Lesson and presentation on the topics: "Arxine. Arcsine table. Formula y=arcsin(x)"

Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions! All materials are checked by an antivirus program.

Manuals and simulators in the online store "Integral" for grade 10 from 1C
Software environment "1C: Mathematical constructor 6.1"
We solve problems in geometry. Interactive tasks for building in space

What will we study:
1. What is the arcsine?
2. Designation of the arcsine.
3. A bit of history.
4. Definition.

6. Examples.

What is arcsine?

Guys, we have already learned how to solve equations for cosine, now let's learn how to solve similar equations for sine. Consider sin(x)= √3/2. To solve this equation, you need to build a straight line y= √3/2 and see: at what points does it intersect the number circle. It can be seen that the line intersects the circle at two points F and G. These points will be the solution to our equation. Rename F as x1 and G as x2. We have already found the solution to this equation and obtained: x1= π/3 + 2πk,
and x2= 2π/3 + 2πk.

Solving this equation is quite simple, but how to solve, for example, the equation
sin(x)=5/6. Obviously, this equation will also have two roots, but what values ​​will correspond to the solution on the number circle? Let's take a closer look at our sin(x)=5/6 equation.
The solution to our equation will be two points: F= x1 + 2πk and G= x2 ​​+ 2πk,
where x1 is the length of the arc AF, x2 is the length of the arc AG.
Note: x2= π - x1, because AF= AC - FC, but FC= AG, AF= AC - AG= π - x1.
But what are these dots?

Faced with a similar situation, mathematicians came up with a new symbol - arcsin (x). It reads like an arcsine.

Then the solution of our equation will be written as follows: x1= arcsin(5/6), x2= π -arcsin(5/6).

And the general solution: x= arcsin(5/6) + 2πk and x= π - arcsin(5/6) + 2πk.
The arcsine is the angle (arc length AF, AG) sine, which is equal to 5/6.

A bit of arcsine history

The history of the origin of our symbol is exactly the same as that of arccos. For the first time, the arcsin symbol appears in the works of the mathematician Scherfer and the famous French scientist J.L. Lagrange. Somewhat earlier, the concept of arcsine was considered by D. Bernuli, though he wrote it down with other symbols.

These symbols became generally accepted only at the end of the 18th century. The prefix "arc" comes from the Latin "arcus" (bow, arc). This is quite consistent with the meaning of the concept: arcsin x is an angle (or you can say an arc), the sine of which is equal to x.

Definition of arcsine

If |а|≤ 1, then arcsin(a) is such a number from the interval [- π/2; π/2], whose sine is a.

If |a|≤ 1, then the equation sin(x)= a has a solution: x= arcsin(a) + 2πk and
x= π - arcsin(a) + 2πk

Let's rewrite:

x= π - arcsin(a) + 2πk = -arcsin(a) + π(1 + 2k).

Guys, look carefully at our two solutions. What do you think: can they be written in a general formula? Note that if there is a plus sign before the arcsine, then π is multiplied by an even number 2πk, and if the sign is minus, then the multiplier is odd 2k+1.
With this in mind, we write the general solution formula for the equation sin(x)=a:

There are three cases in which one prefers to write solutions in a simpler way:

sin(x)=0, then x= πk,

sin(x)=1, then x= π/2 + 2πk,

sin(x)=-1, then x= -π/2 + 2πk.

For any -1 ≤ a ≤ 1, the following equality holds: arcsin(-a)=-arcsin(a).

Let's write a table of cosine values ​​in reverse and get a table for the arcsine.

Examples

1. Calculate: arcsin(√3/2).
Solution: Let arcsin(√3/2)= x, then sin(x)= √3/2. By definition: - π/2 ≤x≤ π/2. Let's look at the values ​​of the sine in the table: x= π/3, because sin(π/3)= √3/2 and –π/2 ≤ π/3 ≤ π/2.
Answer: arcsin(√3/2)= π/3.

