent is equal to a.

Properties of the arc cotangent function:

The function definition interval is infinity.
The range of admissible values is the interval (0; π).
F(x) is neither even nor odd.
Throughout its length, the graph of the function decreases.

Comparing ctg x and arctg x is very simple, you just need to draw two drawings and describe the behavior of the curves.

Task 2. Correlate the graph and the form of the function.

Logically, the graphs show that both functions are increasing. Therefore, both figures display some arctg function. It is known from the properties of the arc tangent that y=0 for x = 0,

Answer: rice. 1 - 1, fig. 2-4.

Trigonometric identities arcsin, arcos, arctg and arcctg

Previously, we have already identified the relationship between arches and the main functions of trigonometry. This dependence can be expressed by a number of formulas that allow expressing, for example, the sine of an argument through its arcsine, arccosine, or vice versa. Knowledge of such identities can be useful in solving specific examples.

There are also ratios for arctg and arcctg:

Another useful pair of formulas sets the value for the sum of the arcsin and arcos and arcctg and arcctg values of the same angle.

Examples of problem solving

Trigonometry tasks can be conditionally divided into four groups: calculate the numerical value of a particular expression, plot a given function, find its domain of definition or ODZ, and perform analytical transformations to solve the example.

When solving the first type of tasks, it is necessary to adhere to the following action plan:

When working with graphs of functions, the main thing is the knowledge of their properties and the appearance of the curve. Tables of identities are needed to solve trigonometric equations and inequalities. The more formulas the student remembers, the easier it is to find the answer to the task.

Suppose in the exam it is necessary to find the answer for an equation of the type:

If you correctly transform the expression and bring it to the desired form, then solving it is very simple and fast. First, let's move arcsin x to the right side of the equation.

If we remember the formula arcsin (sinα) = α, then we can reduce the search for answers to solving a system of two equations:

The constraint on the model x arose, again from the properties of arcsin: ODZ for x [-1; one]. When a ≠ 0, part of the system is a quadratic equation with roots x1 = 1 and x2 = - 1/a. With a = 0, x will be equal to 1.

Since trigonometric functions are periodic, the functions inverse to them are not single-valued. So, the equation y = sin x, for given , has infinitely many roots. Indeed, due to the periodicity of the sine, if x is such a root, then x + 2n(where n is an integer) will also be the root of the equation. In this way, inverse trigonometric functions are multivalued. To make it easier to work with them, the concept of their main values is introduced. Consider, for example, the sine: y = sin x. If we limit the argument x to the interval , then on it the function y = sin x increases monotonically. Therefore, it has a single-valued inverse function, which is called the arcsine: x = arcsin y.

Unless otherwise stated, inverse trigonometric functions mean their principal values, which are defined by the following definitions.

Arcsine ( y= arcsin x) is the inverse function of the sine ( x= siny
Arc cosine ( y= arccos x) is the inverse function of the cosine ( x= cos y) that has a domain of definition and a set of values .
Arctangent ( y= arctg x) is the inverse function of the tangent ( x= tg y) that has a domain of definition and a set of values .
Arc tangent ( y= arcctg x) is the inverse function of the cotangent ( x= ctg y) that has a domain of definition and a set of values .

Graphs of inverse trigonometric functions

Graphs of inverse trigonometric functions are obtained from graphs of trigonometric functions by mirror reflection with respect to the straight line y = x. See sections Sine, cosine, Tangent, cotangent.

y= arcsin x

y= arccos x