Inverse trigonometric functions and their graphs. Formation of concepts of inverse trigonometric functions among students in algebra lessons List the inverse trigonometric functions definition

Lessons 32-33. Inverse trigonometric functions

09.07.2015 8495 0

Target: 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, thenLet'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 inequalitywhich is equivalent to the system of inequalitiesThe 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 changes within the specified limits, from the table data we obtain thatSo the area of ​​change

Example 6

Let us prove that the function y = arctg x odd. LetThen 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

Similarly 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 formSolutions 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 solution

for the equation cos x = -1 solution

Example 14

Let's solve the equation

Since in this example there is a special case of the equation, we will write the solution using the appropriate formula:where can we find it from?

III. Control questions (frontal survey)

1. Define and list the main properties of inverse trigonometric functions.

2. Give graphs of inverse trigonometric functions.

3. Solving simple trigonometric equations.

IV. Lesson assignment

§ 15, No. 3 (a, b); 4 (c, d); 7(a); 8(a); 12 (b); 13(a); 15 (c); 16(a); 18 (a, b); 19 (c); 21;

§ 16, No. 4 (a, b); 7(a); 8 (b); 16 (a, b); 18(a); 19 (c, d);

§ 17, No. 3 (a, b); 4 (c, d); 5 (a, b); 7 (c, d); 9 (b); 10 (a, c).

V. Homework

§ 15, No. 3 (c, d); 4 (a, b); 7 (c); 8 (b); 12(a); 13(b); 15 (g); 16 (b); 18 (c, d); 19 (g); 22;

§ 16, No. 4 (c, d); 7(b); 8(a); 16 (c, d); 18 (b); 19 (a, b);

§ 17, No. 3 (c, d); 4 (a, b); 5 (c, d); 7 (a, b); 9 (d); 10 (b, d).

VI. Creative tasks

1. Find the domain of the function:

Answers:

2. Find the range of the function:

Answers:

3. Plot a graph of the function:

VII. Summing up the lessons

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 inverse function of the 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 take values ​​only 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 segment $\left[−\frac(π)(2),+\frac(π)(2)\right].$ The graph of the function is shown in Fig. $2.$

Under the condition $−1 ≤ a ≤ 1$, 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

In this lesson we will look at the features inverse functions and repeat inverse trigonometric functions. The properties of all basic inverse trigonometric functions will be considered separately: arcsine, arccosine, arctangent and arccotangent.

This lesson will help you prepare for one of the types of tasks AT 7 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 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 a periodic function, unlike the 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:

In this article we will look at such important concepts in trigonometry as arcsine, arccosine, arctangent and arccotangent. We can find the values ​​of numbers (angles) if we know the data of trigonometric functions; this is the very problem that leads us to inverse functions.

Below we will not only give definitions of basic concepts and generally accepted notations, but also provide calculations from which it will be clear what they are. Finally, we will try to connect the concepts of arccotangent, arctangent, arccosine and arcsine with the concept of the unit circle.

Basic definitions

All of the concepts listed above - arcsine, arccosine, arctangent and arccotangent - can be considered both as a number and as an angle. Earlier we already talked about the same duality in the perception of direct functions (sine, cosine, etc.). Let's consider both approaches separately.

Arcsine and other inverse functions as angle

Let's say we have a certain angle whose sine is equal to 1 2. Let's denote it by the letter alpha.

So sin α = 1 2 . An infinite number of angles can have such a sine value: α = (− 1) k · 30 ° + 180 ° · k (α = (− 1) k · π / 6 + π · k), where k ∈ Z. Therefore, we will need to introduce additional conditions. Let the alpha angle be no less than - 90 and no more than 90 degrees (i.e. (in radians it will belong to the segment [ − π 2 , π 2 ]),). In this case, our equality sin α = 1 2 will allow us to designate the alpha angle more clearly: in such conditions there will be only one angle - 30 degrees (π 6 radians).

Based on this equality, we can conclude that the alpha angle is determined subject to any number a ∈ [− 1, 1] and the condition − 90 ° ≤ α ≤ 90 °. This angle is the arcsine of the number a.

Let us formulate the basic definitions.

Definition 1

• arcsine is the inverse function of sin. For a certain number a, it represents an angle from -90 to 90 degrees, the sin of which is equal to a.
• arc cosine- function inverse to cosine. For the number a, this is an angle whose cos is equal to a, and which is in the range from 0 to 180 degrees.
• Arctangent-trigonometric function, inverse to tangent. For a certain number a u 1 is an angle whose value is in the range from - 90 to 90 degrees, the tangent of which is equal to a.
• Arccotangent the number a is also an angle ranging from 0 to 190 degrees, the cotangent of which is equal to a.

Let's summarize: so, the notation a r c sin 0, 3 means just an angle whose sine is equal to 0, 3; a r c cos 0, 7 - angle with cosine 0, 7 and so on.

Signatures of the form a r c sin , a r c cos , a r c t g and a r c c t g are generally accepted for writing inverse trigonometric functions. Sometimes in reference books, especially those compiled in English, you can find slightly different designations for arccotangent and arctangent - a r c tan and a r c c o t. They mean the same thing, but they are not common among us, so we will not use them.