2. Calculate: arcsin(-1/2).
Solution: Let arcsin(-1/2)= x, then sin(x)= -1/2. By definition: - π/2 ≤x≤ π/2. Let's look at the values ​​of the sine in the table: x= -π/6, because sin(-π/6)= -1/2 and -π/2 ≤-π/6≤ π/2.
Answer: arcsin(-1/2)=-π/6.

3. Calculate: arcsin(0).
Solution: Let arcsin(0)= x, then sin(x)= 0. By definition: - π/2 ≤x≤ π/2. Let's look at the values ​​of the sine in the table: it means x = 0, because sin(0)= 0 and - π/2 ≤ 0 ≤ π/2. Answer: arcsin(0)=0.

4. Solve the equation: sin(x) = -√2/2.
x= arcsin(-√2/2) + 2πk and x= π - arcsin(-√2/2) + 2πk.
Let's look at the value in the table: arcsin (-√2/2)= -π/4.
Answer: x= -π/4 + 2πk and x= 5π/4 + 2πk.

5. Solve the equation: sin(x) = 0.
Solution: Let's use the definition, then the solution will be written in the form:
x= arcsin(0) + 2πk and x= π - arcsin(0) + 2πk. Let's look at the value in the table: arcsin(0)= 0.
Answer: x= 2πk and x= π + 2πk

6. Solve the equation: sin(x) = 3/5.
Solution: Let's use the definition, then the solution will be written in the form:
x= arcsin(3/5) + 2πk and x= π - arcsin(3/5) + 2πk.
Answer: x= (-1) n - arcsin(3/5) + πk.

7. Solve the inequality sin(x) Solution: The sine is the ordinate of the point of the numerical circle. So: we need to find such points, the ordinate of which is less than 0.7. Let's draw a straight line y=0.7. It intersects the number circle at two points. Inequality y Then the solution of the inequality will be: -π – arcsin(0.7) + 2πk

Problems on the arcsine for independent solution

1) Calculate: a) arcsin(√2/2), b) arcsin(1/2), c) arcsin(1), d) arcsin(-0.8).
2) Solve the equation: a) sin(x) = 1/2, b) sin(x) = 1, c) sin(x) = √3/2, d) sin(x) = 0.25,
e) sin(x) = -1.2.
3) Solve the inequality: a) sin (x)> 0.6, b) sin (x) ≤ 1/2.

A method for deriving formulas for inverse trigonometric functions is presented. Formulas for negative arguments, expressions relating the arcsine, arccosine, arctangent and arccotangent are obtained. A method for deriving formulas for the sum of arcsines, arccosines, arctangents and arccotangents is indicated.

Basic formulas

The derivation of formulas for inverse trigonometric functions is simple, but requires control over the values ​​of the arguments of direct functions. This is due to the fact that trigonometric functions are periodic and, therefore, their inverse functions are multivalued. Unless otherwise stated, inverse trigonometric functions mean their principal values. To determine the main value, the domain of definition of the trigonometric function is narrowed to the interval on which it is monotonic and continuous. The derivation of formulas for inverse trigonometric functions is based on the formulas of trigonometric functions and the properties of inverse functions as such. The properties of inverse functions can be divided into two groups.

The first group includes formulas that are valid throughout the entire domain of inverse functions:
sin(arcsin x) = x
cos(arccos x) = x
tg(arctg x) = x (-∞ < x < +∞ )
ctg(arctg x) = x (-∞ < x < +∞ )

The second group includes formulas that are valid only on the set of values ​​of inverse functions.
arcsin(sin x) = x at
arccos(cos x) = x at
arctg(tg x) = x at
arcctg(ctg x) = x at

If the variable x does not fall into the above interval, then it should be reduced to it using the formulas of trigonometric functions (hereinafter n is an integer):
sinx = sin(-x-π); sinx = sin(π-x); sinx = sin(x+2πn);
cos x = cos(-x); cosx = cos(2π-x); cosx = cos(x+2πn);
tgx = tg(x+πn); ctgx = ctg(x+πn)

For example, if it is known that
arcsin(sin x) = arcsin(sin( π - x )) = π - x .

It is easy to see that for π - x falls within the required interval. To do this, multiply by -1: and add π: or Everything is correct.

