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00:00:00

we continue to prepare trigonometry under the

00:00:03

fourth number and in this form we

00:00:05

will consider the use of the main 3

00:00:07

genetic identity and

00:00:09

double angle formulas and so the main this

00:00:12

exotic identity is perhaps the most

00:00:13

popular ratio in trigonometry, in

00:00:16

principle it is given to us even in the

00:00:18

reference material before the option, but

00:00:21

it is always very easy to recognize because that

00:00:24

our tasks will involve sums

00:00:27

for genetic functions, of course, in general,

00:00:29

this does not guarantee, but if we

00:00:32

are on games within the framework of the profile, we

00:00:35

can immediately state that the sum here, this, this,

00:00:37

this, every time will go

00:00:40

to one, well, this is still

00:00:44

unofficially the way to solve it, let's

00:00:46

show why it will

00:00:48

happen exactly like this every time, in fact, we

00:00:51

will now rely on the previous video where

00:00:53

we have already looked at the reduction formulas in detail,

00:00:55

we again notice that

00:00:57

here the angles are crap, which means we need to

00:00:59

either add them or subtract them, but if we are from

00:01:02

2 subtract 1, this will be exactly one hundred and

00:01:05

eighty, this apparently has a trick,

00:01:07

which means we can

00:01:11

imagine the second angle as the sum of

00:01:14

180

00:01:19

plus 27,

00:01:22

well, let’s follow our algorithm 180 plus,

00:01:25

let’s show for clarity that

00:01:28

we have 180

00:01:31

on the left side and add,

00:01:34

that is, we get into the third quarter the

00:01:37

third quarter, in principle, we are

00:01:39

not particularly interested in quarters here

00:01:42

because it will lose its sign, but our function

00:01:45

is in a square, that is, it is a cassano

00:01:46

square, as if from a minus, what to

00:01:49

make a plus out of a plus, so well, you can even

00:01:51

indicate that we have a bone from three quarters

00:01:54

negative, the most important thing is that

00:01:56

we don’t change the function and the minus, in fact, does

00:01:59

n’t interest us here because the cosine

00:02:01

s is squared, which means we

00:02:03

can use the reduction formulas to simply

00:02:06

remove this value 180, that is,

00:02:09

we will have a sine squared

00:02:12

and

00:02:14

27 plus a cosine squared from 27 we

00:02:19

see that it is worth the sum of

00:02:20

genetic functions from the same

00:02:22

argument, which means it will

00:02:24

always reduce to one, so the answer

00:02:27

will simply be the numerator 12, this can be

00:02:30

done very easily and from the remaining

00:02:33

examples the only one, only of course

00:02:35

you notice that here we have a feast cosine

00:02:37

cosine, well, that means, according to the reduction formulas,

00:02:40

one of the cosines turns into a sine

00:02:42

so that it’s still combined,

00:02:45

let’s make sure of this, of course, when we

00:02:49

are similar in defending the options, we

00:02:52

can immediately try to remember this

00:02:54

technique and

00:02:55

name only the numerator of the answer, but

00:02:59

of course, taking into account this minus of course, well,

00:03:01

again we see that the first cosine

00:03:04

can be left unchanged, but the

00:03:06

second can be represented as 90 plus 34,

00:03:10

because just 90 will change the function.

00:03:13

We’re not particularly interested in a quarter here

00:03:15

because the cosine squared can

00:03:18

be made from it plus, but below it will be

00:03:25

the main trigonometric identity is already danced out,

00:03:27

cosine squared theta 4 and already plus sine

00:03:31

squared of four,

00:03:33

this collapses into one and therefore

00:03:35

only minus 22 remains, that is, we

00:03:39

can immediately call -12, but in these

00:03:42

examples try to guess that we

00:03:45

can also notice here

00:03:47

that these two comrades

00:03:50

we have the

00:03:52

main identity, that is, they will turn into an

00:03:54

identity, then and then there was another one,

00:03:57

so the denominator will already be two

00:04:00

24 divided by two, this is two in the same way, and

00:04:04

here it will turn into an identity and also

00:04:07

plus one,

00:04:10

that is, we get the

00:04:12

answer minus 8

00:04:18

example 11 very important examples, in fact, are

00:04:22

also considered textbook ones when

00:04:25

the topic concerns trigonometry,

00:04:27

find the sine if given the cosine, of course,

00:04:31

this refers us exactly to who the

00:04:34

main formula from which is easy to

00:04:36

express either sine or cosine, just

00:04:39

pay attention to the plus minus,

00:04:41

this is due to the fact that we

00:04:43

still extract the root and sine

00:04:46

squared, and in order to accurately determine

00:04:49

which sign will take the final

00:04:51

answer for this, and

00:04:54

every time a quarter is indicated in this problem,

00:04:56

read here, here we have an angle from pi

00:05:00

to 2n, that is this is the lower 3rd and 4th quarter,

00:05:05

namely in these quarters we have the sine

00:05:08

that can be found takes

00:05:10

negative values, that is, I

00:05:12

understand for sure that the sine will be taken with a

00:05:15

minus sign, and under the root there will be 1

00:05:19

minus cosine squared, let's try to

00:05:21

learn how to immediately raise this number to

00:05:23

squared root of 7 squared is

00:05:26

simply 7 and below there will be 4, give 16,

00:05:30

let's bring to a common denominator an

00:05:32

additional resident of

00:05:34

1616 -79, that is, under the root 916 will be rejected,

00:05:38

note that if we work with

00:05:41

roots, the numbers must

00:05:43

be extracted, that is, the radical

00:05:46

expression

00:05:47

ideally represents there are

00:05:50

916 of them, that is, the root of it will be three-

00:05:52

quarters and this will be the

00:05:55

final answer, which is

00:05:58

superfluous; similar, we need to find the cosine,

00:06:01

which means we immediately understand that it will be

00:06:03

plus minus one minus sine square;

