Extracurricular lesson - arcsine

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

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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

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

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

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

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

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

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

What does the expression mean

arcsin 0.4?

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

And that's it.

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

arc sin 0,4
corner, whose sine equals 0.4

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

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

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

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

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

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

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

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

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

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

For example: what is arcsin 0.5?

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

arcsin 0.5 = 30°

Or, more solidly, in terms of radians:

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

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

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

It is enough to consider that:

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

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

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

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

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

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

Inside the arccosine on the right already positive meaning. What

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

That's all.

Restrictions on arcsine, arccosine, arctangent, arccotangent.

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

All of these examples, from 1st to 9th, are carefully sorted out on the shelves in Section 555. What, how and why. With all the secret traps and tricks. Plus ways to dramatically simplify the solution. By the way, this section contains a lot of useful information and practical 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.

This article discusses the issues of finding the values ​​of the arcsine, arccosine, arctangent and arccotangent of a given number. To begin with, the concepts of arcsine, arccosine, arctangent and arccotangent are introduced. We consider their main values, according to the tables, including Bradis, finding these functions.

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Values ​​for arcsine, arccosine, arctangent, and arccotangent

It is necessary to understand the concepts of "the values ​​of the arcsine, arccosine, arctangent, arccotangent".

The definitions of the arcsine, arccosine, arctangent and arccotangent of a number will help you understand the calculation of given functions. The value of the trigonometric functions of the angle is equal to the number a, then it is automatically considered the value of this angle. If a is a number, then this is the value of the function.

For a clear understanding, let's look at an example.

If we have the arc cosine of an angle equal to π 3, then the value of the cosine from here is 1 2 according to the table of cosines. This angle is in the range from zero to pi, which means that the value of the arc cosine 1 2 will be π by 3. Such a trigonometric expression is written as a r cos (1 2) = π 3 .

Angle can be either degrees or radians. The value of the angle π 3 is equal to an angle of 60 degrees (detailed in the topic converting degrees to radians and vice versa). This example with the arc cosine 1 2 has a value of 60 degrees. Such a trigonometric notation has the form a r c cos 1 2 = 60 °

Basic values ​​of arcsin, arccos, arctg and arctg

Thanks to table of sines, cosines, tangents and cotangents, we have exact angle values ​​at 0 , ± 30 , ± 45 , ± 60 , ± 90 , ± 120 , ± 135 , ± 150 , ± 180 degrees. The table is quite convenient and from it you can get some values ​​for the arc functions, which are called the basic values ​​of the arc sine, arc cosine, arc tangent and arc tangent.

The table of sines of the main angles offers the following results of the angle values:

sin (- π 2) \u003d - 1, sin (- π 3) \u003d - 3 2, sin (- π 4) \u003d - 2 2, sin (- π 6) \u003d - 1 2, sin 0 \u003d 0, sin π 6 \u003d 1 2, sin π 4 \u003d 2 2, sin π 3 \u003d 3 2, sin π 2 \u003d 1

Given them, one can easily calculate the arcsine of the number of all standard values, starting from - 1 and ending with 1, also values ​​​​from - π 2 to + π 2 radians, following its basic definition value. This is the main values ​​of the arcsine.

For convenient use of the values ​​of the arcsine, we will enter it in the table. Over time, you will have to learn these values, since in practice you often have to refer to them. Below is a table of the arcsine with radian and degree angles.

To obtain the basic values ​​of the arccosine, you must refer to the table of cosines of the main angles. Then we have:

cos 0 = 1 , cos π 6 = 3 2 , cos π 4 = 2 2 , cos π 3 = 1 2 , cos π 2 = 0 , cos 2 π 3 = - 1 2 , cos 3 π 4 = - 2 2 , cos 5 π 6 = - 3 2 , cos π = - 1

Following from the table, we find the values ​​of the arc cosine:

a r c cos (- 1) = π , arccos (- 3 2) = 5 π 6 , arcocos (- 2 2) = 3 π 4 , arccos - 1 2 = 2 π 3 , arccos 0 = π 2 , arccos 1 2 = π 3 , arccos 2 2 = π 4 , arccos 3 2 = π 6 , arccos 1 = 0

Arc cosine table.

In the same way, based on the definition and standard tables, the values ​​of arc tangent and arc tangent are found, which are shown in the table of arc tangents and arc tangents below.

a r c sin , a r c cos , a r c t g and a r c c t g

For the exact value of a r c sin, a r c cos, a r c t g and a r c c t g of the number a, you need to know the value of the angle. This was mentioned in the previous paragraph. However, we do not know the exact value of the function. If it is necessary to find a numerical approximate value of arc functions, apply t table of sines, cosines, tangents and cotangents of Bradys.

