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математика

тригонометрия

арктангенс

арккотангенс

егэ

огэ

00:00:02

trigonometric functions of the arctangent

00:00:04

arccotangent

00:00:05

and start with the arctangent, for starters I’ll argue

00:00:09

what a tangent is,

00:00:11

look here on the unit circle

00:00:13

I’ll mark several angles and on 3 m 4 pin 6

00:00:16

if we draw a ray from the origin

00:00:19

through these points then when they intersect with the

00:00:22

tangent axis, they will give the tangent of the

00:00:25

corresponding angles,

00:00:26

for example, if we draw a ray from the

00:00:29

origin through y Kolpino 3, then we will get

00:00:32

the values root of 3,

00:00:34

that is, the tangent of 3 by 3 is the root of 3

00:00:37

and so on for all angles, and now

00:00:40

suppose we want to perform then the inverse

00:00:42

operation, and knowing that the tangent is equal to the root of 3,

00:00:46

find the corresponding angle. In this case,

00:00:48

we must make the reverse movement

00:00:51

and get to the point foam 3. This is exactly the

00:00:55

operation that the arctangent function performs,

00:00:58

that is, for this level, the root of 3,

00:01:01

we get the angle pin 3

00:01:04

this can be written as the arctangent from the

00:01:07

root of 3 at stage 3 or the ideal

00:01:11

inverse side of the level root of 3

00:01:13

at 3 and we get the angle pin 6 which means the

00:01:18

arctangent from the root of 3 at stage 3 at

00:01:20

6 and so on you may ask

00:01:24

why we stop at these

00:01:26

first points why don’t we go further,

00:01:29

for example, if we continue, better from

00:01:32

point 1 we

00:01:33

first get foam point 4 and then

00:01:36

we continue it,

00:01:38

we can reach a point with an angle of 514

00:01:41

why do we take cinema 4 they are 5 foam 4

00:01:45

the remark is correct and she says that that

00:01:48

here there is uncertainty about what

00:01:51

angle to choose so that this does not happen and the

00:01:53

arctangent function is unambiguous;

00:01:56

the mathematicians agreed that for this

00:01:59

function we take angles only of the right

00:02:02

semicircle here, that

00:02:05

is, angles from the range from minus pin 2

00:02:08

to 2, that is, from this

00:02:12

range minus cinema 2

00:02:14

da pino 2 and the boundary value is

00:02:17

not included here because there is no

00:02:21

pin 2 and minus pin two days for angles,

00:02:24

so in general

00:02:27

the arctangent for any value a can be

00:02:30

written so this is the alpha angle

00:02:33

which is in the range from minus

00:02:35

pin 2 to pi on 2 do not include the boundary

00:02:39

value, the argument itself can

00:02:41

represent any real number,

00:02:44

that’s what arctangent is, by analogy, the

00:02:48

concept of cotangent is introduced, let’s look

00:02:51

at the unit circle again,

00:02:53

done how to plot with cotangent,

00:02:59

rays are drawn through the points of the corresponding angles, but already on the cotangent axis

00:03:03

and gets the corresponding value

00:03:06

for example when we pass through the point

00:03:09

pin 4 we get the value 1 this means

00:03:13

that the cotangent Atkin 4 is 1 so for

00:03:17

any angle for which there is a cotangent now we will

00:03:21

perform the reverse operation,

00:03:22

take the value of the cotangent root of 3

00:03:25

and going in the opposite direction

00:03:28

we get angles of 6 here reverse

00:03:33

movement

00:03:34

such a function is performed by the

00:03:36

cotangent function and in this case

00:03:39

the arccotangent of the root of 3 at stage 6

00:03:42

or for the level the root of 3 by 3

00:03:45

the arccotangent of the root of 3 by 3 we go in the

00:03:48

opposite direction, we get the

00:03:51

point foam 3 means

00:03:53

arcade ansat the root of 3 on 3 this is foam

00:03:56

three and so on for any angle you can do this

00:03:59

so you feel the arccotangent

00:04:00

that is the arc is not there this is the

00:04:03

inverse function of the tangent here you can also

00:04:05

notice that continue the ray you can

00:04:08

get several angles

00:04:10

for the same tangent value

00:04:12

to eliminate this uncertainty

00:04:14

mathematicians decided to limit themselves to the

00:04:16

upper block,

00:04:19

that is, the inverse tangent can be given a

00:04:22

value from the range from 0 to 3, from

00:04:25

this range from 0 to pi radians,

00:04:28

and the boundary value is not taken.