00:06:07

you can

00:06:09

look at it at first, but in essence it’s kind of

00:06:11

direct should

00:06:13

I think you carry out taxes from the same

00:06:15

Pythagorean theorem then we express, for

00:06:17

example, some kind of rolls through the other

00:06:20

side and the hypotenuse is similar, let's

00:06:23

note that here alpha belongs to the angle

00:06:25

from 0 to pi by 2, this is the first quarter

00:06:27

and in it all the functions are positive, that

00:06:30

is, space it will definitely already have a

00:06:32

plus sign, and here under the root

00:06:34

we will also square it and

00:06:37

already under the root there will be simply 916 of them,

00:06:42

which is three-quarters of them, only the answer

00:06:45

will already have a plus sign,

00:06:47

let’s take another example, find the sine, and

00:06:53

we again understand that it will be necessary under take the

00:06:56

root of this number from one, subtract

00:06:59

the square of this fraction, the root of all 1

00:07:03

squared is just c1 and below it will be 25,

00:07:07

let's determine what sign

00:07:09

our sine will have by duplicating 3 pin two is

00:07:14

here and 2pi from here you can

00:07:17

use

00:07:18

if you are on your own At the beginning, we tape a pair

00:07:21

with a tick to quickly get used to

00:07:22

these angles, these are 4 quarters and

00:07:26

the sine of 4 quarters has a minus sign, let me

00:07:30

remind you not of us, so we will definitely have an

00:07:33

answer with a minus, I’ll leave a minus in front of the

00:07:36

root, but under the root, what remains is an

00:07:38

additional factor of

00:07:42

2525 minus 121

00:07:46

it will be 4 25 because how ideally

00:07:50

the roots are extracted 4 and at the bottom 5 but this

00:07:54

will determine the

00:07:55

final answer - 0.4

00:07:59

variations of this problem also find these

00:08:03

sine values only a little bit give a

00:08:07

bad argument, that is, we

00:08:09

must first deal with it using the reduction formulas

00:08:13

let's let's try

00:08:18

to show some kind of thing where we

00:08:21

have 7 pin 27 pin let's easily

00:08:24

calculate directly,

00:08:27

namely, let's move to 2

00:08:31

2 pin 2 3 pin 2 4 5 6 7 that is, we

00:08:35

will be at the bottom and we will show here we have

00:08:40

7 pin 2 the angle is taken away from us, that is, we

00:08:44

go back, we end up

00:08:47

in the third quarter

00:08:50

in the 3rd quarter,

00:08:53

we will show the 3 quarters of us in it is negative,

00:08:56

well, we need to calculate the functions, which means we need to

00:08:58

change them to the opposite, that is, in

00:09:01

fact, we need to find the minus cosine alpha,

00:09:03

notice the minus comes forward and

00:09:06

we will change the functions, so we need to find the minus cosine

00:09:10

if we were given the sine, which quarter of us is from

00:09:14

pin 2 to pi,

00:09:16

so as not to get confused with these

00:09:18

minuses, we’ll just first find the cosine, even the

00:09:20

alpha, and then

00:09:23

we’ll just put a minus sign in front of it, so from pin 2

00:09:30

from pin 2 to pi pi by 2 here, this

00:09:35

is essentially

00:09:37

2 quarter and the

00:09:41

cosine of the second quarter has a minus sign,

00:09:44

so it will definitely be a minus, well, under the

00:09:48

root of one we subtract the sine

00:09:50

squared

00:09:51

0 8 squared is 064

00:09:55

because how ideally under the root it is formed

00:09:57

but 36 smoke from it will be 06 only

00:10:02

the cosine itself, then we will have a negative one, but

00:10:04

here we have the form of

00:10:06

conducting a minus, so a

00:10:08

minus by a minus will finally give 06

00:10:13

this is the catch with such tasks, it’s very

00:10:16

easy to make a mistake with these minuses,

00:10:18

so I recommend that you first

00:10:19

encourage just the function itself, but then

00:10:22

take into account the fact that there is a minus sign in front of it,

00:10:25

change it, let’s

00:10:27

do a similar task,

00:10:30

so first we’ll deal with this

00:10:32

argument,

00:10:39

some quarter you have

00:10:42

pi over 2 minus alpha pin, yes, this is us, the

00:10:45

angle is here,

00:10:49

subtract that is, we go back, alpha

00:10:51

we end up in the first quarter in the first

00:10:52

quarter, all functions have a + sign, but

00:10:56

the only thing is that since pin 2 means vum su,

00:10:57

let’s change the sine to a cosine, that is, in

00:11:00

fact, we need to go 8 cosine alpha,

00:11:03

but now let’s separately

00:11:06

find this cosine alpha

00:11:08

with a plus or minus and take it under

00:11:11

this root and let's look at

00:11:13

quarters one and a half pieta to 2p one and a half pieta,

00:11:17

that is, right here at the bottom three second

00:11:20

pa2 pieta here, that is, it turns out 4

00:11:24

quarters and the cosine is not positive, which

00:11:27

means they will still be plus, but under the

00:11:29

root we put one minus sine

00:11:32

squared

00:11:33

06 squared, this is 036

00:11:40

036,

00:11:43

well, and under our root, 064 is

00:11:47

formed very well, that is, 08 and

00:11:50

therefore the cosine itself will be zero

00:11:52

eight, we remember that we need to

00:11:55

remember to multiply it by another eight,

00:11:58

eight by eight, this is how much we have

00:12:00

6 4 a once it stood, with the same sign, it means

00:12:04

it will be 6 and 4,

00:12:08

well, now let’s move on to the

00:12:11

double angle formulas;

