you
okay I will show you guys how to find the exact value for sine of 18° with a special
right triangle and we will call that to be the 18 72 90 special right triangle I will show you guys
how to cuss try it maybe you have a CMAs before but definitely check this out because this is
going to be really really cool so first of all on the start with a triangle with the angle to be 36
degrees and then two of the sides to be the same all right and hopefully my picture looks
okay right here so I will put down a 36 degree angle right here I don't want the angle to be 18
yet alright I want this angle to be 36 degrees and I want these two sides to be the same and
the easiest number form to consider it will be just 1 and 1 I can totally do this because I'm
constructing the triangle because these two sides are the same we know that once I connect a third
type which is right here this two angles will be the same as well right and of course we can
just go ahead and do 180 minus 36 which is 144 and then divided by 2 that means we can get this to be
72 degrees likewise this will also be 72 degrees and of course now what we did I saw this triangle
which is prettier here and now ideally speaking if I can figure out what this is I will actually
be done because even though this is now right triangle the moment I cut this crying happening
to the middle I will get the 1872 90 special right triangle for you guys right so I really
want to figure out how long this is and now they may get a quote for my subscriber right here which
he says that the greatest trick in mathematics is that the greatest trick in mathematics that
whenever we don't know something we are just going to go ahead and cut out to be X so right
here because I don't know what this is let me just go ahead and say this is X right here okay
and now what else can we do well as we discussed the earlier I'm going to cut this triangle right
in the middle like this in a minute right that's how we can get the 18 degrees but
what if I decided to cut this right here right in the middle as well let's see what do we get here
we have 72 degrees notice when I cut this right in the middle like this we can watch this angle
which is going to be half of 72 and that's going to be 36 degrees right and more importantly this
side will be what will also be X isn't it because you see we have 36 degrees right here this is 72
and you know this angle it's also going to be 72 degrees right because in fact the triangles are
happening in red almost in red this right here is similar to the original triangle this is so
cool and now let me draw the red triangle on the side for you guys so right here you see X and X
let me just put this down right here X and X and this right here I was just keeping black which I
don't know what it is yet and you see this angle was the 36 degrees and then these two angles are
still 72 degrees right so let me just indicate that 72 degrees here likewise 72 degrees here
this two triangles are indeed similar because all the angles are the same isn't it and now
what else can we do from here if you look back to the train going black right here we have 1 1 and X
right here we have X X and this right here I don't know but remember this piece comes from here right
what other information can we put down though check this out here is the beauty of using 36
72 72 because the moment that we cut this into half here we get 36 likewise here we also have
36 degree right and check this out this is 36 that's also 36 once again these two angles are
the same that means these two sides would be the same isn't it here is X that means this die will
also be X isn't it so now up to this time blue this right here it's also X okay even though my
picture is not perfect but just a number trust a math all right believe in the math I'm not using
Wolfe on help on this case so just believe in the math okay so here is X remember the whole thing
is 1 I know this much is X already so how big is this let me just do this in black as well oh go to
the simple point naught how BIG's this the whole thing is 1 this is X so I know this right here
has to be 1 minus X isn't it this is not this is blue by the way this is 1 minus X ok now you know
this piece it's the same as that so you must have this piece right here being 1 minus X as well so
we have two similar triangle what can we do we can stop this by setting up proportion isn't it
and I will just go ahead and do not for you guys depends on how you want to do it as long as you
do it consistently you'll be okay so we set out a proportion I will just do this die over that side
namely x over 1 okay we must have x over 1 from here to here it has to be this over that okay so
of 1 minus x over x once against this over that so we have the 1 minus x over that X and that's
the equation that we have of course you can just go ahead and solve the equation you can cross
multiply if you will like so this is going to be X square and then that would be equal to 1 times
2 which is 1 minus X put everything on one side so you have x squared plus X minus 1 is equal
to 0 and now what quadratic equation isn't it so you go ahead and say this is X equal to negative
B which is 1 so you have negative 1 like this plus minus square root of B squared which is 1 squared
minus 4ac a is 1 and C is negative 1 and this is what we have inside all over 2 times a which is
1 all right so that's what we have right and you will notice ok we have negative 1 plus minus no
problem at the moment open the square root and check this out 1 square is 1 minus times minus
is plus and 4 times 1 times 1 just numbers wise it's four all together is 1 plus 4 which is 5 so
in size 5 over 2 times 1 which is of course just a 2 we have two answers for X but only one makes
sense because we are talking about triangles if you have negative 1 minus square root of
5 it doesn't make sense because this right here will give us a negative result we cannot have a
negative side in a triangle right so we are going to ignore negative so only consider only counts
either X equal to negative 1 plus square root of 5 over 2 what is this this is X which is the same as
that and now I'm just going to draw that picture again so the same picture over there we like this
put this down in red this right here it's a 36 let me not do that actually though why not let
me just do that 36 degrees 72 degrees 72 degrees this right here is that which is negative one plus
square root of five over to this size what that's is one because I got it from there now what as I
said earlier I will cut this dragging the middle so now cut this right in the middle this is my
ninety degree and now check this out I don't know what this is you could figure it out by
using the Pythagorean theorem or two sine cosine whatsoever whatever you would like right okay I
will have this triangle here this is right here which is now 18 degrees and this that is still
equal to 1 and right here this is half of the whole thing earlier so instantly buddy by 2 only
we will have to divide this by 4 altogether right so altogether you have negative 1 plus square root
of 5 over what because this is just half that so it's not just over 2 it's actually over 4 and you
have the 72 degrees right here and here you have it this is almost the complete version of the 1872
90 special right triangle we're all from here you can go ahead and do sine of 18° how do we do that
here is the 18 degrees you can go ahead and say that's the opposite which is right here which
is this negative 1 plus square root of 5 over 4 that's the opposite side over the hypotenuse which
is over 1 right of course the open one doesn't matter so ladies and gentlemen let me just write
this down finally for you guys sine of 18° equal to just that we negative 1 plus square root of 5
or over 4 and I know somebody is about to comment down below and tell me what this number is related
to the golden ratio right this is so cool so amazing so much fun and as usual that it's it