let's do some math for fun and this time we'll do some trigonometry questions we are going to
evaluate cosine of 20 degrees times cosine 40 degrees times cosine of 80 degrees huh of course
we are not going to use the calculator otherwise you will defeat this great question right and in
same side this question its well-designed it otherwise nobody can do it right I don't think
if you just want me to do cosine of 20 degrees I will be able to figure out whatever use I don't
think that a nice number in terms of radical what in terms of decimal what not right but all these
trees thing is the any connection yeah because we see that 20 degrees 40 degrees and 80 degrees
just doubled right maybe we should look at the double angle formula and that's talked about
it right here on the side first of all we know the top angle formula for cosine Seoul anybody
talked Rafael to theta this like here we have to be fresh little bit and then just regular first
pressure which is cosine square theta minus sine square theta but this is not good because yes I
am dealing with cosine however if you look at the right hand side we are working with
squares that's no good that me avoid that but we also have another power angle formula and up for
sine let's look at final two theta we know this is going to be 2 sine theta times cosine theta
but in here we don't care about sign right we care about cosine it's okay let's divide both
side by 2 sine theta in another word we can look at this equation here and we can say this is the
same as cosine theta equals two sine two theta on the top over 2 sine theta like this and I think I
would like to use this one instead I will change all the cosines into the science right here and I
see what happens so for example for the first one theta it's going to be the 20 decrease right here
in terms of the side this is how we're going to get so I will plug in 20 decrease into the data
here on the top we will have sine of two times 20 degrees that would be 40 degrees so this is going
to be sine of 40 degrees right and then with the body by 2 sine and once again the theta is
20 degrees like this and we multiply by next here this angle is 40 degrees and we'll plug into here
and here right so that means we have sine of 2 times 40 which is 80 degrees over 2 sine of
this angle which is all he decrease to see already huh because this and that cancel that's very nice
right and let's look at the third one and here we have angle theta is equal to 80 degrees plug
into here in here so we will have sine of 160 degrees 80 times 2 right so that's 160 degrees
over 2 times sine of 80 degrees like this what's happening is this and that cancel out listen that
can solve so nice right I didn't even know what sine 40 degrees by it's okay because it doesn't
matter anymore and that's the altogether what we have on the top we have sine of 160 degrees over
on the bottom we have 2 times 2 times 2 which is going to be an age and then this register
survives oh my god sine of 20 degrees and now watch what this compare to that look at this is
one 260 degrees and this is 20 degrees they add up to be 180 right in fact they are the same and
there's our identity I want to put it down for you guys whenever you have sine of theta if this right
the odd angle when you add this and that if the angles about to be 180 it's actually simple sine
theta is the same as saying sine of 180 degrees minus which ever the angle was and this is also
another identity we have to use so you can look at this right here's the data we respond to 60
degree right and let me register you note this right here it will be the same as sine of 180
degrees minus 160 degrees which is 20 degrees so all together on a tablet style also 20 degrees
over 8 sine of 20 degrees and so of course final answer 1 over 80 without a talk later cool huh