00:12:13

essentially, these four formulas are very

00:12:15

important; they are important because if we have a

00:12:19

sine that has one formula, then the

00:12:22

cosine itself appears to be three

00:12:24

different I mean that the reference

00:12:27

material that we will have before the

00:12:29

option will only contain

00:12:31

these two formulas out of these four, I

00:12:34

highly recommend that you always remember that the

00:12:36

cosine of this angle is represented by

00:12:38

these triumph formulas, what is the benefit, the fact is

00:12:41

that these three formulas allow you to

00:12:45

see immediately

00:12:47

move the idea of solving problems look what is

00:12:50

remarkable about the last last

00:12:52

formula it expresses the platoon

00:12:54

glad dice only through a sine and the

00:12:57

penultimate one expresses only through a

00:12:59

cosine and this can be very convenient and

00:13:02

indeed in the problems these formulas are

00:13:04

immediately visible now we

00:13:06

will see it, well, needless to say which of these

00:13:09

formulas, we can even separately express

00:13:11

sine square and cosine square, and such

00:13:13

formulas, even in their own name, are

00:13:16

called the forum for reducing the degree, well, this will

00:13:19

not be useful to us in the first part, and

00:13:22

so

00:13:24

either we keep them in front of us and help, if

00:13:27

possible, at first we will

00:13:29

peep here we see that in

00:13:31

the numerator there is since square minus

00:13:33

cosine square let's not confuse it with

00:13:35

identity because there we also

00:13:37

had squares only between them style

00:13:39

plus and if it is plus then we would immediately

00:13:41

take this to one to identity in this

00:13:44

case it is minus which means this and hints

00:13:47

at this formula for the

00:13:49

cosine of a double angle, just see in

00:13:52

our form the tests are from cosine squared

00:13:54

minus sine rest on the contrary, yes, it does

00:13:57

n’t complicate things at all, we’ll swap

00:13:59

places, just a minus sign will come out in front, so

00:14:10

be careful, so

00:14:11

let’s do that, swap

00:14:14

places, that is, it will come out in front - 9 well,

00:14:18

here the usual difference will already be there,

00:14:22

which will now go

00:14:26

into the cosine of the double angle, the cosine of the double

00:14:30

angle but double from 86, see how

00:14:34

ideally the lower number on suggests

00:14:36

this and will be 172, that is, this is a successful

00:14:40

goodbye and the answer will be simply

00:14:43

-9, that is, now we can immediately

00:14:45

assume that, for example, in this

00:14:47

example,

00:14:48

when we swap places, a minus sign will appear

00:14:52

and these machinations themselves will be reduced, that

00:14:54

is, the answer will be minus 30

00:14:59

example 14 find the value of the expression,

00:15:02

too, as soon as we see the sine of one

00:15:05

argument is multiplied by the cosine of the

00:15:08

same argument, then this is always hints at

00:15:11

the sine of a double angle, this needs to be recognized

00:15:13

immediately, the unity, you just need to remember that

00:15:15

in order to collapse them, there must be

00:15:18

a two in front, so even if there is no art, we will

00:15:21

add any art, or in

00:15:23

this case, we will take it away from the four,

00:15:25

that is, we can write like this, well,

00:15:29

once in detail let's write four, let's take it

00:15:32

as two and two,

00:15:36

but at the bottom we'll rewrite

00:15:40

the cosine 56,

00:15:44

these are the names of the factors, we have to return

00:15:47

to the sine of the double angle, only

00:15:51

two remains, and the double angle is what 17 by 2,

00:15:55

multiply 34 and we

00:15:58

get this familiar problem to us,

00:16:02

we started ours with it trigonometry in

00:16:04

the room and maybe we can immediately understand that

00:16:07

the answer here will be either 2 or minus 2,

00:16:09

well, God bless him and let's look at the angles,

00:16:12

the sum of the angles is 90, which means we can again

00:16:14

imagine the top angle as

00:16:18

90 minus 56

00:16:20

at the bottom, leaving 90 minus 1 quarter there

00:16:24

funk and positive, the sine will be replaced

00:16:26

by a cosine and 90 will be removed, that is, there will

00:16:29

also be a die of 56, so

00:16:32

the answer will simply be 2,

00:16:36

that is, we will have the

00:16:41

same expressions, so this goes away and

00:16:45

remains simply 2,

00:16:48

let's try the

00:16:51

same thing here, let's immediately imagine that the

00:16:54

numerator on the spheres collapses into the sine of the

00:16:56

double angle is only from 48 from from we take

00:17:00

two and what remains is 24 and the sine will already be

00:17:04

well one hundred fifty two by two multiply 304

00:17:08

these lower number by just tells us that

00:17:11

it all goes away safely and remains

00:17:14

just 24 here we can immediately give the

00:17:17

answer what it will be 9 that is, from 18

00:17:21

we will only take a two, but here what the

00:17:24

answer will

00:17:25

really be 14

00:17:31

example 15 so, but here we also see

00:17:35

the sine and cosine are taken in the same

00:17:37

argument, which means it will collapse into the sine of a

00:17:39

double angle from the four, from the four we take a

00:17:42

two will remain, which means 2 roots of 2 and the

00:17:45

sine will be already a double angle, that is,

00:17:48

double from the original one, of course,

00:17:51

in the future we can immediately reduce this because

00:17:52

we see that 2 and 8 will give a four at the bottom,

00:17:56

that is, in fact, we need to find the sine of 7

00:18:00

by 4, but we started with this, too, there are

00:18:03

reduction formulas, you can of

00:18:06

course find this point on a circle

00:18:10

and

00:18:13

can be represented again in the form of the sum and

00:18:16

difference 7 pin 4 what number is closest to

00:18:19

seven is divisible by 4 8 8 divided by 4

00:18:22

this will be 2 times we went more to the side of

00:18:25

the shield we will subtract this small fraction pin

00:18:28

4 but here let’s reduce the insurance 72

00:18:32

ppi here is where we take away we go

00:18:35

back we get 4 quarter sine is

00:18:38

non-negative and we won’t change the function,

00:18:41

that is, the minus sign is already coming forward, we do

00:18:44

n’t change the function, but the reduction formula

00:18:47

allows us to remove 2pi with a minus and

00:18:51

what remains is just sine pi by 4,

00:18:55

as we see that the root of two is two

00:18:59

will ideally remove the two and

00:19:02

only remain in the answer -2. A

00:19:08

similar example is only bad in that the

00:19:11

five does not seem to contain

00:19:13

the two, what should we do, but we need to

00:19:15

collapse it and we will make an artificial

00:19:18

reception, multiply the first coefficients and

00:19:22

divide, that is, we have just a two

00:19:25

sine and cosine are folded into sin from the

00:19:28

double angle but remains ahead then there are

00:19:31

already five second to the sine there will already be a

00:19:34

double angle from pi to 12

00:19:38

5 second let it remain so,

00:19:42

but here we see that

00:19:44

inside there will be just a sine taken from

00:19:47

cinema 6 and this is already the zenith table

00:19:49

value 1 2 means there will be five fourths

00:19:53

this is

00:19:54

1.25

00:20:00

let's take another example here

00:20:04

they will be just a two so

00:20:06

we can calmly and fold and

00:20:08

the sine will be already double the angle from

00:20:14

23 p to 12 of course this disappears for us and

00:20:21

twenty-three pin 6 remains here, of course,

00:20:24

we will definitely fall into ghost form because

00:20:26

now let’s imagine this fraction through an

00:20:30

integer part, what number of the nearest

00:20:32

twenty-three is divisible by six

00:20:35

2424 divided by 6 this will be 4, that is,

00:20:39

four integer n and really since we went on the

00:20:42

larger side, subtract pin 6

00:20:46

again I repeat, you can check every time like this

00:20:48

using our additional one

00:20:50

or

00:20:51

so, but let’s give them a circle of 4

00:20:55

pi on mine, we have it on the right, once we subtract it,

00:20:58

we go back, we get 4 quarters, the sine is

00:21:01

negative there and we won’t change the function,

00:21:03

that is, we have a minus in front and

00:21:06

forms carrying out this removes 1 table

00:21:10

angle and what remains is simply the sine of pi on 6 but

00:21:14

your strong back 6 is one second

00:21:16

just let’s not forget there is still ahead minus

00:21:22

one more example these again we have

00:21:26

the root that is it is not a single rack of course in

00:21:28

no case we already have art

00:21:30

multiply and divide by a pure two, and

00:21:33

then this two by sine cosine

00:21:36

will collapse into the cosine of a double angle and the

00:21:39

feathers of the village, the root of two divided by

00:21:42

2,

00:21:43

the sine will be already a

00:21:46

double angle from the original one,

00:21:49

we again see that this is let’s

00:21:52

cancel with the figure eight and

00:21:55

get

00:21:58

13 pin 4

00:22:01

Well,

00:22:03

we can again imagine in the form of

00:22:07

a sum what we have and the difference the number of the

00:22:11

nearest 13 and divide 412 12 divided

00:22:14

by 4 this will be 3 times we went to the smaller

00:22:17

side then add

00:22:20

3t plus pin 4 where o on the strip is

00:22:26

on the left side here we have odd times

00:22:29

we add, we go further, we end up in the

00:22:31

third quarter of the hay, are the quarters worth

00:22:33

negative, then we will definitely already

00:22:36

have a minus in front, well, we don’t change the function

00:22:39

and the back is strongly left 4 and

00:22:42

this is the

00:22:44

root of two by two,

00:22:47

we see that the root of 2 by the root of 2

00:22:50

two will cancel with one of these twos

00:22:52

and there remains generally minus one second, that

00:22:56

is, minus

00:22:58

0.5

00:23:02

16 and 17 and 18 examples are very important these are

00:23:06

some of the most popular and frequently

00:23:07

encountered examples, pay attention to them,

00:23:17

well, thank God because they are precisely what

00:23:21

hints

00:23:22

at these three formulas

00:23:25

you know how you can immediately recognize them,

00:23:29

let’s even go over

00:23:31

these formulas 1 formula expresses the

00:23:33

double angle through the difference 2 through

00:23:37

only the cosine and the third only through the

00:23:39

sine because like these formulas where the

00:23:43

difference is the difference that is, from a double angle the

00:23:46

next type is expressed only through

00:23:49

cosine, well, here are 3 types of these tasks

00:23:53

through sine, that is, this will always

00:23:56

tell us that we are working with simply the

00:23:58

cosine of a double angle because

00:24:00

there are ways to do these tasks through

00:24:04

reducing the degree, but in my opinion it takes a

00:24:06

long time and we can do the only thing right away I’m

00:24:11

so glad to endure these bad roots, that

00:24:14

is, here we have

00:24:16

the root of 18 coming forward and what remains is

00:24:19

cosine square minus sine square, that

00:24:21

is, this formula

00:24:24

is folded into a bone from a double angle,

00:24:26

so let’s immediately fold the double

00:24:29

angle

00:24:31

from 7 pin 8,

00:24:35

that is in fact, it came down to the

00:24:39

sine 7 pin 4 but we already represented this

00:24:44

angle

00:24:46

7 pe this is the nearest eight, that is,

00:24:49

where the whole climbs and minus pin 4

00:24:52

let's imagine two peaks we

00:24:55

have here

00:25:01

we take away we go back we get 4 quarters

00:25:06

4 read the bone for 4 count and is

00:25:09

positive, well, we don’t change the function, that is, what

00:25:12

remains is 1,

00:25:14

just the cosine of pi by 4,

00:25:18

but the cosine of pi by 4 is the root of two

00:25:21

by two,

00:25:23

because how fortunately the root of 18 by the root

00:25:26

of two will give the root of 36, that is,

00:25:32

we will definitely have the root extracted six by six divided by 2 is already three,

