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00:00:00

Hello and welcome to another video of easy matte in this video we are going to demonstrate some

00:00:06

trigonometric identities specifically we are going to demonstrate the Pythagorean trigonometric identities

00:00:13

these three trigonometric identities are called thus the one that relates to the sine and the tangent

00:00:19

and secant cosine and ccoo tangent and co secant To demonstrate these identities, we are going to start

00:00:24

with the first one, which is the most elementary, the one that is used the most, which tells us that sine squared

00:00:30

theta plus cosine squared theta is equal to 1 for any real number theta. Well, I

00:00:38

will demonstrate this identity. In two ways we are going to start with the classic proof that has

00:00:44

to do with a right triangle, which is how we start by defining the sine and the sine and then we will

00:00:50

extend the proof to include more angles than can be built in

00:00:56

a triangle, then we go to start with a right triangle like this one here we are going to

00:01:03

write the angle theta in one of these two angles the east of here remember it is the angle

00:01:09

of 90 degrees is often represented as a cuadradito then we will choose as angle

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theta one of these two'll put this one here it will then be our theta angle and

00:01:21

let's remember what are the definitions of the sine and the cosine the sine of an angle from

00:01:27

a right triangle is defined as the opposite leg on the hypotenuse while the cosine

00:01:33

of the angle is defined as the adjacent leg between the hypotenuse let's quickly remember

00:01:39

how it is that The sides of the triangle are called as we said this here the angle of 90 degrees and

00:01:46

this angle is formed by these two sides, these sides are called legs while the third side

00:01:52

that is the largest of the right triangle is called hypotenuse now well if we choose

00:01:58

as our theta angle is from here then the leg that is next to the theta angle is

00:02:04

called the adjacent leg while the leg that is in front of the theta angle it is called opposite

00:02:11

leg, well with that in mind we can start with the demonstration and for that we are going to suppose that

00:02:18

this right triangle has hypotenuse 1 we are going to put it like this for simplicity then we say that

00:02:25

the hypotenuse of this triangle is 1 now to the other two sides of the triangle we are going to put variables

00:02:32

in them we are going to say that this side measures x and we are going to say that this side measures y we could

00:02:40

put any other letter but well the usual thing is to put x and now using these formulas we have

00:02:48

that the sine of the angle theta will be opposite legs on hypotenuse that is to say it will make

00:02:54

between 1 then sine of theta is since it is the opposite leg on the hypotenuse that is worth 1

00:03:03

I have divided by 1s simply and when we divide a number by 1 it gives us exactly

00:03:09

the same then we leave it directly as ye we do the same with the cosine cosine of theta it

00:03:15

will be adjacent leg which in this case adjacent leg is x divided by hypotenuse which is 1

00:03:23

and again here x between e 1 is simply x now what we are going to do is use the

00:03:31

pythagorean theorem the pythagorean theorem remember that it tells us that the hypotenuse squared is equal

00:03:39

to the sum of the squared legs so in this case the legs are x and so If

00:03:47

we square them and add them that gives us the result of the hypotenuse squared which

00:03:53

is the 1 squared good then we have that x squared more than squared is equal to 1 squared

00:03:59

but from what we obtained up here sine of tts cosine of theta is x so we substitute here

00:04:07

instead of x we put cosine detect instead of we put sine so we have that cosine

00:04:13

squared of theta plus sine squared of theta is equal to 1 squared which is simply one

00:04:19

is from here is the Pythagorean identity, although in this case the sine appears first and then

00:04:25

the cosine, because remember that the sum does not matter the order in which we do it, it is the same to put

00:04:31

the cost first and then the sine than first the sine and then the cosine because they

00:04:35

are finally being added with this from here we have demonstrated the first of the Pythagorean

00:04:40

trigonometric identities and we see why they are called like that because they arise precisely from the Pythagorean theorem but

00:04:47

notice something here this angle theta is an angle of the right triangle and remember that The three

00:04:54

angles of a triangle add up to 180 degrees, we already have an angle of 90 here, therefore

00:05:01

this angle plus this angle must add another 90 so that the three give us 180 that means that

00:05:07

this angle theta cannot be an angle greater than 90 degrees. because this more this here should

00:05:14

give us 90 then this idea we have just demonstrated but only for angles that are greater

00:05:21

than 0 and less than 90 degrees, however we can demonstrate the identity for any angle both

00:05:28

positive or negative and it can therefore be a greater angle Let 90 or any real number, we are going

00:05:35

to do that proof, but for that we must remember the definition of sine and cosines

00:05:40

for any angle. Then we are going to start by remembering that definition, the definition of sine

00:05:47

and cosine for any angle is made from a circle, we start by drawing our

00:05:55

Cartesian plane, we have the x-axis and the y-axis and we draw a circle centered on the

00:06:01

origin that has radius 1, then if we We draw an angle here, always starting to measure

00:06:08

it from the positive x axis, for example this angle here this angle begins to be measured from

00:06:15

the positive x axis and ends here, do not give this blue point, we are going to call that angle theta

00:06:22

then this point in blue It is a point of the Cartesian plane, we are going to call it py, it has

00:06:28

its coordinates x comma, then the sine of the theta angle is defined as the y-coordinate and

00:06:36

the cosine of the theta angle as the x-coordinate, that is, the cosine of theta is equal to the sine of theta is

00:06:43

equal. a and e that is to say that if we draw here any angle on the circumference knowing

00:06:49

the coordinates of that point we can determine the cosine of the angle that s e formed and the sine of the

00:06:55

angle that was formed well this is simply to remember the definition of sines and cosines for

00:07:01

any angle I presented here the angle in the first quadrant but we could

00:07:07

draw the angle in any other quadrant and the definition is still valid Well with this in

