In a number of problems of mathematics and its applications, it is required to find the corresponding value of the angle, expressed in degree or in radian measure, from the known value of the trigonometric function. It is known that an infinite set of angles corresponds to the same sine value, for example, if $ \\ sin α \u003d 1/2, $ then the angle $ α $ can be equal to both $ 30 ° $ and $ 150 °, $ or in radian measure $ π / 6 $ and $ 5π / 6, $ and any of the angles obtained from these by adding a term of the form $ 360 ° ⋅k, $ or, respectively, $ 2πk, $ where $ k $ is any integer. This becomes clear from considering the graph of the function $ y \u003d \\ sin x $ on the whole number line (see Fig. $ 1 $): if on the $ Oy $ axis we put off a segment of length $ 1/2 $ and draw a straight line parallel to the $ Ox axis, $ then it will intersect a sinusoid at an infinite number of points. In order to avoid the possible variety of answers, reverse trigonometric functions, otherwise called circular, or arc functions (from latin word arcus - "arc").
The four basic trigonometric functions $ \\ sin x, $ $ \\ cos x, $ $ \\ mathrm (tg) \\, x $ and $ \\ mathrm (ctg) \\, x $ correspond to four arc functions $ \\ arcsin x, $ $ \\ arccos x , $ $ \\ mathrm (arctg) \\, x $ and $ \\ mathrm (arcctg) \\, x $ (read: arcsine, arccosine, arctangent, arccotangent). Consider the functions \\ arcsin x and \\ mathrm (arctg) \\, x, since the other two are expressed in terms of them by the formulas:
$ \\ arccos x \u003d \\ frac (π) (2) - \\ arcsin x, $ $ \\ mathrm (arcctg) \\, x \u003d \\ frac (π) (2) - \\ mathrm (arctg) \\, x. $
Equality $ y \u003d \\ arcsin x $ by definition means such an angle $ y, $ expressed in radian measure and enclosed in the range from $ - \\ frac (π) (2) $ to $ \\ frac (π) (2), $ sine which is equal to $ x, $ that is, $ \\ sin y \u003d x. $ The function $ \\ arcsin x $ is a function, inverse function $ \\ sin x, $ considered on the segment $ \\ left [- \\ frac (π) (2), + \\ frac (π) (2) \\ right], $ where this function monotonically increases and takes all values \u200b\u200bfrom $ −1 $ to $ + 1. $ Obviously, the argument $ y $ of the function $ \\ arcsin x $ can take values \u200b\u200bonly from the segment $ \\ left [−1, + 1 \\ right]. $ So, the function $ y \u003d \\ arcsin x $ is defined on the segment $ \\ left [−1, + 1 \\ right], $ is monotonically increasing, and its values \u200b\u200bfill the segment $ \\ left [- \\ frac (π) (2), + \\ frac (π) (2) \\ $ 2. $
Under the condition $ −1 ≤ a ≤ 1 $ all solutions of the equation $ \\ sin x \u003d a $ can be represented as $ x \u003d (- 1) ^ n \\ arcsin a + πn, $ $ n \u003d 0, ± 1, ± 2, …. $ For example, if
$ \\ sin x \u003d \\ frac (\\ sqrt (2)) (2) $ then $ x \u003d (−1) ^ n \\ frac (π) (4) + πn, $ $ n \u003d 0, ± 1, ± 2 ,…. $
The relation $ y \u003d \\ mathrm (arcctg) \\, x $ is defined for all values \u200b\u200bof $ x $ and by definition means that the angle $ y, $ expressed in radian measure, is within the limits
$ - \\ frac (π) (2)
and the tangent of this angle is x, that is, $ \\ mathrm (tg) \\, y \u003d x. $ The function $ \\ mathrm (arctg) \\, x $ is defined on the whole number line, is a function inverse to the function $ \\ mathrm ( tg) \\, x $, which is considered only on the interval
$ - \\ frac (π) (2)
The function $ y \u003d \\ mathrm (arctg) \\, x $ is monotonically increasing, its graph is shown in Fig. $ 3. $
All solutions of the equation $ \\ mathrm (tg) \\, x \u003d a $ can be written as $ x \u003d \\ mathrm (arctg) \\, a + πn, $ $ n \u003d 0, ± 1, ± 2,…. $
Note that inverse trigonometric functions are widely used in mathematical analysis. For example, one of the first functions for which a representation of an infinite power series was obtained was the function $ \\ mathrm (arctg) \\, x. $ From this series, G. Leibniz, for a fixed value of the argument $ x \u003d 1 $, obtained the famous representation of the number to infinite nearby
Inverse trigonometric functions (circular functions, arc functions) - mathematical functions that are inverse to trigonometric functions.