Such a table allows you to perform fairly accurate calculations, since the values ​​\u200b\u200bare given with four decimal places. Thanks to this, the numbers come out accurate to the minute. The values ​​of a r c sin , a r c cos , a r c t g and a r c c t g of negative and positive numbers is reduced to finding formulas a r c sin , a r c cos , a r c t g and a r c c t g of opposite numbers of the form a r c sin (- α) = - a r c sin α , a r c cos (- α) = π - a r c cos α , a r c t g (- α) = - a r c t g α , a r c c t g (- α) = π - a r c c t g α .

Consider the solution of finding the values ​​a r c sin , a r c cos , a r c t g and a r c c t g using the Bradis table.

If we need to find the value of the arcsine 0 , 2857 , we are looking for the value by finding the table of sines. We see that this number corresponds to the value of the angle sin 16 degrees and 36 minutes. This means that the arcsine of the number 0, 2857 is the desired angle of 16 degrees and 36 minutes. Consider the figure below.

To the right of degrees there are columns called corrections. With the desired arcsine of 0.2863, the same amendment of 0.0006 is used, since the nearest number will be 0.2857. So, we get a sine of 16 degrees 38 minutes and 2 minutes, thanks to the correction. Let's consider a drawing depicting the Bradys table.

There are situations when the desired number is not in the table and even with amendments it cannot be found, then the two closest values ​​\u200b\u200bof the sines are found. If the desired number is 0.2861573, then the numbers 0.2860 and 0.2863 are its closest values. These numbers correspond to the values ​​of the sine of 16 degrees 37 minutes and 16 degrees and 38 minutes. Then the approximate value of this number can be determined to the nearest minute.

Thus, the values ​​a r c sin , a r c cos , a r c t g and a r c c t g are found.

To find the arcsine through the known arccosine of a given number, you need to apply the trigonometric formulas a r c sin α + a r c cos α \u003d π 2, a r c t g α + a r c c t g α \u003d π 2 (you need to look at topic of sum formulassarccosine and arcsine, the sum of the arctangent and arccotangent).

With known a r c sin α \u003d - π 12, it is necessary to find the value a r c cos α, then it is necessary to calculate the arc cosine using the formula:

a r c cos α = π 2 − a r c sin α = π 2 − (− π 12) = 7 π 12 .

If you need to find the value of the arctangent or arccotangent of a number a using the known arcsine or arccosine, you need to make long calculations, since there are no standard formulas. Let's look at an example.

If the arccosine of the number a is given and equal to π 10, and the table of tangents will help to calculate the arctangent of this number. The angle π 10 radians is 18 degrees, then from the table of cosines we see that the cosine of 18 degrees has a value of 0, 9511, after which we look into the Bradis table.

When looking for the value of the arc tangent 0, 9511, we determine that the value of the angle is 43 degrees and 34 minutes. Let's look at the table below.

In fact, the Bradis table helps in finding the required angle value and, given the angle value, allows you to determine the number of degrees.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Arcsine translated from Latin means arc and sine. This is the reverse function.

In other words:

Explanatory example:
Let's find arcsin 1/2.

Solution .
The expression arcsin 1/2 shows that the sine of angle t is 1/2 (sin t = 1/2).

point 1/2 located on the axis at, corresponds to the point π/6 on the number circle.
So arcsin 1/2 = π/6.

Note:

if sin π/6 = 1/2, then arcsin 1/2 = π/6.

That is, in the first case, by a point on the numerical circle, we find the value of the sine, and in the second, on the contrary, by the value of the sine, we find a point on the numerical circle. Movement in the opposite direction. This is the arcsine.

Formulas.

(2)

√2
Example 1 : Calculate arcsin (- --).
2

Solution .

When solving an example, we literally follow the table above our example.

√2
a = - --.
2

√2
Then sin t = – –, while t is included in the segment [–π/2; π/2]
2

π
So t = – -- (included in the segment [–π/2; π/2])
4

√2π
Answer: arcsin (- --) = - -
2 4

We focus your attention: the sine of the number –π/4 is -√2/2, and the arcsine of -√2/2 is –π/4. Reverse movement. The sine of a number is a point on the coordinate axis, and the arcsine is a point on the number circle.

√3
Example 2 : Calculate arcsin --
2

Solution .

√3
Let arcsin -- = t.
2

√3
Then sint = --.
2

The point t is in the segment [–π/2; π/2]. We calculate the value of t.