00:25:35

that is, in all these problems, of course,

00:25:37

the root must go, and now let’s

00:25:40

show you as much food as possible in the future you

00:25:43

need to go the faster way, that is,

00:25:45

skip some steps, which steps in the

00:25:47

first one we can immediately

00:25:49

reduce by two the bottom number is eight

00:25:52

twelve twelve and already go like

00:25:56

this,

00:25:57

be sure to throw out this root and

00:26:00

then we will

00:26:03

already have the cosine of the double angle of the double, that

00:26:07

is, 12 will be reduced to 7 pin 6 well, and

00:26:10

we already have

00:26:13

one of the vegetables peeling off, the six is

00:26:16

divisible by 6 1 integer and plus another

00:26:20

pin 6

00:26:22

where we have saw on the left side

00:26:25

since we add, go further, we get into the

00:26:28

third quarter

00:26:29

cosine three quarters to negate or that means it

00:26:32

will definitely be minus

00:26:35

75 we don’t change the function since pi and

00:26:38

it’s just cosine pi at 6 a

00:26:41

cosine pi by 6 is the root of 3 by 2,

00:26:44

we see that now the roots will

00:26:48

go away safely because under the root we

00:26:50

will have 225,

00:26:52

this will be 15

00:26:55

divided by 2, well, not an integer, of course,

00:27:00

but in any case we get a

00:27:03

good answer,

00:27:07

let's try here,

00:27:11

apply the root

00:27:13

remains the cosine, let's immediately by 2

00:27:16

we multiply this initial argument

00:27:23

5 by 6 12 and 6 will be reduced and of

00:27:29

course 5 by 6 is very well known in the corner

00:27:32

it is located on the left side from pi to 6

00:27:35

that is in the second quarter

00:27:37

but the cosine of the second quarter is

00:27:40

negative, that is, we can already to say

00:27:42

that this will be a

00:27:43

value with a minus, well, let’s not even

00:27:46

deviate from the general outline 5pin 6 that for us

00:27:50

this is essentially subtracted from pi by pin 6, as

00:27:55

we said, it’s 2, read the cosine there is

00:27:57

negative, which means the answer will be

00:28:00

exactly with a minus

00:28:02

root of 3, but the cosine pi by 6 points

00:28:06

has not changed, but this is the root of 3 by 2,

00:28:09

because the root of 3 by the root of 3 in dos

00:28:11

C in the case of 2 and we get the

00:28:15

final answer of

00:28:17

one and a half,

00:28:22

I specifically consider a lot of

00:28:24

similar examples to

00:28:27

consolidate this, and they are the ones you should always

00:28:31

recognize and

00:28:33

work out these reduction formulas

00:28:36

and immediately see the double angle formulas, the

00:28:38

same root of 32, we take out and inside there

00:28:42

remains the bone from the double angle, that is,

00:28:44

three pi is already on 4, well, three pin 4 is also 2

00:28:50

quarter cosine there is negative and

00:28:53

we will have that it will be the same thing

00:28:56

as the cosine of pi over 4 only with a

00:28:58

minus sign and this will be minus the root of two

00:29:01

over two because how fortunately we will have 64 under the roots,

00:29:07

that is, minus 8 divided by 2

00:29:10

minus 4

00:29:16

example 17,

00:29:18

also from the same opera, we understand that

00:29:21

this will be cosine of a double angle, why,

00:29:23

but according to the form of notation and the condition, a dice is

00:29:27

subtracted from a square,

00:29:29

what formula from 3 is this similar to, but

00:29:32

of course, like the previous cosine

00:29:34

squared minus 1, and at the end we have a

00:29:37

root here, and this will give us ideas for

00:29:39

solutions that is, we will take this interfering

00:29:41

root out of brackets, then it

00:29:44

remains in brackets so from 32 we will take out

00:29:48

the eight, there under the root there will already be 4, but

00:29:51

because 35 by 8 it will be 4 and the root

00:29:54

from the 4th floor 2 will still be two

00:29:57

cosine squared pin 8 minus 1 so

00:30:00

this will always be what remains of the parentheses,

00:30:02

this is the cosine of the double angle, well,

00:30:06

the double angle, which we will now

00:30:09

write as the cosine of the double angle from

00:30:12

the original one, and initially it was pin 8, because

00:30:15

how well it will be reduced,

00:30:19

that is, in general, the cosine of pi by 4 remains

00:30:22

this is actually the table value of the

00:30:24

root of two by two then in our

00:30:27

numerator there will be a root of 16, as we

00:30:30

can see it is well extracted it will be 4 and

00:30:33

four subtrees by 2 and we will get the

00:30:36

final answer we

00:30:40

will do the same in a similar example so that at the end of the table the 1 will

00:30:44

immediately tell us this the root of 50 and

00:30:48

how ideally if there are 2 hundred parts of them, output

00:30:51

50, that is, 200 losses per 50, it will be 4,

00:30:54

that is, the cosine of the double angle will always remain there,

00:30:56

and therefore in the

00:30:58

future we can even immediately write

00:31:00

something will be the cosine of the double angle from the

00:31:02

original 5pin

00:31:06

828 will be reduced by 4,

00:31:10

that is, it turns out that we have a cosine of 5 pin 4,

00:31:14

well, let’s we are already familiar with the

00:31:17

reduction form, let’s imagine the angle as the

00:31:20

sum or difference of what number of the nearest

00:31:23

50 4 but 4 will be, that is, it will be 1 plus

00:31:29

pin 4, let’s find out some let's

00:31:33

show the quarters on the whitefish we have pi on the left side

00:31:37

once we add, we go further 3 quarter of the

00:31:40

dice from thousand quarters to deny or that

00:31:43

is, the answer will definitely be with a minus, well, we do

00:31:45

n’t change the function because it’s worth drinking,

00:31:49

let’s free it here and

00:31:52

then we get

00:31:56

equal to minus root from 5 10 we multiply by the

00:32:01

cosine of pi by 4

00:32:05

cosine of pi by 4 this is the root of two by

00:32:07

two stages of the table and how ideally

00:32:10

from above we get the root of one hundred and this

00:32:13

is extracted -10 let the numerator be divided

00:32:16

by 2 minus 5 that is, we get a very

00:32:19

good answer without roots let's

00:32:24

make 2 more similar ones,

00:32:28

take out the root of 3 so that at the end there is

00:32:30

one, this will always be reduced to the

00:32:32

cosine of a double angle, double, that is,

00:32:35

from the original 5pin 12 we can immediately

00:32:39

reduce 211 and we are left with

00:32:44

the cosine of 5pin 6 and 5 by 6, this is a very

00:32:49

popular angle here here it

00:32:51

is in the second quarter,

00:32:52

so we can immediately understand that the

00:32:54

cosine of the second quarter

00:32:56

is negative, that is, the answer will be

00:32:58

minus,

00:32:59

even in this way,

00:33:02

let 1 root of 3 remain, but the

00:33:04

cosine of 5 by 6 is

00:33:06

minus and

00:33:07

just the cosine of pi by 6, what value

00:33:10

does the

00:33:11

root of 3 by 2 and as you can see at

00:33:14

the top we will have the root of 3 to the root of 3

00:33:17

completion and at the bottom 2 the value will be minus

00:33:20

one and a half and

00:33:24

in the last example we will also

00:33:26

take out the

00:33:28

root of 27 and inside there remains the cosine of the

00:33:31

double angle double

00:33:34

from the original 23a by 12 as you can see and

00:33:40

five will be reduced, of course, you can

00:33:41

immediately reduce it in your mind and write the

00:33:44

original angle has already been multiplied by 2,

00:33:47

so then you can do it, but now

00:33:50

you need to represent twenty-three pin 6

00:33:53

as a

00:33:55

sum or difference, but we already know our

00:33:59

secret what number the next 23 is divisible

00:34:02

by six

00:34:03

2424 will appear on 6 this will be 4 so

00:34:07

it will be 4 p minus pin 6 minus because

00:34:11

we 24 took the larger side, which means we need to

00:34:14

subtract this smaller google table and

00:34:16

then we get the

00:34:22

root 27 remains 4 pi subtract

00:34:27

the angle of some quarter 4 pi we have is on the

00:34:30

right subtract that means we go

00:34:33

backwards, this is 4 quarters of the cosine is not

00:34:36

positive,

00:34:38

which means it will just be the cosine of pi by

00:34:41

6 because we do not change the function itself, the

00:34:45

root of 27, but the cosine of pi by 6 is the

00:34:48

root of 3 by 2, because in the numerator

00:34:51

it turns out very well under the root of 81 this

00:34:55

will be 9a 9 divide by 2

00:34:57

we get the final answer

00:35:00

4 and a half

00:35:05

example 18 from the same opera because we will

00:35:09

again see what formula

00:35:11

is visible here in the condition itself, of

00:35:13

course the last these 1 minus sine 2 sine

00:35:17

square only instead of units there is a

00:35:19

root of 3 this and suggests the idea that we

00:35:22

need to throw out this root and then we will

00:35:25

now do the first in detail, what will remain is

00:35:28

if 12 we derive three under the root there will be

00:35:31

4 left and this will give two, then

00:35:34

for now we leave the angle the same and

00:35:38

this again will always be the

00:35:43

cosine of a double angle

00:35:46

so the root of 3 multiply by the cosine of the

00:35:50

double angle from the original 5p by 12

00:35:55

211, reduce it to 6 and get the root of

00:36:00

3 by the cosine of 5 by 6 5 by 6 this is the second

00:36:06

quarter, you can again do something according to the

00:36:07

form of the conduction, but in general the cosine of three

00:36:09

quarters takes a negative

00:36:11

value and itself whatever the value is

00:36:15

and what is the cosine of pi equal to 6 root of 3

00:36:18

by 2 then

00:36:20

we will have a minus in the answer the

00:36:23

roots on top will give 3 and below 2 that is, we

00:36:27

will finally get minus one and a half

00:36:30

let's do similar examples we

00:36:33

'll immediately take overboard minus the root of 32 and

00:36:37

inside What remains is the cosine of the double angle of the

00:36:39

double from this

00:36:42

original 3 pin 8,

00:36:45

you can immediately reduce it and

00:36:49

we will have the cosine from 3 pin 4, well, three

00:36:54

pin 4, this is again our 2nd quarter if we

00:36:58

don’t want to go to the form of carrying out the ato not

00:37:02

the second quarter, the cosine there is

00:37:03

negative, that is it will definitely be with a

00:37:05

minus, but it’s just

00:37:07

that the cosine of pi by 4 is equal to the root of two by two,

00:37:11

which means it will definitely be a minus where, as

00:37:14

well as on top, there will already be 64 under the root,

00:37:18

and this is extracted about 7 -8, divide by 2 and

00:37:23

get the final answer minus 4

00:37:28

we’ll do more similar examples, let’s take the

00:37:32

root of 2 overboard when the

00:37:34

cosine of the double angle of the double

00:37:38

from 7 pin 8 remains

00:37:40

below there will be four already

00:37:43

so the root of two

00:37:50

is multiplied by the cosine of

00:37:55

7 pin 4 well now let’s play it safe and

00:37:58

solve it using the formula for ghosts 7 pin 4

00:38:02

which closest seven is divisible by

00:38:04

four number 8 8.42 means a will be 2

00:38:07

integers and subtract pin 4

00:38:11

subtract where this angle of us is

00:38:14

2pi I subtract here we go back we

00:38:18

get 4 quarters the bone is counted and the

00:38:20

quarters are positive well and we don’t change the function,

00:38:24

that is, we will remain like this the

00:38:27

root of 2, well, the cosine of pi by 4 is the

00:38:30

root of two by two, and

00:38:32

as you can see, ideally the numerator will simply turn out to be

00:38:35

your k and below two 2, divide

00:38:38

by 2, we’ll just get 1 and

00:38:43

the last example is a big root, it doesn’t matter, we

00:38:47

’ll also throw it out root of 75 and here What

00:38:51

remains is the cosine of the double angle of the double

00:38:54

from 13 5 by 12,

00:38:58

you could immediately write that this

00:39:00

will be

00:39:03

the cosine

00:39:04

from 13 to 6, and let’s again

00:39:09

present it in the form of a sum or difference, what

00:39:13

number of the nearest 13 is divisible by 6 12 12 9

00:39:16

by 6 it will be 2, that is two pi since we

00:39:20

went to the smaller side, now we add

00:39:22

this smaller tabular one to the corner and we get the

00:39:29

root of 75 remains 2pi + pin 6

00:39:35

2pi + this will already end up in the first quarter

00:39:40

in the first quarter in the first quarter all

00:39:43

functions have a plus sign, well, we don’t change the function

00:39:45

that is, it will simply be

00:39:48

the cosine of pi by 6

00:39:53

root of 75, we multiply the tabular knowledge

00:39:57

root of 3 by 2 and now we can

00:40:02

calculate in the numerator, we can, of course,

00:40:04

directly multiply 7 5 by 3 this is

00:40:06

225, but if it turns out to be a large number, then

00:40:10

it is convenient to do this with let's calculate it

00:40:13

will be 15 measles from it divided by 2 well, and

00:40:17

already 15 tree 2

00:40:19

we get

00:40:21

7 and a half that's how else it could be

00:40:24

if we want to multiply the root of 75 by 3

00:40:29

then we can generally speaking split it into

00:40:31

smaller factors 75 can be

00:40:34

represented as 25 multiply by 3 and another

00:40:37

3 because how conveniently from 25 the root

00:40:40

is extracted 5 and from 3 to 39 the root is also

00:40:44

extracted and we get the root 15 well, this

00:40:48

works well if the number under the ring is

00:40:50

large enough and we cannot

00:40:52

remember what uranium is immediately from it the root is

00:40:58

example 19 find four cosine 2 alpha

00:41:02

if the sine alpharo is -05 and this is a direct

00:41:06

hint at what formula if we were given a

00:41:09

sine, then of course it is reasonable to use the

00:41:11

last form because it is in

00:41:14

it that the cosine of a double angle is expressed

00:41:16

only through the sine, so we need and

00:41:20

we will formulate the survey like this: 4 multiply instead of a

00:41:24

bone from a double angle, let's take this third

00:41:26

formula and

00:41:28

how fortunately we will now have a

00:41:31

strong squared this number

00:41:34

squared, it is clear that this will be a

00:41:37

minus will disappear it will simply be zero 25

00:41:41

and

00:41:43

equal to this will be

00:41:45

let's first figure it out with a bracket 2

00:41:48

multiplied by 25 this will be 0 5 1 minus 0 5

00:41:52

parentheses will give 05 and 4 multiplied by 0 5 that

00:41:56

is, the same thing that is needed 2 we’ll just

00:41:58

get 2

00:42:01

give now let’s consolidate this technique,

00:42:05

again we see that they gave us a sine, so we

00:42:07

take the last formula, then somewhere 17 of

00:42:10

course we throw it out of the bracket 1 -2 and

00:42:14

instead of sine, let’s immediately put 0 8

00:42:16

squared

00:42:18

then of course, first we’ll figure out what

00:42:22

’s going on in brackets

00:42:25

064 here you just need to calculate carefully,

00:42:30

so if we multiply 064 by 2, well, just

00:42:34

64 on the 2nd floor 128 just a comma and

00:42:37

pork by 2 digits then there we

00:42:41

will have 1.28

00:42:45

so then in brackets it turns out minus zero point

00:42:48

twenty eight

00:42:50

the answer will definitely be with a minus well and of

00:42:54

course

00:42:55

we multiply this

00:42:58

in a column but let’s

00:43:00

reproduce it once that is, we want to multiply 17 by

00:43:05

0 28

00:43:08

so we take it for everyone on 8

00:43:11

56 56 we write 5 to mind 81 88 plus 513

00:43:19

further 2 by 7 14

00:43:22

2 by 1 2 and plus 13 that is, it will be

00:43:28

674 only we had two commas, two

00:43:31

decimal places, so the answer will be

00:43:33

- 4.76

00:43:37

here we just go carefully on a person

00:43:39

we count the column

00:43:42

further, here we have further in the condition

00:43:45

not sine and cosine, this is somewhere that we

00:43:48

will use the previous formula,

00:43:51

this one because it expresses the

00:43:53

cosine of the angle, only through the cosine we will

00:43:56

of course take the six overboard and

00:43:59

remain in brackets 2 dice you squared

00:44:03

minus 1 we can immediately, of course, instead of dice

