okay let's do some math for fun here we're going to calculate sine of 10 degrees times sine of 30
degrees times sine of 50 degrees times sine of 70 degrees and yes we are using degrees and also if
you are like you can put up sine of 90 degrees but that's just equal to 1 so it doesn't really matter
right but anyway you should pause the video and try this first okay how did you get to it
alright let's see 10 30 50 70 yes there's a nice pattern for that so hopefully there's a way we can
figure this out right and of course I'm not gonna use a calculator otherwise you defeat the whole
purpose of doing math hmm I don't suggest any like good identity that we can use at the moment
I don't use the product to sum whatever identity the only thing I know at the moment that sounds
dirty kweisi good 1/2 okay but no I don't know I know we have the tap or anko identity and maybe
the triple angle identity right that's a triple and go pad no doesn't really help because this is
to that it's not a point oh it's not your point and go so yeah let's straight on the top enjoy
tunity on the side to see what we can do we know that sigh because we're talking about science is
U sine sine of two theta this right here is equal to two sine theta times cosine theta okay right
and then that's a double angle but this is they are not twice of each other whatever okay keep
in mind whenever we're sine cosine should also come out to help us out right whenever necessary
and this is the body call function identity so for example if you're talking about sine theta
we can rewrite it in terms of cosine as long as the angles are to be 90 degrees because they are
the complementary angles of each other that's a con function identity so when we have sine of 10
degrees we can write this down as cosine of 80 degrees 10 plus 80 is 90 okay so this and that
are equivalent likewise thus to the same for this I will write this times cosine of say degrees and
then I will write this as cosine of well I can do 19 minus this which is 40 degrees and nasty this
is the same as cosine of 20 degrees and now when we look at the cosine version of this question 20
40 60 80 now this is double this is was two times two times right not read a path from here to here
stab Oh from here to here is double okay then I skip this because I know this actually this
is one half so I can't care this about this right but yeah this is top of that this is top of that
and maybe I should write down the top Oh inca formula for cosine but no because i will have
to deal with a square write cosine two theta in for some kind of sign square what cosine square
right let's focus on this first well I'm looking at this question in terms of cosine now so that's
I saw like the cosine right here and I can just divide this on both sides so in another work I
can look at cosine theta is equal to sine of 2 theta divided by 2 sine theta right and now here
is the deal let's look at the first one right here for example 80 degrees right that would be
the theta and I will just plug in daytime to here and here and I was set up and cook them for for
that okay so this right here when theta is equal to 80 degrees I will plug in right here and here
on the top two times 80 we get 160 so this is first sign of 160 degrees over 2 times sine of
this angle 2 on outside and the sine of this angle right here okay and now let's do the same for all
of them even though I know this is 1/2 but up to you guys if you guys want to say this is 1/2 let's
go ahead I don't have why not uh ok let's do this right here cosine of 40 degrees that means theta
is 40 in this case right 40 so I'll plug in 40 40 there I will get sine of 2 times 40 which is
18 now over 2 times sine of this angle which is 42 we see if happens this and ah ha ha they are
the same we can cancel them out right but let's write this down first as well 20 degrees that's
the theta now so you know the deal on the top is shy of 2 times this angle which is going to be
40 degrees on the bottom it's going to be 2 times sine of just this angle which is 20 degrees here
is the fun part sign up 80 degrees and this now 80 degrees cancel each other out shy of
40 degrees cos always that sine of 40 degrees okay now what do we have left sign up 160 degrees as 20
degrees hmm what can we do well in fact they are the same let me show you here is the picture for
it remember signs the white value on the unit circle okay so if you just look at this right
here and I say we have 20 degrees this is right here okay I know the picture is not perfect but
let's say this is 20 degrees this right here the Y value it's a sine of 20 degrees well 160 degrees
is exactly just like on the other side like that okay and the truth is they will have the same y
value 160 degree means you go from here to here and that's just a nice reflection of Leven right
because the 20 happens to dry here as well so this is sine of 160 if you look at the wife over here
and it's a same height same y-value so in another work sine of 20 degrees actually equals to sign of
160 degrees and a Delius if this two angles add to be 180 degrees then the ascent values will be
the same okay if you want to see a identity that's this one sine of theta it's equal to sine of 180
degrees minus y over the theta is okay but anyway this and that can be canceled as well that's all
I'm trying to say because of this okay now on the top we have nothing but just one on the bottom we
have two times two times two times two of course that will give us 60 finally the answer is just 1
over 16 that's it it's so much fun but all this right so yeah that's it hopefully you guys like
this video and if you're new to my channel please subscribe and let me know what you can stick and
that is not the way to do this but I will save that for another time by the moment that's it