√3
The number -- corresponds to the value of sin π/3, while π/3 is in the segment [–π/2; π/2].
2

Outcome:

√3
arcsin --= π/3.
2

This article is about finding the values ​​of the arcsine, arccosine, arctangent and arccotangent given number. First, we will clarify what is called the value of the arcsine, arccosine, arctangent and arccotangent. Next, we get the main values ​​​​of these arc functions, after which we will figure out how the values ​​\u200b\u200bof the arc sine, arc cosine, arc tangent and arc tangent are found from the tables of sines, cosines, tangents and cotangents of Bradys. Finally, let's talk about finding the arcsine of a number when the arccosine, arctangent or arccotangent of this number is known, etc.

Page navigation.

Values ​​for arcsine, arccosine, arctangent, and arccotangent

First, you need to figure out what is value of arcsine, arccosine, arctangent and arccotangent».

Tables of sines and cosines, as well as tangents and cotangents of Bradys, allow you to find the value of the arcsine, arccosine, arctangent and arccotangent of a positive number in degrees with an accuracy of one minute. It is worth mentioning here that finding the values ​​of the arcsine, arccosine, arctangent and arccotangent of negative numbers can be reduced to finding the values ​​of the corresponding arcfunctions of positive numbers by referring to the formulas arcsin, arccos, arctg and arcctg of opposite numbers of the form arcsin(−a)=−arcsin a , arccos (−a)=π−arccos a , arctg(−a)=−arctg a , and arcctg(−a)=π−arcctg a .

Let's deal with finding the values ​​of the arcsine, arccosine, arctangent and arccotangent using the Bradis tables. We will do this with examples.

Suppose we need to find the value of the arcsine 0.2857. We find this value in the table of sines (cases when this value is not in the table, we will analyze below). It corresponds to the sine of 16 degrees 36 minutes. Therefore, the desired value of the arcsine of the number 0.2857 is an angle of 16 degrees 36 minutes.

Often it is necessary to take into account the corrections from the three columns on the right of the table. For example, if we need to find the arcsine of 0.2863. According to the table of sines, this value is obtained as 0.2857 plus a correction of 0.0006, that is, the value of 0.2863 corresponds to a sine of 16 degrees 38 minutes (16 degrees 36 minutes plus 2 minutes of correction).

If the number whose arcsine is of interest to us is not in the table and cannot even be obtained, taking into account the corrections, then in the table you need to find the two values ​​\u200b\u200bof the sines closest to it, between which this number is enclosed. For example, we are looking for the value of the arcsine of the number 0.2861573 . This number is not in the table; with the help of amendments, this number cannot be obtained either. Then we find the two closest values ​​\u200b\u200bof 0.2860 and 0.2863, between which the original number is enclosed, these numbers correspond to the sines of 16 degrees 37 minutes and 16 degrees 38 minutes. The desired value of the arcsine 0.2861573 lies between them, that is, any of these angle values ​​can be taken as an approximate value of the arcsine with an accuracy of 1 minute.

The values ​​of the arc cosine, and the values ​​of the arc tangent and the values ​​of the arc cotangent are absolutely similar (in this case, of course, tables of cosines, tangents and cotangents are used, respectively).

Finding the value of arcsin through arccos, arctg, arcctg, etc.

For example, let's say we know that arcsin a=−π/12 , but we need to find the value of arccos a . We calculate the value of the arccosine we need: arccos a=π/2−arcsin a=π/2−(−π/12)=7π/12.

The situation is much more interesting when, from the known value of the arcsine or arccosine of the number a, it is required to find the value of the arctangent or arccotangent of this number a, or vice versa. Unfortunately, we do not know the formulas that define such relationships. How to be? Let's deal with this with an example.

Let us know that the arc cosine of the number a is equal to π / 10, and we need to calculate the value of the arc tangent of this number a. You can solve the problem as follows: find the number a from the known value of the arc cosine, and then find the arc tangent of this number. To do this, we first need a table of cosines, and then a table of tangents.

The angle π / 10 radians is an angle of 18 degrees, according to the table of cosines we find that the cosine of 18 degrees is approximately equal to 0.9511, then the number a in our example is 0.9511.

It remains to turn to the table of tangents, and with its help find the value of the arc tangent we need 0.9511, it is approximately equal to 43 degrees 34 minutes.

This topic is logically continued by the material of the article evaluate expressions containing arcsin, arccos, arctg, and arcctg.

Bibliography.

• Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
• Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
• Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
• I. V. Boikov, L. D. Romanova. Collection of tasks for preparing for the exam, part 1, Penza 2003.
• Bradis V. M. Four-digit mathematical tables: For general education. textbook establishments. - 2nd ed. - M.: Bustard, 1999.- 96 p.: ill. ISBN 5-7107-2667-2