The above definitions can be formulated in a more concise and symbolic form:

Definition 2

• arcsin numbers a in the range from minus one to one is an angle with sin α = a of magnitude − 90 ° ≤ α ≤ 90 ° (− π 2 ≤ α ≤ π 2)
• arccos numbers a in the range from minus one to one there is an angle with cos = a with a value of 0 ° ≤ α ≤ 180 ° (0 ≤ α ≤ π)
• arctg any number a is an angle with t g α = a of magnitude − 90 °< α < 90 ° (− π 2 < α < π 2)
• arctg of any number a there is an angle with c t g α = a value that is 0 °< α < 180 ° (0 < α < π)

Please note that in the definitions of arcsin and arccos the range is from minus one to plus one, but for the other two functions a can be any number. It turns out that arcsine 3 is an erroneous notation, because three does not belong to the specified range. Also meaningless are the entries a r c sin 5, a r c cos - 7, a r c sin - 3, 7 2 3 and with any other values ​​that go beyond the boundaries of the segment we need, because sine and cosine cannot be greater than one and less than minus one. In the case of arctangent and arccotangent, there is no such problem; any real number is suitable for them, including zero, pi, and so on.

Example 1

Now let's look at examples of inverse functions of a number. First, let's take the arcsine. From its basic definition it follows that the angle π 3 is the arcsine of the number 3 2, thus (in this case α = 3 2 and α = π 3).

3 2 is a number that is less than one and greater than minus one, and the angle π 3 is in the range from - π 2 to π 2 and sin π 3 = 3 2.

Example 2

Other examples of a r c sin are records of the form a r c sin (− 1) = − 90 °, a r c sin (0, 5) = π 6, a r c sin (- 2 2) = - π 4. In this case, π 10 cannot be a r c sin 1 2, because sin (π 10) ≠ 1 2.

Example 3

Let's take the following example: sin 270 degrees - minus one, but the opposite is not true: angle 270 is not arcsine - 1, because a r c sin should be no more than 90 degrees. An angle of 270 degrees is not the arcsine of any number because it lies outside the required range.

Example 4

Let's find examples of other inverse functions. So, an angle of 0 radians is arc cosine 1, i.e., a r c cos 1 = 0. Here all arc cosine conditions are met, the number belongs to the desired segment, the angle of a given value is in the range from zero to pi and cos 0 = 1. Angle π 2 - arc cosine of zero: a r c cos 0 = π 2.

Example 5

According to the definition of arctangent, the values ​​are a r c t g (− 1) = − π 4 or a r c t g (− 1) = − 45 °. The arctangent of the root of three is 60 degrees (π 3 rad). From this we can conclude that a r c c t g 0 = π 2, since the angle π 2 lies within the range from 0 to π and c t g (π 2) = 0.

If you want to study this approach to defining inverse trigonometric functions in more detail, we recommend Kochetkov’s textbook (part 1, pp. 260-278)

Arcsine and other inverse functions as numbers

If the problem concerns, say, the sine of an angle, then it is logical to perceive its arcsine as an angle. If we need, for example, to calculate the cosine of a certain number, then it is important to take a different point of view and consider inverse functions as numbers. Based on the second approach, we can slightly reformulate the definitions:

Definition 3

• arcsine and there is some number, t ∈ [ − π 2 , π 2 ], whose sine is equal to a.
• arc cosine number a ∈ [−1, 1] is some number t ∈ [0, π] whose cosine is equal to a.
• Arctangent number a ∈ (− ∞, + ∞) is a number t ∈ (− π 2, π 2) whose tangent is equal to a.
• Arccotangent number a ∈ (− ∞, + ∞) is a number t ∈ (0, π) whose cotangent is equal to a.

Such formulations are typical for most modern mathematics textbooks.

Example 6

Which approach should you choose? How do you understand when it is better to consider the values ​​of the arcsine and other functions as angles, and when - as numbers? This can be understood from the context of the task. Usually if it mentions, say, a r c sin a - 11°, then it is an angle. If we see a notation of the form π − a r c t g a , then most likely it is just a number or an angle measured in radians. If there are simply formulations of the form a r c sin, a r c c t g, etc. without indicating numbers and values, then we are free to choose any approach we want.

The inverse functions of a number can be more clearly represented geometrically: after all, if these are angles, they can be depicted in a drawing. This is easy to do if you haven't forgotten the basic definitions of basic direct functions.

To do this, we need the unit circle that is already familiar to us. Its arcs connecting the main angles will correspond to the values ​​of the inverse functions.

For example, let's take an arc that will illustrate the arcsine of a certain number a. Let's draw a line of sines and indicate a point on it in accordance with the value of a. From this point we now need to get to the x-axis (take the positive direction). We have a ray that will intersect the circle at a special point. The arcsine of the number a is the part of the arc of a circle from this point to the origin. Let us recall two approaches to considering functions: as an angle and as a number. The angle corresponding to the arc is an illustration of the arcsine in the first approach, and the length of the arc, expressed quantitatively, illustrates the arcsine in the second.

Now let's draw arcs that will illustrate the remaining inverse functions for us. In the second graph they are marked with blue lines. Take a look at how you can graphically display the concepts a r c sin , a r c cos , a r c t g , a r c c t g for an arbitrary number a (in the above ranges):

Conclusion: what are arc functions

As a result, we can formulate the following: for any number a a ∈ [ − 1, 1 ] we can calculate the angles - arcsine and arccosine, and for each real number - the angles arctangent and arccotangent. This point of view allows you to compare the numerical value of the argument and the specific angle, which is the value of the function.

We can look at the concepts a r c sin , a r c cos , a r c t g and a r c c t g as numbers and as angles. If we take them as numbers, then they are numerical functions: each value of a corresponds to a number.

Let's summarize: all these four concepts are inverse trigonometric functions. The name is clear: arcsine is opposed to sine, arccosine is opposed to cosine, arctangent is opposed to tangent, arccotangent is opposed to cotangent. Therefore, another common collective name for them is arc functions.

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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 functions

Sum 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 formulas

Expressions 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 .

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

See also: