Inverse trigonometric functions. you can get acquainted with functions and derivatives
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 satisfying 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 interval [- 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). Highest value it takes 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 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 the drawing of a unit circle, which graphically displays the values 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. Otherwise, however, some elementary type USE assignments can confuse students.
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 under test conditions, the time spent on drawing for a simple task required for 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.
Here is a brief comparative analysis tg x and arctg x as 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.
- Region allowed 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 divided into four groups: calculate numerical value a specific expression, build a graph of this 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 function graphs, the main thing is the knowledge of their properties and appearance crooked. For solutions trigonometric equations and inequalities tables of identities are needed. 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 quadratic equation with roots x1 = 1 and x2 = - 1/a. With a = 0, x will be equal to 1.
Assignments related to inverse trigonometric functions are often offered at school final exams and on 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: to 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 of 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 (arctgx) = .
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) =
4)
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
Definition and notation
Arcsine (y = arcsin x) is the inverse function of the sine (x = siny -1 ≤ x ≤ 1 and the set of values -π /2 ≤ y ≤ π/2.sin(arcsin x) = x ;
arcsin(sin x) = x .
The arcsine is sometimes referred to as:
.
Graph of the arcsine function
Graph of the function y = arcsin x
The arcsine plot is obtained from the sine plot by interchanging the abscissa and ordinate axes. To eliminate the ambiguity, the range of values is limited to the interval on which the function is monotonic. This definition is called the main value of the arcsine.
Arccosine, arccos
Definition and notation
Arc cosine (y = arccos x) is the inverse of the cosine (x = cos y). It has scope -1 ≤ x ≤ 1 and many values 0 ≤ y ≤ π.cos(arccos x) = x ;
arccos(cos x) = x .
The arccosine is sometimes referred to as:
.
Graph of the arccosine function
Graph of the function y = arccos x
The arccosine plot is obtained from the cosine plot by interchanging the abscissa and ordinate axes. To eliminate the ambiguity, the range of values is limited to the interval on which the function is monotonic. This definition is called the main value of the arc cosine.
Parity
The arcsine function is odd:
arcsin(-x) = arcsin(-sin arcsin x) = arcsin(sin(-arcsin x)) = - arcsin x
The arccosine 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 arcsine and arccosine functions are continuous on their domain of definition (see the proof of continuity). The main properties of the 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 | increases monotonically | decreases monotonically |
Maximums | ||
Lows | ||
Zeros, y= 0 | x= 0 | x= 1 |
Points of intersection with the y-axis, x = 0 | y= 0 | y = π/ 2 |
Table of arcsines and arccosines
This table shows the values of arcsines and arccosines, in degrees and radians, for some values of the argument.
x | arcsin x | arccos x | ||
deg. | glad. | deg. | glad. | |
- 1 | - 90° | - | 180° | π |
- | - 60° | - | 150° | |
- | - 45° | - | 135° | |
- | - 30° | - | 120° | |
0 | 0° | 0 | 90° | |
30° | 60° | |||
45° | 45° | |||
60° | 30° | |||
1 | 90° | 0° | 0 |
≈ 0,7071067811865476
≈ 0,8660254037844386
Formulas
See also: Derivation of formulas for inverse trigonometric functionsSum and difference formulas
at or
at and
at and
at or
at and
at and
at
at
at
at
Expressions in terms of logarithm, complex numbers
See also: Derivation of formulasExpressions in terms of hyperbolic functions
Derivatives
;
.
See Derivation of arcsine and arccosine derivatives > > >
Derivatives of higher orders:
,
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 a substitution x = sin t. We integrate by parts, taking into account that -π/ 2 ≤ t ≤ π/2,
cos t ≥ 0:
.
We express the arccosine in terms of the arcsine:
.
Expansion in series
For |x|< 1
the following decomposition takes place:
;
.
Inverse functions
The inverses of the arcsine and arccosine are sine and cosine, respectively.
The following formulas are valid throughout the domain of definition:
sin(arcsin x) = x
cos(arccos x) = x .
The following formulas are valid only on the set of values of the arcsine and arccosine:
arcsin(sin x) = x at
arccos(cos x) = x at .
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.
Inverse trigonometric functions are arcsine, arccosine, arctangent and arccotangent.
Let's give definitions first.
arcsine Or, we can say that this is such an angle belonging to the 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
Arc tangent number a is called 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 the number b to the 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 the logarithm turned out to be necessary in order to write solutions, for example, to such an equation: The solution to this equation is an irrational number. This is the exponent to which 2 must be raised to get 7.
It's the same with trigonometric equations. For example, we want to solve the equation
It is clear that its solutions correspond to points on the trigonometric circle, the ordinate of which is equal to And it is clear that this is not a 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 the arcsine.
The angle belonging to the segment whose sine is equal is the arcsine of one fourth. And 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
More about solving trigonometric equations -.
It remains to be clarified - why is it indicated in the definition of the arcsine that this is an angle belonging to the segment?
The fact is that there are infinitely many angles whose sine is, 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 value of the sine, 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 the segment you can define a function that takes values from to
Let's repeat the definition again:
The arcsine of a is the number , such that
Designation: The area of definition of the arcsine is a segment. The range of values is a segment.
You can remember the phrase "arxins live on the right." We only do not forget that not just on the right, but also on the segment .
We are ready to graph the function
As usual, we mark the x-values on the horizontal axis and the y-values on the vertical axis.
Since , therefore, x lies between -1 and 1.
Hence, the domain of the function y = arcsin x is the segment
We said that y belongs to the segment . This means that the range of the function y = arcsin x is the segment .
Note that the graph of the function y=arcsinx is all placed in the area bounded by lines and
As always when plotting an unfamiliar function, let's start with a table.
By definition, the arcsine of zero is a number from the segment whose sine is zero. What is this number? - It is clear that this is zero.
Similarly, the arcsine of one is the number from the segment whose sine is equal to one. Obviously this
We continue: - this is a number from the segment, the sine of which is equal to. Yes it
0 | |||||
0 |
We build a function graph
Function Properties
1. Domain of definition
2. Range of values
3. , that is, this function is odd. Its graph is symmetrical with respect to the origin.
4. The function is monotonically increasing. Its smallest value, equal to - , is achieved at , and its largest value, equal to , at
5. What do graphs of functions and have in common? Don't you think that they are "made according to the same pattern" - just like the right branch of the function and the graph of the function, or like the graphs of the exponential and logarithmic functions?
Imagine that we cut out a small fragment from to from an ordinary sine wave, and then turned it vertically - and we get the arcsine graph.
The fact that for the 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 for and , and the 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 the segment we need is one on which each value of the angle corresponds to its own cosine value, and knowing the cosine, we can uniquely find the angle. We need a cut
The arc cosine of a is the number , such that
It is easy to remember: “arc cosines live from above”, and not just from above, but on a segment
Designation: Area of definition of the arc cosine - segment Range of values - segment
Obviously, the segment is 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
The arccosine is neither an even nor an odd function. Instead, we can use the following obvious relation:
Let's plot the function
We need a part of the function where it is monotonic, that is, it takes each of its values exactly once.
Let's choose a segment. On this segment, the function monotonically decreases, that is, the correspondence between the sets and is one-to-one. Each x value has its own y value. On this segment, there is a function inverse to the cosine, that is, the function y \u003d arccosx.
Fill in the table using the definition of the arc cosine.
The arccosine of the number x belonging to the interval will be such a number y belonging to the interval that
So, because ;
Because ;
Because ,
Because ,
0 | |||||
0 |
Here is the plot of the arccosine:
Function Properties
1. Domain of definition
2. Range of values
This is a generic function - it is neither even nor odd.
4. The function is strictly decreasing. The function y \u003d arccosx takes the largest value, equal to , at , and the smallest value, equal to zero, takes at
5. The functions and are mutually inverse.
The next ones are arctangent and arccotangent.
The arc tangent of a is the number , such that
Designation: . The area of definition of the arc tangent is the interval. The range of values is the interval.
Why are the ends of the interval - points excluded in the definition of the arc tangent? 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 plot the arc tangent. According to the definition, the arc tangent of a number x is a number y belonging to the 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 section of the function graph, 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 , the domain of definition will be the entire number line, from to and the range of values is the interval
Means,
Means,
Means,
But what happens if x is infinitely large? 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 value of the tangent tend to infinity? - Obviously, this
So, for infinitely large values of x, the plot of the arc tangent approaches the horizontal asymptote
Similarly, as x tends to minus infinity, the plot of the arc tangent approaches the horizontal asymptote
In the figure - a graph of the function
Function Properties
1. Domain of definition
2. Range of values
3. The function is odd.
4. The function is strictly increasing.
6. The functions and are mutually inverse - of course, when the function is considered on the interval
Similarly, we define the function of the arc cotangent and plot its graph.
The arc tangent of a is the number , such that
Function Graph:
Function Properties
1. Domain of definition
2. Range of values
3. The function is of a general form, that is, neither even nor odd.
4. The function is strictly decreasing.
5. Direct and - horizontal asymptotes of the given function.
6. Functions and are mutually inverse if considered on the interval