00:44:06

from the square in 08, we will square 064

00:44:11

and -1 let's again deal

00:44:15

with the buying first only, that is,

00:44:19

let's multiply 064 by 2, we have already met with this

00:44:23

number

00:44:24

128 -1 about the bracket we will have already 028

00:44:29

well, here we go again

00:44:33

in a column and already get the final

00:44:36

answer

00:44:37

1.68 and

00:44:42

here the answer is negative in the

00:44:44

previous one the answer turned out to be positive

00:44:46

negative it’s as if it can happen here every time

00:44:49

let it not surprise you

00:44:53

19:03 again since they give us a cosine

00:44:56

that means we choose which formula is

00:44:58

really so which is expressed

00:45:00

through cosine here it is the previous

00:45:03

last nine we throw out in brackets

00:45:07

there will be two space squared minus

00:45:11

12 multiplied by one third we will immediately

00:45:15

square it it will be 1 9 minus 1

00:45:18

here it is reasonable to calculate not the bracket

00:45:21

separately but right away so nine

00:45:23

is multiplied by the

00:45:25

distribution by this difference, these

00:45:28

discharge in the first term will be reduced,

00:45:31

it will be simply 2 and there 9 and therefore the

00:45:34

answer will finally be minus 7

00:45:40

example 20

00:45:42

find this friend if we were given

00:45:45

the sine of a double angle here the whole idea is

00:45:48

that on top of there is sie das 4 alpha

00:45:50

and at the bottom there are 2 alphas, so we can

00:45:53

imagine in just 4 alphas how two

00:45:55

multiply by two and let’s

00:45:58

write it down in detail,

00:45:59

that is, I can say that 4 alpha is a

00:46:03

double angle from two alphas, but at the bottom

00:46:06

let it be rewritten like this for now this is a

00:46:09

synthesis of a double angle double from two

00:46:12

alphas and at the bottom there is a wolf, so

00:46:14

we’ll use the formula,

00:46:15

but it’s the only one we have, the

00:46:18

two comes forward, that is, we will already

00:46:20

get them not 5 but 10

00:46:24

sine 2 alpha and cosine 2 alpha, that is,

00:46:28

our original angle 4 alpha has decreased

00:46:30

twice below we’ll just rewrite as you can see,

00:46:34

we have ideally called the bone fa will be reduced

00:46:37

and what’s next in the condition is

00:46:39

sine 2 alpha and that is just 06 and everything is

00:46:44

ideally considered 10 multiply by 0 6 and

00:46:47

divide by 3

00:46:50

effective 6 is just 66 divided by 3 and

00:46:53

we get the answer 2,

00:46:57

now let’s try here also to do

00:47:01

only 4 alpha at once, we represent it

00:47:05

as a double angle from two alphas, so

00:47:07

the troop just climbs forward and this will already

00:47:10

be the numerator 4

00:47:13

sine 2 alpha cosine 2 alpha, well, we’ll

00:47:17

just rewrite the substitute

00:47:20

as you can see, we have it safely

00:47:22

multipliers are abbreviated axes us two alpha

00:47:26

and that is just 02 so 4 multiply

00:47:29

by 0 2 and

00:47:32

divide by another 5 well, how can we

00:47:38

do this faster here, let’s for example

00:47:41

get rid of the

00:47:44

ten with a comma, that is, multiply for

00:47:47

example the numerator is initially 10, we

00:47:49

will have 4 2 and below 50, in principle, we already

00:47:53

see that this will be 8

00:47:56

fiftieths if we multiply the top and bottom

00:47:59

by 2 to get a hundred below, this

00:48:03

will give us the final answer of

00:48:04

sixteen hundredths, that is,

00:48:09

0.16,

00:48:12

exactly the same idea will be in a similar

00:48:15

example only when the angles

00:48:17

3 alpha and 6 alpha already appear, but really 6

00:48:20

alpha can be represented as a double

00:48:22

angle from 3 alpha, that is, the idea again is that

00:48:28

we imagine 6 alpha as a double angle

00:48:30

from 3 alpha

00:48:32

downwards, we don’t touch it now, the top sen of

00:48:36

the ride on the loot we write forward

00:48:39

the deuce climbs out, this already means there will be 6 first

00:48:42

sine 2 alpha and then cosine volvo

00:48:45

just didn’t give the fatf of course 3 alpha and

00:48:49

below we have

00:48:51

these rewritten similarly to the previous

00:48:54

examples the

00:48:55

axis of us 3 alpha disappears according to our

00:48:58

conditions it is equal to 08 so 6 multiplied by 0

00:49:02

8 and divide by 5, for example, we can

00:49:06

even go directly, but let’s

00:49:10

imagine 6 multiplied by 0 8 6 8 48 again

00:49:14

, that is, 48, well, here let’s

00:49:18

get rid of the comma,

00:49:21

multiply, or even like this, you can do it,

00:49:24

multiply the top and bottom by what number so that the

00:49:26

bottom turns out to be two tens by 2,

00:49:31

which means we already have 10 at the bottom, and

00:49:34

on top

00:49:35

482 is like 96 only with a comma 9

00:49:40

6 and this just says that when

00:49:43

we divide 9 and 6 by 10, we shift the comma

00:49:45

by one more the sign, that is, will be 096, but

00:49:48

let's go more in this classical

00:49:50

way, let's do something else again,

00:49:54

here we multiplied the top and

00:49:57

bottom by 10, as in the previous examples, that is, you

00:49:59

will have 6 and 8 on top and 56 848 on the bottom,

00:50:03

well, we understand that now

00:50:08

it’s easy to multiply the numerator and know or by

00:50:10

2 and then at the bottom there will be a hundred

00:50:13

and it will be exactly 96 and at the bottom of a hundred it

00:50:17

will be 0.96 we

00:50:19

got the same answer and the

00:50:25

last block of the trace form is related to the

00:50:28

definition of tangent let's remember

00:50:30

how we define tangent, this is the

00:50:33

ratio of sine to cosine, of course,

00:50:36

this function itself has about d d m, that

00:50:39

is, you always need to understand that the cosine is not

00:50:41

equal to zero, well, the idea here will be

00:50:43

precisely in defining

00:50:46

example 21, you need the tangent, if

00:50:49

we further divide such a fraction, the numerator and

00:50:52

denominator by the cosine, why

00:50:55

exactly by cosine, but because it is precisely the

00:50:58

cosine that is in the denominator in

00:51:02

the pipe which determines the tangent,

00:51:04

look, we take all the sine alphas,

00:51:07

divide them by cosine, and we actually

00:51:09

get seven tangent alphas plus

00:51:13

if we divide 13 cosine by cosine but

00:51:16

cosine by dice already share and all

00:51:17

that remains will be reduced is only 13,

00:51:20

also at the bottom people five will already be

00:51:24

close to the tangent, but here the cosine will

00:51:27

disappear and it will simply be -17 and all this is

00:51:29

equal to 3, that is, we have obtained an equation

00:51:31

for the tangent which is very

00:51:33

easy to solve, this denominator and

00:51:35

transfer to right to the three and

00:51:39

will wake up on the left side 7 tangent alpha

00:51:41

+ 13 and

00:51:44

on the right side let's immediately go to the

00:51:47

parentheses 3 multiplied by 5 tangent this

00:51:51

will already be 15, well, let's multiply 17 by 351 but

00:51:56

now the tangent is flipped from one

00:51:58

side to the other and we have

00:52:01

it turns out that on the left side there are 64 and on the right there are

00:52:05

15 tanks minus 7

00:52:07

8 and how ideally it turns out that the

00:52:10

tangent alpha will be equal to 64 times

00:52:14

88

00:52:16

now we can just

00:52:19

use this idea to solve

00:52:21

similar problems, but as you can see 21 1 boom

00:52:25

we will do exactly the same way and

00:52:27

get the equation

00:52:28

but let's try to take

00:52:32

21 2

00:52:34

definitely times also on the titanium guide, in any

00:52:38

case, we will divide the top and bottom of this first fraction

00:52:40

by cosine, so we

00:52:43

get

00:52:45

three tangent

00:52:49

alpha plus 2-minus 4 divided by cosine

00:52:54

alpha, that is, these first two

00:52:56

terms well surprised 1 it turned out

00:52:59

tangent 2 cosine was missing only two, but

00:53:03

four when it comes to the bone os

00:53:04

of course it will go down but it doesn’t matter let’s

00:53:07

go down

00:53:09