00:07:14

mind we must remember what is the equation of a circle of radius 1 centered at the origin the

00:07:23

equation of a circle of radius 1 and center at the origin, that is, at 0 0 is x squared plus and

00:07:29

squared equal to 1 well here if we know that x is the cost not sucking and breast boob to the

00:07:38

substitute here on Xs and air because we have again the Pythagorean identity good is here

00:07:44

to remember that the equation of the circle is obtained by applying the Pythagorean theorem and that is

00:07:52

why this identity is called the Pythagorean identity, it must be remembered that the equation of

00:07:58

the circumference is obtained from calculating all the points in the Cartesian plane whose distance

00:08:05

to the origin n is equal to 1 and to calculate that distance is that the Pythagorean

00:08:10

theorem is used that I will explain in some other video in which we will find the equation of

00:08:16

circumference for any radius and any good center so with this we have already demonstrated

00:08:23

the first Pythagorean identity for any angle because in this case we can already represent

00:08:28

positive angles and negative angles of any value and we still need to prove two more Pythagorean

00:08:35

Pitot identities but those other two identities already emerge very quickly. From

00:08:41

here we are going to start. demonstrating the first of those identities for that what we are going to do is

00:08:47

divide this identity which is finally an equation we are going to divide it by cosine

00:08:53

squared of theta that means that each term of the equation we are going to divide by cosine

00:09:00

squared of theta then we have square sine of t is divided by square cosine of theta

00:09:07

then square cosine of t is divided by square cosine of theta and on the other side of the

00:09:12

equal sign we have 1 divided by square cosine of theta and now we are going to use the following that

00:09:20

is not detected between cosine of theta is equal to tangent and that 1 enzo the non-theta cost is equal

00:09:25

a secant beteta then here as we have sine between cosine that will be tangent but since

00:09:32

the trigonometric functions are squared then this will give us tangent to the square of theta

00:09:38

then here we have squared cosine between squared cosine as we are dividing the same quantity

00:09:43

by the same amount because that gives us equal to 1 and on the right side we have one between the cosine

00:09:49

one between cost is not secant and since it is squared of the cosine because it will be secant

00:09:55

to the square and thus we have obtained the second of the Pythagorean identities for us A third

00:10:01

identity is missing and that we are going to obtain by dividing this identity by the square sine of theta

00:10:09

then again starting from this identity we are now going to divide by se not square of

00:10:16

theta each of the terms of the equation and we have the following sine squared between

00:10:22

0 squared then here cosine squared between sine squared and here one between sine squared and here

00:10:29

we are going to use now that they do not detect between sine of theta is co tangent of theta and

00:10:36

that one between the sine of theta is co secant of theta so here square sine of theta between

00:10:42

square sine of theta gives us 1 because we are dividing the a quantity by itself

00:10:48

that gives us 1 then cosine between sine as We said it is co tangent and since here they are

00:10:53

squared because it is with a square tangent of theta and then here one between squared sine

00:10:59

because as one between sine is with secant because we will be here with secant squared

00:11:05

then this is the third identity and so We have finished if you want to see other trigonometric

00:11:11

proofs in the description of this video you can find the link to the complete list

00:11:18

of proofs of trigonometric identities so I invite you to Let them see it and if you liked

00:11:25

this video, please support me by giving me a like, subscribe to my channel and share my videos

00:11:30

and remember that if you have any question or suggestion, you can leave it in the comments.

Description:

▼ ВАЖНО ▼ В этом видео я продемонстрирую 3 тригонометрических тождества Пифагора, квадратный синус плюс квадратный косинус, тангенс, котангенс, секанс, косеканс, используя прямоугольный треугольник, а также из определения единичной окружности в декартовой плоскости, все объяснено пошагово Он прошел. # тригонометрия # геометрия # демонстрация ---------- ** ВАЖНЫЕ ССЫЛКИ ** Курс тригонометрии: https://www.youtube.com/playlist?list=PL9SnRnlzoyX3kLYWUsrmrq0qy2Bye1JF4 Специальные видео: https://www.youtube.com/playlist?list=UUMOHwtud9tX_26eNKyZVoKfjA Обзорный курс математики (довузовской) https://www.youtube.com/playlist?list=PL9SnRnlzoyX1-FFtFcUupLSdnTRvs8B5K ---------- ** СМОТРЕТЬ ВСЕ МОИ КУРСЫ ЗДЕСЬ ** https://www.youtube.com/c/Arquimedes1075/playlists ---------- ** СПИСОК ИСПОЛЬЗУЕМОЙ ЛИТЕРАТУРЫ ** - Тригонометрия, Своковски Коул - Геометрия и тригонометрия, Baldor - Упрощенная математика, Конамат ---------- ** ПОЖЕРТВОВАНИЯ ** - Paypal: https://www.paypal.com/donate/ - Членство в каналах: https://www.youtube.com/channel/UCHwtud9tX_26eNKyZVoKfjA/join - Патреон: https://www.patreon.com/matefacil ---------- ** МОИ ДРУГИЕ КАНАЛЫ И СОЦИАЛЬНЫЕ СЕТИ ** - Канал по физике: https://www.youtube.com/channel/UCeFNpG-n8diSNszUAKaqM_A - Канал видеоигр: https://www.youtube.com/channel/UClSpw-rlRdygJmI33x1YagA - Twitch: https://www.twitch.tv/matefacil - Приложение MateFacil: https://educup.io/apps/matefacil - Facebook (Страница): https://www.facebook.com/unsupportedbrowser - Twitter: @Matefacilx - Instagram: @Matefacilx - Раздор: https://discord.com/invite/Gmb7sF9 ---------- # Math #tutorial #tutor #tutoriales #profesor ----

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