okay video will be proving both the law of sine and the law of cosine and we'll just be focusing
on acute angles and the same for any angles in triangle right anyway let's start with law of
sine first let me show you guys a picture right here and I'm gonna label this to be capital a
capital B and capital C and this right here will be just it'll be and this right here little C and
this right here PDT oh hey all right so now this is how we are going to start first we pick a side
to be the base so let's go ahead and just pick this to be the base then we draw a perpendicular
line from this point and of course this is going to give us the height so I'll put on each right
here and actually let me write yes h1 right so now we'll have this two triangles right here and
because we are mentioning law of sine so focus on the side you see this right here is a right
triangle now so if you look at this angle a we can just figure out the sign for you for that so let's
see sine of capital a this right here of course we know sine is the opposite over hypotenuse so
in our case it would be H 1 over C and there's a reason why I'm using H 1 you'll see why all right
so this is what we have of course we can multiply the C on both sides so based on this we know that
H 1 is equal to C times sine of angle a that's good then we will be doing the same thing that
was this triangle do catch sine of the C right here well in this case this is Jo opposite and
this is the hypotenuse now so we have h 1 over a and again multiply the a on both sides we get H
1 is equal to little a times sine of the angle C very good now have a look H 1 is equal to this H
1 is equal to that of course this and that they have to equal right so if I make them equal to
each other this right here we will get C times a that will be little a times sigh see like that and
then of course we can just deep IDC on both sides and divide the little a on both sides then we can
say sine a over little a is equal to sine C over D toasty which is very good and we're almost done
but you know we are only doing this to the angle a and also and go see hopin copy and copies to
getting jealous that's not good so have a look here is angle a now let's go ahead pick this to
be the base and again with your perpendicular line but this time will be from here so this
is going to be another height isn't it well this time our label is sh2 this is Phi so this is h1
area right now we can do the same thing have a doctor if we look at a right here in this blue
triangle on the bottom right in this blue triangle on the button right here well let's go ahead and
figure out sign a this is going to be what well a person is h2 over the hypotenuse is little P
so we have this right here yeah similarly we can look at this be right here and we can say sign
capital B this right here will be equal to this time the a person's to h2 but the hypnosis dito
a so we have this right here and you can imagine we can do the same thing and yes we will be able
to get sang a over D to a to be equal to sine B over D topi this again as I said it's equal
to sine B over Ito B and this is equal to that so of course everybody's equal to each other so
that's it so in the enemy just put it down in the usual water right here so ladies and
gentlemen this right here is our law off side now right here we'll be looking at the law of
cosine so let's go ahead and draw the triangle yeah so you have a B and C and let's go ahead
and I bought these little B little C and little a the step it's pretty much the same but you had
to use what we call the Pythagorean theorem so let me remind you guys of course right here if
we have a triangle let's say and you'll see is the right triangle so here we have of course a
right triangle and let's put it down like this let's say this is little C this is D 2 a this
is C Toby well as we all know this right here C square is equal to a squared plus B squared just
like this very good this is the classic pass out the real zero and you do not have to use the law
of cosine to prove the Pythagorean theorem you can use similar triangle in fact can do over 100 ways
to prove this right by the way here we go this is how we are going to do the law of cosine first we
do exactly the same as that pick a base I will put this down right here to be our base I will
show you guys what C squared equal to in our case right okay and then I'm gonna draw a perpendicular
line from here so again this is going to be the hide this time dope I'm going to label this as b1
and this part is p2 I so b1 is going from here to here and b2 is going from here to here and next we
are going to focus on this popped triangle and do pretty much what it over there here's the angle C
let's go ahead and figure out what sine C is of course it's just the opposite over hypotenuse
which is H over a and again multiply on both sides we get H is equal to a sine C all right good well
this is the law of cosine so we will also have to figure what's the cosine of C well still focus on
this triangle cosine is adjacent over hypotenuse so we get B 2 over a and again that's multiplied
the a on both sides so we get B 2 being equal to a times cosine C now we use this triangle ready
that's go ahead and use this one and again the law of cosine is a generalized version of the
Pythagorean theorem for this triangle here we will be using the Pythagorean theorem again you
do not have to noted love cosine first in order to prove that so you can legitimately use this
in this triangle here right so by looking at this triangle we know that C square has to be equal to
this square plus a square right well they mean right down the H squared first right here and
then we add it with d1 square like this okay H of course is this already so let me just write
it down but what's be wonder what's p1 well p1 is only the difference here to here we know the
hosting from here to here B and this much is p2 which we know that right he already so I'm going
to look at the b-1s the whole thing which is B and then minus b2 and then of course we still
have the square and this is so wonderful because we do know this and we do know that and you will
see C 2 will be based on a and B and also cosine you will see all right let me write this down here
for you guys first we put on H and then that's this so I'll put on parentheses a sign see this
is the capital C all right and then we square that next we add B is just B right B is just B minus B
2 is this namely a cosine C and then right here we square that good Square this we get a square
sine square c plus well this thing squared we have to multiply the out so we get this thing
squared which is P squared let me actually write this part in red to make it clear for you guys
next we do minus 2 times this and that which we have to a B and then cosine see like this right
and lastly we have to add and we just put down we have to add a square and then cosine square see
like this again the first thing squared minus two times this and that and then the third part is we
have to add this quantity square which is a square plus cosine squared now have a look here we have
a square a square we can factor it out so we just left with sine square plus cosine square and the
best thing is that their input is both C so sine square c plus cosine square c is nice t equal to 1
so in another word with this part it's just going to be a squared times 1 oh this right here is that
a square all right and then for the rest you can see that this is just plus B Square and then minus
2 a B and then of course in the yet we have the cosine C like this and of course there are other
versions of the law of cosine depends on which side you want to find out earlier I found out the
C square which we get a square plus B Square minus 2 a B cosine C you can do the same thing with a
square and B Square right and just how to change the variables and change the arguments of the
picture like that okay that's it and as you can see C square is equal to a square plus B Square
with this part it doesn't for the cosine if C is 90 degrees cosine 90 degrees is 0 that will be
making this part equal to 0 and we get our usual passacaglia serie but yeah this is a generalized
version this right here can be done without using the passage without using the law of cosine like