Inverse Functions of Negative Argument

Applying the above formulas and properties of trigonometric functions, we obtain formulas for the inverse functions of a negative argument.

arcsin(-x) = arcsin(-sin arcsin x) = arcsin(sin(-arcsin x)) = - arcsin x

Since then multiplying by -1 , we have: or
The sine argument falls within the allowable range of the arcsine range. Therefore the formula is correct.

Similarly for other functions.
arccos(-x) = arccos(-cos arccos x) = arccos(cos(π-arccos x)) = π - arccos x

arctan(-x) = arctg(-tg arctg x) = arctg(tg(-arctg x)) = - arctg x

arcctg(-x) = arcctg(-ctg arcctg x) = arcctg(ctg(π-arcctg x)) = π - arcctg x

Expression of the arcsine in terms of the arccosine and the arctangent in terms of the arccotangent

We express the arcsine in terms of the arccosine.

The formula is valid for These inequalities hold because

To verify this, we multiply the inequalities by -1 : and add π/2 : or Everything is correct.

Similarly, we express the arctangent through the arccotangent.

Expression of the arcsine through the arctangent, the arccosine through the arccotangent and vice versa

We proceed in a similar way.

Sum and difference formulas

In a similar way, we obtain the formula for the sum of arcsines.

Let us establish the limits of applicability of the formula. In order not to deal with cumbersome expressions, we introduce the notation: X = arcsin x, Y = arcsin y. The formula is applicable when
. Further, we note that, since arcsin(- x) = - arcsin x, arcsin(- y) = - arcsin y, then for different signs, x and y, X and Y also have different signs, and therefore the inequalities hold. The condition of different signs for x and y can be written with one inequality: . That is, when the formula is valid.

Now consider the case x > 0 and y > 0 , or X > 0 and Y > 0 . Then the condition for the applicability of the formula is the fulfillment of the inequality: . Since the cosine monotonically decreases for values ​​of the argument in the interval from 0 , to π , then we take the cosine of the left and right sides of this inequality and transform the expression:
;
;
;
.
Since and ; then the cosines included here are not negative. Both parts of the inequality are positive. We square them and convert the cosines through the sines:
;
.
Substitute sin X = sin arc sin x = x:
;
;
;
.

So, the resulting formula is valid for or .

Now consider the case x > 0, y > 0 and x 2 + y 2 > 1 . Here the sine argument takes the values: . It needs to be reduced to the interval of the arcsine value area:

So,

at i.

Replacing x and y with - x and - y , we have

at i.
We perform transformations:

at i.
Or

at i.

So, we got the following expressions for the sum of arcsines:

at or ;

for and ;

at and .

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
injection, whose sine equals 0.4

As it is written, so it is heard.) Almost. Prefix 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 tips on 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.

Definitions of inverse trigonometric functions and their graphs are given. As well as formulas relating inverse trigonometric functions, formulas for sums and differences.

Definition of inverse trigonometric functions

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. Thus, 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

y= arctg x

y= arcctg x

Basic formulas

Here, special attention should be paid to the intervals for which the formulas are valid.

arcsin(sin x) = x at
sin(arcsin x) = x
arccos(cos x) = x at
cos(arccos x) = x

arctg(tg x) = x at
tg(arctg x) = x
arcctg(ctg x) = x at
ctg(arctg x) = x

Formulas relating inverse trigonometric functions

Sum and difference formulas

at or

at and

at and

at or

at and

at and

at

at

at

at

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 ​​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:

1. The function is defined on the segment [-1; one].
2. ODZ for arccos - .
3. The graph is entirely located in the I and II quarters, and the function itself is neither even nor odd.
4. Y = 0 for x = 1.
5. 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 under test conditions, 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:

1. The graph is infinite and defined on the interval (- ∞; + ∞).
2. Arctangent is an odd function, therefore, arctan (- x) = - arctan x.
3. Y = 0 for x = 0.
4. 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:

1. The function definition interval is infinity.
2. The range of admissible values ​​is the interval (0; π).
3. F(x) is neither even nor odd.
4. 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.