They usually include 6 functions:
- arcsine (designation: arcsin x; arcsin x Is the angle sin which is x),
- arccosine (designation: arccos x; arccos x Is the angle whose cosine is x etc),
- arctangent (designation: arctg x or arctan x),
- arccotangent (designation: arcctg x or arccot \u200b\u200bx or arccotan x),
- arcsecant (designation: arcsec x),
- arcsecant (designation: arccosec x or arccsc x).
Arcsine (y \u003d arcsin x) is the inverse function to sin (x \u003d sin y 
... In other words, it returns the angle by its value sin.
Arccosine (y \u003d arccos x) is the inverse function to cos (x \u003d cos y cos.
Arctangent (y \u003d arctan x) is the inverse function to tg (x \u003d tg y), which has a domain and a set of values 
... In other words, it returns the angle by its value tg.
Arccotangent (y \u003d arcctg x) is the inverse function to ctg (x \u003d ctg y), which has a domain and many values. In other words, it returns the angle by its value ctg.
arcsec - arcsecant, returns the angle by the value of its secant.
arccosec - arcsecant, returns an angle by its cosecant value.
When the inverse trigonometric function is not defined at the specified point, then its value will not appear in the resulting table. Functions arcsec and arccosec are not defined on the segment (-1,1), but arcsin and arccos are determined only on the segment [-1,1].
The name of the inverse trigonometric function is derived 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 contracts this arc), which corresponds to one or another segment.
Sometimes in foreign literature, as in scientific / engineering calculators, they use notations like sin −1, cos −1 for arcsine, arccosine and the like, this is not considered completely accurate, since likely confusion with raising a function to a power −1 (« −1 »(Minus the first degree) defines the function x \u003d f -1 (y), the inverse of the function y \u003d 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 connecting inverse trigonometric functions.
We denote any of the values \u200b\u200bof the inverse trigonometric functions by Arcsin x, Arccos x, Arctan x, Arccot \u200b\u200bx and keep the notation: arcsin x, arcos x, arctan x, arccot \u200b\u200bx for their main meanings, then the relationship between them is expressed by such ratios.
What is arcsine, arccosine? What is arc tangent, arc cotangent?
Attention!
There are additional
materials in Special Section 555.
For those who are "not very ..."
And for those who "very much ...")
To concepts arcsine, arccosine, arctangent, arccotangent learning people are wary. He does not understand these terms and, therefore, does not trust this nice family.) But in vain. These are very simple concepts. Which, by the way, make life enormously easier knowledgeable person when solving trigonometric equations!
Doubt 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 tabular values \u200b\u200bfor some angles ... At least in the most general terms. Then there will be no problems here either.
So, we are surprised, but remember: arc sine, arc cosine, arc tangent and arc cotangent are just some angles.No more, no less. There is an angle, say 30 °. And there is an angle arcsin 0.4. Or arctg (-1.3). There are all kinds of angles.) You can simply write down the 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 expression mean
arcsin 0.4?
This is the angle whose sine is 0.4 ! Yes Yes. This is the meaning of the arcsine. I will specifically repeat: arcsin 0.4 is the angle whose sine is 0.4.
And that's all.
To keep this simple thought in my head for a long time, I will even give a breakdown of this terrible term - arcsine:
arc sin 0,4
angle, whose sine is equal to 0.4
As it is written, it is heard.) Almost. Prefix arc means arc (word arch know?) ancient people used arcs instead of angles, but this does not change the essence of the matter. Remember this elementary decoding of a mathematical term! Moreover, for the arccosine, arctangent and arccotangent, the decoding differs only in the name of the function.
What is arccos 0.8?
This is the angle whose cosine is 0.8.
What is arctg (-1,3)?
This is the angle whose tangent is -1.3.
What is arcctg 12?
This is an angle whose cotangent is 12.
By the way, such an elementary decoding allows avoiding epic blunders.) For example, the expression arccos1,8 looks quite solid. We start decoding: arccos1,8 is the angle whose cosine is 1.8 ... Dop-Dop !? 1.8 !? The cosine cannot be more than one !!!
Right. The arccos1,8 expression is meaningless. And writing such an expression in some answer will greatly amuse the examiner.)
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, arc tangents and arc cotangents are intended. Further, I will call this whole family diminutive - arches. To print less.)
Attention! Elementary verbal and conscious decoding arches allows you to calmly and confidently solve a variety of tasks. And in unusual tasks only she and saves.