5 tango with alpha plus 5 and minus 6

00:53:14

divided by cosine alpha and all this is

00:53:17

equal to 2 3 it

00:53:19

would seem like how the tangent appeared,

00:53:21

well, and these are still fractions that strain

00:53:25

a little, well, what is straining, let’s

00:53:28

solve it like an equation, that is,

00:53:30

let’s try to throw it somewhere here and the

00:53:33

dust, by the way, so that there is no fraction and

00:53:36

vice versa to the left side up, well, that

00:53:38

is, as it were we use our

00:53:40

cross method and what we will immediately have is let’s

00:53:43

multiply the numerator of the first group

00:53:45

by 3,

00:53:46

that is, it will be 9 tangent alpha

00:53:49

+ 6 and minus 12 divided by the cosine alpha

00:53:54

on the right side, we are the denominator of the left

00:53:57

/ and we will multiply by 2, that is, it will be 10

00:54:01

tables alpha plus 10 and minus 12 divided

00:54:05

by cosine alpha because this was

00:54:07

the calculation that these bad fractions are just

00:54:10

destroyed tangents will be thrown to one

00:54:13

side of course the best ones will be thrown to the right

00:54:17

because 10 tangent -9 tangent this

00:54:21

will already be 1 tangent and ten is the opposite

00:54:24

here here 6 minus 10 this will be exactly the

00:54:28

answer minus 4 you immediately got an

00:54:32

expression for the tangent which is what we

00:54:34

asked for

00:54:38

example 22 here it’s like the inverse problem

00:54:41

the fraction already needs to be found if they gave a tangent

00:54:44

equal to one but Rastan didn’t pass the exam and

00:54:46

we do the same

00:54:48

we divide the numerator and denominator by the cosine and then

00:54:50

we will have a numerator of 4 and with a five

00:54:54

we will have a tangent below 4 tangent alpha

00:54:59

-6 because the cosine of the six

00:55:02

will be reduced and we were waiting on the condition that the

00:55:05

tangent of one therefore simply the place

00:55:07

of the tangent everywhere can be

00:55:10

represented by one and then us

00:55:12

will be in the numerator 4 minus 5 minus 1 and

00:55:16

in the denominator -2 and thus minus

00:55:20

us zaz + 1 2 is 0.5

00:55:26

now we can easily do a

00:55:29

similar task, divide the numerator and

00:55:31

denominator by the cosine and we will have 7

00:55:35

minus 6 tangent alpha and below 3 tangent

00:55:39

alpha plus four tanks is equal to two according to the

00:55:42

conditions,

00:55:45

let’s substitute it and

00:55:47

it will be in the numerator 7 minus 12 minus 5

00:55:53

in the denominator 6 ps4 10 as you can see minus

00:55:58

five tenths this will give the

00:56:02

answer -0.5

00:56:07

20 22

00:56:10

we also see similar examples here what

00:56:12

next do we need the tangent and to find the desired

00:56:15

expression, these are even more sophisticated

00:56:18

fractions, well, there is only one idea here,

00:56:20

as if there are no options, so let’s

00:56:23

divide the numerator and know, or by cosine

00:56:25

we will have 8 minus 2 tangent alpha plus

00:56:30

12 divided by cosine alpha, let it remain that way

00:56:34

below is the

00:56:35

tangent alpha minus 4 plus 4 divided by the

00:56:41

cosine alpha, it seems like they’re bad again, these

00:56:45

last fractions seem to spoil everything,

00:56:48

but let’s try to substitute a

00:56:52

four instead of the tangent, maybe most

00:56:55

often and we’ll get 8 minus 2

00:57:00

multiplied by 4 and

00:57:03

we have the fraction here will be 4 minus 4,

00:57:08

well, I think you’ve already noticed what

00:57:11

the idea here is that these first terms

00:57:14

go to zero and what remains is

00:57:17

12 by the cosine divided by 4, that is,

00:57:21

these distributions

00:57:22

will cancel out to the cosine of

00:57:24

12 divided by 4, quite specifically

00:57:27

determine the number 3,

00:57:33

example 23 a

00:57:35

very important task,

00:57:37

pay attention to it

00:57:40

on the child tangent if further the sine

00:57:44

itself is tango, even for us it

00:57:46

is defined as a sine such a sine

00:57:48

therefore we have a sine we need an influx

00:57:51

cosine and it is very important to immediately look at

00:57:54

this angle, we have an angle from 0 to pi we need

00:57:56

this is the first quarter of the first quarter, all

00:57:58

functions have a plus sign, that is, our

00:58:00

tangent will definitely be positive,

00:58:03

but since our tangent is determined by the

00:58:07

ratio of

00:58:09

sine to cosine of the night, the whole task

00:58:11

comes down to knowing the sine to find

00:58:13

the cosine, we already did this in the first videos

00:58:16

through the main identity that is, we are under the

00:58:20

root and amide of using

00:58:23

these first corollaries from the main

00:58:26

identity, well, plus and minus y, and only we

00:58:28

understand that the first quarter is that

00:58:31

the cosine is ours and the tangent will be

00:58:33

positive, and so, under the root of

00:58:36

unity, we subtract the square of the sine,

00:58:39

because like this It’s unfortunate that

00:58:41

such a number is Darren,

00:58:44

if we now

00:58:46

raise the square to the unit 29, it still

00:58:50

takes some of our efforts, and

00:58:53

you can even notice that if I

00:58:55

remove the root 29 from above, then below this root

00:58:58

is formed for what reasons, well at

00:59:01

least for those that 29 can be represented as

00:59:05

multiplying the roots 2 times,

00:59:07

if we remove it, then essentially the number

00:59:12

that was given to us is the same as two

00:59:14

divided by the root of 29,

00:59:16

it’s just beneficial because now it’s

00:59:20

very easy to square this number, here we have the roots

00:59:24

add the factor 29 and

00:59:26

then the underground expression will already

00:59:29

represent the form 25 divide by 29 from

00:59:34

the numerator the root a is extracted and

00:59:36

there is no denominator, but it doesn’t matter

00:59:38

because this is still not the final

00:59:40

answer,

00:59:41

we now need to substitute the sine

00:59:43

into the tangent,

00:59:45

let’s look at the sine as we already we took

00:59:50

this two divided by the root of 29

00:59:54

and the cosine, so we found this fraction, but

00:59:56

let’s immediately divide by the fraction,

00:59:59

this means multiplying by the inverted one, that

01:00:01

is, we take the inverted fraction and see

01:00:04

that these bad roots are successfully

01:00:06

reduced and in general two-fifths remain,

01:00:09

that is,

01:00:11

0.4

01:00:13

that is, it is very advantageous to keep in mind that in

01:00:16

this way it would seem

01:00:17

at first glance to transform cumbersome

01:00:20

expressions, of course it was possible

01:00:23

to square 29 but it would be more

01:00:26

difficult purely

01:00:29

numerically and so find where

01:00:32

the cosine was held here we already have what

01:00:35

quarter let's play it safe we'll see it

01:00:38

from pin 2

01:00:40

top this is the second quarter of the tangent of the second

01:00:45

quarter takes a minus sign here,

01:00:48

that is, we understand for sure that the answer

01:00:49

will be a minus answer, well, now we

01:00:56

’ll do the same thing for the tangent as

01:00:59

dividing the sine by the cosine, well, we need to find the sine then

01:01:04

that the cosine below

01:01:07

can easily be taken from the condition, let's separately

01:01:10

calculate Sirius by the way of the second quarter

01:01:13

it will be positive for us ip through the

01:01:15

main identity

01:01:17

from unity we subtract the square of our

01:01:21

cosine but tell me the minuses try it

01:01:24

will disappear and a / this squared will be one

01:01:27

tenth and then we will have in the numerator

01:01:29

9

01:01:30

voters 10 from the numerator the root is

01:01:33

extracted,

01:01:34

this is three and from the bottom there is no, well, to hell with it, the

01:01:37

most important thing is that we now

01:01:39

substitute sitelove the tangent, these roots

01:01:44

will cancel out and therefore we will have 3

01:01:47

divided by minus 1 minus 3 this will be

01:01:55

another answer let's take

01:01:58