00:04:31

there is no for them does not exist and in the general

00:04:34

case we can write that for any

00:04:36

real number the inverse cotangent is the

00:04:39

angle alpha from the range from 0 to pi,

00:04:42

from this range this is what the inverse

00:04:45

cotangent function is in the

00:04:48

second part of our lesson, we will look at

00:04:50

several consequences from the functions from

00:04:52

cotangent and arctangent, namely such

00:04:55

equality, for example, tangent from

00:04:59

arctangent a is the level and why is this

00:05:03

so, let's look at the unit

00:05:06

semicircle with the axis of tangents,

00:05:09

suppose that in our case a is the

00:05:12

root of 3 and there a is the root of 3 then this

00:05:16

expression is the arctangent from the root of 3 and that

00:05:20

is, the corners on 3, here is the arctangent from the

00:05:24

root of 3, this is about Kolpino 3 and then

00:05:27

we take the tangent from this angle pin 3 and

00:05:30

get the level of the root of 3,

00:05:33

that is, we go back and get

00:05:36

the value of the root of 3,

00:05:37

so we got that the tangent under the

00:05:39

arctangent a this is, and that is, we

00:05:42

first went in one direction and got

00:05:45

3 from Kolpino and then went in the opposite

00:05:47

direction and again got the desired level,

00:05:49

but that’s why this equality is true for

00:05:52

any and the second expression is arctangent from

00:05:55

tangent alpha and that is, alpha again, let’s

00:05:58

take a certain angle and on 3 here it is the

00:06:02

angle pin 3 you first take the tangent from

00:06:05

this angle we get the level root of 3

00:06:08

so we go in this direction

00:06:10

then we take the arctangent from this level

00:06:13

root of 3 and we get the angle pin 3 that is

00:06:17

we go in the opposite direction so it

00:06:21

turns out that arctangent from alliance alpha

00:06:24

is alpha that is, you first went here and then

00:06:28

went in the opposite direction and returned to the

00:06:30

same corner, therefore this

00:06:32

equality is true. The following formula says that

00:06:35

the arctangent of minus and is minus

00:06:37

the arctangent and as an example,

00:06:40

consider the values of minus a by equal to

00:06:43

minus 1, this is the level

00:06:46

minus 1 for which the arc tangent is equal to

00:06:49

minus pin 4, here we move from this

00:06:53

level in the opposite direction and

00:06:56

get an angle of -14,

00:06:58

so the arc tangent from minus one is

00:07:01

minus foam 4 but since the arc tangent is from

00:07:05

one to foam 4 from the arctangent of

00:07:08

one at stage 4, then this equality

00:07:12

can be rewritten and so the arctangent of

00:07:15

minus one is minus the arctangent of

00:07:19

one, you see, but instead of pen 4

00:07:23

we substituted this expression, minus

00:07:25

naturally remains,

00:07:27

so it turns out that the arctangent of

00:07:29

minus one is minus the arctangent of

00:07:32

one and this is true for any

00:07:35

arctangent argument, that is, in the general

00:07:38

case, arctangent from minus and this is minus

00:07:40

arctangent from a similar expression

00:07:43

can be written or arccotangent,

00:07:45

namely attorneys at our cotangent and this

00:07:49

is and why is this so again, let's look

00:07:52

at the unit circle and assume

00:07:55

that our level

00:07:56

and this is the root of 3, first we take

00:07:58

the arc cotangent

00:07:59

from a that is, from the root of 3 we go get in the

00:08:02

opposite direction we get the angle foam

00:08:05

6 that is the arc cotangent from the root of 3

00:08:08

attacks by 6 and then we take the cotangent from

00:08:11

this angle api by 6 we go back and

00:08:13

we get the same level

00:08:15

and that is, katana tens by 60 root of

00:08:18

3 or chief level and therefore

00:08:21

this expression is valid for 2 equal to steel we

00:08:24

get the following let's say we take

00:08:26

point pin 6 here it is and calculate

00:08:30

the cotangent of this point that is we get

00:08:33

this level root of 3

00:08:34

and then the attachments are at 6 from the root of 3, then

00:08:38

we take the arccotangent from this level, we go

00:08:40

back, we get the angle again and at 6,

00:08:43

therefore it turns out that the arch dance is from the

00:08:46

cotangent alpha and that is, the angle alpha,

00:08:49

the following equality says that the arch

00:08:51

dance is from the crown of the stage minus the arccotangent

00:08:54

and why this is so, let’s assume that we

00:08:57

choose the level minus x is equal to minus

00:08:59

one, here it is, and from the figure you can see that

00:09:02

the inverse tangent

00:09:03

of minus one is trippin 4, so we

00:09:06

go in the opposite direction and

00:09:09

get this angle 3 by 4, but this

00:09:13

very angle trippin 4 can be written as a

00:09:15

peak minus 1 and 4 even this angle and

00:09:18

this pin 4 and drink minus this angle just

00:09:21

turns out to be 3 movie 4

00:09:23

and at the same time the angle of foam 4 is the

00:09:27

inverse cotangent

00:09:28

of unity, here it is, therefore we

00:09:32

can write this inverse cotangent instead of foam 4

00:09:35

and get

00:09:39

the value and minus the arc cotangent of one

00:09:42

and then all this can be rewritten so that the

00:09:45

arc cotangent of minus one this one

00:09:48

here is nothing more than minus

00:09:51

this arc cotangent of one and this

00:09:54

expression is valid for any

00:09:56

argument, that is, in the general case,

00:09:57

the arc cotangent of minus

00:09:59

stage minus arch dance but if all

00:10:03

this is clear then you now know

00:10:05

what arctangent arccotangent is and for

00:10:08

self-test complete the following task

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Понятия арктангенса и арккотангенса. Основные соотношения. Примеры.

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