Is it possible to switch from arches to ordinary degrees or radians? - I hear a cautious question.)
Why not!? Easy. And you can go there and back. Moreover, sometimes it must be done. Arches are a simple thing, but without them it's somehow calmer, right?)
For example: what is arcsin 0.5?
We recall the decryption: arcsin 0.5 is the angle whose sine is 0.5. Now we turn on the head (or Google)) and remember at what angle the sine is 0.5? The sine is 0.5 y an angle of 30 degrees... That's all there is to it: arcsin 0.5 is an angle of 30 °. You can safely write:
arcsin 0.5 \u003d 30 °
Or, more solidly, in radians:
That's it, you can forget about the arcsine and continue working with the usual degrees or radians.
If you realized what is arcsine, arccosine ... What is arctangent, arccotangent ... You can easily deal with such a monster, for example.)
An ignorant person will recoil in horror, yes ...) remember the decryption: arcsine is an angle whose sine ... And so on. If a knowledgeable person also knows the table of sines ... Table of cosines. The table of tangents and cotangents, then there are no problems at all!
It is enough to realize that:
I will decipher, i.e. I will translate the formula into words: angle whose tangent is 1 (arctg1) is an angle of 45 °. Or, which is one, Pi / 4. Similarly:
and that's it ... We replace all the arches with values \u200b\u200bin radians, everything will shrink, it remains to calculate how much 1 + 1 will be. This will be 2.) Which is the correct answer.
This is how you can (and should) go from arcsines, arccosines, arctangents and arc cotangents to ordinary degrees and radians. This simplifies scary examples a lot!
Often, in such examples, there are negative values. Like arctg (-1.3), or arccos (-0.8) ... that's not a problem. Here are some simple formulas for going from negative to positive values:
You need, say, to define the value of an expression:
This can be done by trigonometric circle decide, but you don't feel like drawing it. Well, okay. Moving from negative values \u200b\u200binside the inverse cosine k positive according to the second formula:
Inside the arccosine on the right already positive value. What
you just have to know. It remains to substitute radians for the arccosine 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 1 through 9 are carefully shelved in Section 555. What, How, and Why. With all the secret traps and tricks. Plus ways to drastically simplify the solution. By the way, this section contains many useful information and practical advice on trigonometry in general. And not just 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.
Inverse cosine function
The range of values \u200b\u200bof the function y \u003d cos x (see Fig. 2) is a segment. On a segment, the function is continuous and decreases monotonically.
Figure: 2
This means that a function is defined on the segment inverse to the function y \u003d cos x. This inverse function is called the inverse cosine and is denoted by y \u003d arccos x.
Definition
The arcosine of the number a, if | a | 1, is the angle whose cosine belongs to the segment; it is denoted by arccos a.
Thus, arccos a is an angle satisfying the following two conditions: cos (arccos a) \u003d a, | a | 1; 0? arccos a? p.
For example, arccos, since cos and; arccos since cosi.
The function y \u003d arccos x (Fig. 3) is defined on a segment, the range of its values \u200b\u200bis a segment. On the segment, the function y \u003d arccos x is continuous and monotonically decreases from p to 0 (since y \u003d cos x is a continuous and monotonically decreasing function on the segment); at the ends of the segment, it reaches its extreme values: arccos (-1) \u003d p, arccos 1 \u003d 0. Note that arccos 0 \u003d. The graph of the function y \u003d arccos x (see Fig. 3) is symmetric to the graph of the function y \u003d cos x relative to the straight line y \u003d x.
Figure: 3
Let us show that the equality arccos (-x) \u003d р-arccos x holds.
Indeed, by definition, 0? arcсos x? R. Multiplying by (-1) all parts of the last double inequality, we get - p? arcсos x? 0. Adding p to all parts of the last inequality, we find that 0? p-arccos x? R.
Thus, the values \u200b\u200bof the angles arccos (-x) and p - arccos x belong to the same segment. Since the cosine decreases monotonically on a segment, there cannot be two different angles on it with equal cosines. Find the cosines of the angles arccos (-x) and p-arccos x. By definition, cos (arccos x) \u003d - x, by the reduction formulas and by definition, we have: cos (p - - arccos x) \u003d - cos (arccos x) \u003d - x. So, the cosines of the angles are equal, which means that the angles themselves are also equal.
Inverse sine function
Consider a function y \u003d sin x (Fig. 6), which is increasing, continuous on the segment [-p / 2; p / 2] and takes values \u200b\u200bfrom the segment [-1; one]. Hence, on the segment [- p / 2; р / 2] a function is defined which is the inverse of the function y \u003d sin x.