find where the goals were given the sine of some

01:02:01

quarter,

01:02:02

let's find out one and a half pieta below and

01:02:06

2pi is here, that is, it

01:02:08

will be the fourth quarter and the tangent does not

01:02:10

understand the minus sign, so

01:02:15

we will definitely have an answer with a minus, and just like that

01:02:19

with alpha for cosine alpha

01:02:23

we already have sine alpha,

01:02:26

so we need to find through the rest of

01:02:28

the identities cosine well, the dice for 4 quarters

01:02:32

are positive, so it will be in front of the root

01:02:35

plus 1 minus, let's square this number

01:02:39

25 and below it will be 41 if the numerator is 45

01:02:44

121

01:02:47

that is 41 from them in 25

01:02:51

we ideally get 16, because how well in the

01:02:54

numerator the root is extracted from not occupied,

01:02:57

well, to hell with it, the definition in the tangent will be

01:02:59

reduced, the numerator of us will be 4 and

01:03:03

at the bottom smoke 41

01:03:07

we substitute our tangent,

01:03:09

safely we will disappear and in general we are

01:03:12

left with minus five quarters

01:03:16

minus 5 4 of course you need to convert the

01:03:19

decimal fraction this is minus 1.25

01:03:28

example 24

01:03:30

also find the tangent only from alpha with the

01:03:33

spin yes if you give the tangent of course this

01:03:36

again essentially refers the problem to the

01:03:38

reduction formulas because when

01:03:41

McPhee needs to add the angle we end up

01:03:43

in the second quarter there is no with it

01:03:45

is negative and since it is spin 2, then you need to

01:03:49

change the function to the opposite

01:03:51

opposite, that is, it will no longer be a

01:03:53

tangent and the cotangent will only

01:03:56

come forward with the minus sign 1 2 quarter, that is, it

01:03:59

will be minus cotangent alpha, and how is

01:04:03

the tangent related to the tangent, which is what

01:04:05

we have the relationship between them, that is,

01:04:07

in essence, these are mutually inverse quantities, that

01:04:10

is, you can always imagine that

01:04:13

tangent alpha is 1 divided by

01:04:16

cotangent alpha and vice versa, that is, they are,

01:04:19

as it were, mutually inverse from each other,

01:04:21

so in order for those cotangent it is necessary

01:04:25

to divide 1 by tangent tangent then

01:04:28

they gave us 051, let's divide 05, that will be two, just

01:04:33

don't forget, of course, minus,

01:04:36

let's take similar ones, here we have a

01:04:39

kind of similar example, but it's a bad

01:04:42

argument, we're used to pi having to be

01:04:45

with a plus, so let's swap

01:04:47

this

01:04:49

difference,

01:04:51

but forward looks like a minus because

01:04:53

the tangent

01:04:55

throws out the minus period and

01:04:57

inside there will already be pin 2 minus alpha,

01:05:00

so let's watch the movie 2 minus alpha,

01:05:03

this will be the first quarter

01:05:05

in it, all the functions are positive, although

01:05:07

the dance will change to cotangent minus

01:05:10

also there remains ahead only there will be already

01:05:13

minus cotangent and cotangent is the same 1

01:05:16

divided by we were waiting for the tangent according to the condition,

01:05:19

it’s two and a half, so we divide 1 by

01:05:23

two and a half,

01:05:25

you can do it this way, two and a half is

01:05:27

then 5 second, that is, it’s like two-fifths,

01:05:30

we turn the fraction over, and two-fifths is

01:05:33

minus

01:05:34

0.4,

01:05:39

well, for clarity, let’s also take it so that

01:05:43

We also need to work out the reduction formulas,

01:05:45

we see that again this is a bad argument,

01:05:47

so we will have to change places, but the

01:05:50

minus sign

01:05:53

7 pin to minus alpha will come forward, let's find where

01:05:56

this angle is 7 pin 2 how to

01:05:59

gradually calculate 1 pi by 2 2 pin 2 3 4

01:06:03

5 6

01:06:05

give everyone one more time two three 4 5 six 7 that

01:06:10

is, we got exactly down, down,

01:06:18

this one. 7 pin 2,

01:06:23

well, now since the angle is subtracted, we go

01:06:26

back and find ourselves in the third quarter 3 4

01:06:30

the tangent there is positive, which means it’s as if

01:06:33

ahead - it will remain so, and since the angle

01:06:35

contains pin 2, then we’ll change the tangent

01:06:37

to cotangent, that is, it will be minus

01:06:40

cotangent alpha and

01:06:42

cotangent is personally the inverse of the tangent and

01:06:45

therefore simply divide -1 by

01:06:49

125

01:06:51

from 125 can be represented as five

01:06:54

fourths

01:06:56

turn it over it will be minus four

01:06:59

fifths amine four fifths is minus 1.8

01:07:08

and the last example is

01:07:11

25

01:07:13

find the tangent squared if we were given

01:07:16

this expression that

01:07:18

would be all right if there was a zero on the right

01:07:20

and you would divide both sides by the

01:07:23

cosine squared, this technique is very well

01:07:26

known and is used in genetic

01:07:28

equations, but here the six on

01:07:30

the right gets in the way as a number, so let’s

01:07:33

again use the basic

01:07:35

metric identity,

01:07:37

nothing will really change if this six is on the

01:07:39

right sign

01:07:42

has the color of a multiplier sine

01:07:46

squared alpha plus cosine squared

01:07:48

olive why won’t it change but these are

01:07:50

identities this is the same unit only the idea is

01:07:53

that if we now throw

01:07:54

a six on this fraction for this purchase

01:07:56

we will have similar expressions we will

01:07:59

throw everything into one side and the most important thing is

01:08:01

that on the right there will already be 0, which is what we

01:08:03

need, that we will have, well, we will rewrite the left side and on the

01:08:11

right

01:08:13

6 us squared plus 6 cosine squared,

01:08:18

let's throw everything to the left

01:08:20

side and we will have some similar terms with

01:08:24

sine from cosine 4 sine -6

01:08:30

sine squared

01:08:32

it will be minus

01:08:34

2sin x squared 7 cosine squared

01:08:38

minus 6 cosine is just cosine squared

01:08:41

and the most important thing on the right will be 0

01:08:44

now we use our familiar trick

01:08:46

to get the tangent we divide by the

01:08:48

cosine only 1 tango target squared and

01:08:51

here is the expression squared, we divide by the

01:08:53

cosine in the square, this will allow us

01:08:55

to get sine by cosine when

01:08:58

the tangent is divided only in the square, but

01:09:01

here it’s one,

01:09:03

because now it’s easy to

01:09:06

divide them into different signs on opposite

01:09:09

sides of the equal sign, and

01:09:11

from here the tangent squared will be equal to

01:09:13

one half that is, 05 is what we

01:09:18

needed to find,

01:09:21

let’s now try

01:09:24

to do it in this similar example,

01:09:27

let’s immediately rewrite the left side as

01:09:31

it is, and

01:09:36

on the right, instead of the

01:09:38

eight, we’ll immediately imagine it as 8 of the

01:09:41

form squared plus 8 cosine squared,

01:09:45

that is, in fact, this is what it is eight

01:09:47

only now all the bikinis are on the left side

01:09:50

and we will have minus 2 sen squared

01:09:53

plus 3 cosine squared, we

01:09:57

divide our artificial technique by cosine

01:10:00

squared to get tangents and

01:10:03

just get tangent squared, and here

01:10:06

the three will be

01:10:08

spread on different sides 2 tangent

01:10:11

squared is equal hold and then the tango

01:10:14

square itself is equal to 3 second,

01:10:16

that is,

01:10:18

one and a half this will be the answer

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4 задание Формулы приведения - Курс ПРОФИЛЬ 2022 от Абеля / Математика ЕГЭ ДЗ №4: https://vk.com/doc81516960_622369515?hash=fea37abd2ee4cc9a38&dl=f978832a8b73e4f03a Плейлист Профиль 2022 / 1 часть: https://www.youtube.com/playlist?list=PLH... ___________________________ Группа в ВК по математике https://vk.com/abel_mat Группа в ВК по физике: https://vk.com/abel_fiz Вопросы предложения: [email protected] ___________________________

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