Figure: 6
This inverse function is called the arcsine and is denoted y \u003d arcsin x. Let us introduce the definition of the arcsine of the number a.
The arcsine of a number a, if called an angle (or arc), the sine of which is equal to the number a and which belongs to the segment [-p / 2; p / 2]; it is denoted by arcsin a.
Thus, arcsin a is an angle satisfying the following conditions: sin (arcsin a) \u003d a, | a | ?one; -p / 2? arcsin huh? p / 2. For example, since sin and [- p / 2; p / 2]; arcsin, since sin \u003d and [- p / 2; p / 2].
The function y \u003d arcsin х (Fig. 7) is defined on the segment [- 1; 1], the range of its values \u200b\u200bis the segment [-p / 2; p / 2]. On the segment [- 1; 1] the function y \u003d arcsin x is continuous and monotonically increases from -p / 2 to p / 2 (this follows from the fact that the function y \u003d sin x on the interval [-p / 2; p / 2] is continuous and monotonically increasing). It takes the greatest value at x \u003d 1: arcsin 1 \u003d p / 2, and the smallest at x \u003d -1: arcsin (-1) \u003d -p / 2. For x \u003d 0, the function is zero: arcsin 0 \u003d 0.
Let us show that the function y \u003d arcsin x is odd, i.e. arcsin (-x) \u003d - arcsin x for any x [ - 1; 1].
Indeed, by definition, if | x | ? 1, we have: - р / 2? arcsin x? ? p / 2. Thus, the angles arcsin (-x) and - arcsin x belong to the same segment [ - p / 2; p / 2].
Find the sinuses of theseangles: sin (arcsin (-x)) \u003d - x (by definition); since the function y \u003d sin x is odd, then sin (-arcsin x) \u003d - sin (arcsin x) \u003d - x. So, the sines of the angles belonging to the same interval [-p / 2; р / 2], are equal, which means that the angles themselves are equal, that is, arcsin (-x) \u003d - arcsin x. Hence, the function y \u003d arcsin x is odd. The graph of the function y \u003d arcsin x is symmetrical about the origin.
Let us show that arcsin (sin x) \u003d x for any x [-p / 2; p / 2].
Indeed, by definition -p / 2? arcsin (sin x)? p / 2, and by condition -p / 2? x? p / 2. This means that the angles x and arcsin (sin x) belong to the same interval of monotonicity of the function y \u003d sin x. If the sines of such angles are equal, then the angles themselves are equal. Let's find the sines of these angles: for the angle x we \u200b\u200bhave sin x, for the angle arcsin (sin x) we have sin (arcsin (sin x)) \u003d sin x. We got that the sines of the angles are equal, therefore, the angles are equal, i.e. arcsin (sin x) \u003d x. ...
Figure: 7
Figure: 8
The graph of the function arcsin (sin | x |) is obtained by the usual transformations associated with the modulus from the graph y \u003d arcsin (sin x) (shown by the dashed line in Fig. 8). The desired graph y \u003d arcsin (sin | x- / 4 |) is obtained from it by shifting / 4 to the right along the abscissa axis (shown by the solid line in Fig. 8)
Inverse of the tangent function
The function y \u003d tg x on the interval takes all numerical values: E (tg x) \u003d. On this interval, it is continuous and increases monotonically. Hence, on the interval, a function is defined that is inverse to the function y \u003d tg x. This inverse function is called the arctangent and is denoted by y \u003d arctan x.
The arc tangent of the number a is the angle from the interval, the tangent of which is equal to a. Thus, arctan a is an angle satisfying the following conditions: tg (arctan a) \u003d a and 0? arctg a? R.
So, any number x always corresponds to a single value of the function y \u003d arctan x (Fig. 9).
Obviously, D (arctan x) \u003d, E (arctan x) \u003d.
The function y \u003d arctan x is increasing because the function y \u003d tan x is increasing in the interval. It is not hard to prove that arctg (-x) \u003d - arctgx, i.e. that the arctangent is an odd function.
Figure: 9
The graph of the function y \u003d arctan x is symmetric to the graph of the function y \u003d tg x relative to the straight line y \u003d x, the graph of y \u003d arctan x passes through the origin (because arctan 0 \u003d 0) and is symmetric about the origin (like the graph of an odd function).
One can prove that arctan (tg x) \u003d x if x.
Inverse cotangent function
The function y \u003d ctg x on the interval takes all numerical values \u200b\u200bfrom the interval. Its range of values \u200b\u200bcoincides with the set of all real numbers. In the interval, the function y \u003d ctg x is continuous and monotonically increasing. Hence, on this interval the function inverse to the function y \u003d ctg x is defined. The inverse function of the cotangent is called the arc cotangent and is denoted y \u003d arcctg x.
The arc cotangent of the number a is the angle belonging to the interval whose cotangent is equal to a.
Thus, arcctg a is an angle satisfying the following conditions: ctg (arcctg a) \u003d a and 0? arcctg a? R.
From the definition of the inverse function and the definition of the arctangent it follows that D (arcctg x) \u003d, E (arcctg x) \u003d. The arc cotangent is a decreasing function, since the function y \u003d ctg x decreases in the interval.
The graph of the function y \u003d arcctg x does not intersect the Ox axis, since y\u003e 0 R. At x \u003d 0 y \u003d arcctg 0 \u003d.
The graph of the function y \u003d arcctg x is shown in Figure 11.
Figure: 11
Note that for all real values \u200b\u200bof x the identity is true: arcctg (-x) \u003d p-arcctg x.
The sin, cos, tg, and ctg functions are always accompanied by inverse sine, inverse cosine, arctangent, and inverse cotangent. One is a consequence of the other, and function pairs are equally important for working with trigonometric expressions.
Consider a drawing of a unit circle, which graphically displays the values \u200b\u200bof trigonometric functions.
If you calculate arcs OA, arcos OC, arctg DE and arcctg MK, then they will all equal 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, you need to consider its function. Schedule has the form of an asymmetric curve passing through the center of coordinates.
Arcsine properties:
If you compare the graphs sin and arcsin, two trigonometric functions can have common patterns.
Arccosine
Arccos of the number a is the value of the angle α, the cosine of which is equal to a.
Curve y \u003d arcos x mirrors the arcsin x graph, with the only difference that it passes through the π / 2 point on the OY axis.
Let's consider the arccosine function in more detail:
- The function is defined on the segment [-1; one].
- ODZ for arccos -.
- The entire graph is located in the first and second quarters, and the function itself is neither even nor odd.
- Y \u003d 0 for x \u003d 1.
- The curve decreases along its entire length. Some properties of the inverse cosine are the same as the cosine function.
Some properties of the inverse cosine are the same as the cosine function.
Perhaps, schoolchildren will find such a "detailed" study of "arches" superfluous. However, otherwise, some elementary type uSE assignments can lead students to a dead end.
Exercise 1. Specify the functions shown in the figure.
Answer: fig. 1 - 4, Fig. 2 - 1.
IN this example the emphasis is on the little things. Usually students are very inattentive about the graphing and appearance of functions. Indeed, why memorize the type of a curve, if it can always be built from the calculated points. Do not forget that under test conditions, the time spent drawing for a simple task will be required to solve more complex tasks.
Arctangent
Arctg of the number a is such a value of the angle α that its tangent is equal to a.
If we consider the arctangent graph, the following properties can be distinguished:
- The graph is infinite and defined on the interval (- ∞; + ∞).
- The arctangent is an odd function, therefore arctan (- x) \u003d - arctan x.
- Y \u003d 0 at x \u003d 0.
- The curve rises over the entire definition area.
Here is a short comparative analysis tg x and arctg x as a table.
Arccotangent
Arcctg of 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 acceptable values \u200b\u200bis the interval (0; π).
- F (x) is neither even nor odd.
- The graph of the function decreases along its entire length.
It is very easy to compare ctg x and arctan x, you just need to draw two pictures and describe the behavior of the curves.
Task 2. Correlate the graph and the form of recording the function.
Logically, the graphs show that both functions are increasing. Therefore, both figures display some function arctg. It is known from the properties of the arctangent that y \u003d 0 for x \u003d 0,
Answer: fig. 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 make it possible to express, 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 a value for the sum of the arcsin and arcos values, and the arcctg and arcctg of the same angle.
Examples of problem solving
Trigonometry tasks can be roughly 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 an 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 to know 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.
Let's say in the exam you need to find an answer for an equation like:
If you convert the expression correctly and lead to the right kind, then solving it is very simple and quick. First, let's move arcsin x to the right side of the equality.
If you recall the formula arcsin (sin α) \u003d α, then the search for answers can be reduced to solving a system of two equations:
The restriction on the model x arose, again from the properties of arcsin: ODZ for x [-1; one]. For a ≠ 0, part of the system is a quadratic equation with roots x1 \u003d 1 and x2 \u003d - 1 / a. For a \u003d 0, x will be equal to 1.