okay so now that we've seen the basics of how the double and the compound angles work now we're going to put it to practice i'm not going to only test you on compound we're going to do everything together now so here's the first question so with this one of course you could go and simplify the bracket inside that's absolutely fine many of you would prefer to do that but what we could also do is think of this as two angles so there's your first part and there's your second part yes i know there's a lot of stuff inside each part but that doesn't matter because what we know is that if we have the cos of a plus b then that can become something so on the formula sheet if you have to look at that that becomes cos a cos b minus sin a sin b so we're going to put we're going to use that template on this question so that's going to become where this part here by the way is a and this part over here is b so that's going to become the cos of 180 minus x times by the cos of 180 minus x and then it says minus it will say minus the sun of 180 minus x times by the sun of 180 minus x and so we know that from grade 11 trigonometry and please if you do struggle with how to reduce these remember when we have to use the cost diagram then you really need to go purchase the grade 11 course where i explain all of that from the basics up to the more complicated type of questions but for now i'm going to assume that you have a fairly good idea of how that works so cos of 180 minus x well we know that the 180 minus x quadrant is over here but that is where sin is positive so cos is going to be negative over this we say negative cos x that'll be the same for this next one then i'm going to say minus now we know sin of 180 minus x because 180 minus x is in the second quadrant then that just becomes positive sin x and so does the next part it's positive sin x now we know that negative cos x times negative cos x well that's just going to be positive cos squared x and sin x times sin x well that's just going to be sin squared x now we know that cos squared x minus sin squared x if you look carefully on your formula sheet at the part where we have all the expansions for cos 2x we can see that there's an expansion that goes cos squared x minus sin squared x yes there's also others like 2 cos squared x minus 1 or we could even use 1 minus 2 sin squared x but what we can see is that this part over here is over here and so if we want to simplify which means make everything more simple we can rewrite it as close to x so this whole green block over here is the same as cos 2x and that's the answer for that question so here's another question cos 75 and they ask us to do this without a trying i mean without a calculator so that typically means special triangles now once again i do cover special triangles in the grade in my grade 11 course i think i even have a grade 10 course on that so with the special triangles you've got a 30-60-90 triangle and if you write the 30 and the 60 in this position then your triangle will look like that then we also have a 45-45 triangle where the two sides there are the same because it's isosceles because those two angles are the same and then this over here is square root two okay so make sure you know that for the exams so cos 75 well we know that 75 is the same as 30 plus 45 so what we can do is we can say that that's the same as 45 plus 30 or 30 plus 45 it doesn't really matter what's nice about this and please at this point do not say cos 45 plus cos 30. it doesn't work like that i've seen students do this a lot instead we know that if we have a cos and we have two angles inside there then we can expand it using the compound angle formula that is on your formula sheet so please follow along on your formula sheet and make sure this is all making sense for you it should say cos 45 then with cos it stays cos cos then we have sin 45 and sim 30. with cos it's always the opposite of the the sign inside the bracket so that'll be a minus now we use our special triangle so cos 45 and also remember that you need to remember okay not everyone uses sokkatoa but this is just my way to remember that cos is the adjacent and the hypotenuse sin is opposite over hypotenuse i've seen other riddles that people use but just remember you need to know that so the cos of 45 so i go to a 45 degree i've got two options i'm just going to choose any one so i'll just go to that one and i know that cos is adjacent which is 1 over hypotenuse which is square root 2. so i replace this part with 1 over square root 2. the cos of 30 so i go to the 30 i know that cos is adjacent over the hypotenuse so that's going to be replaced with the square root of 3 over 2 minus the center of 45 i'm just going to speed things up a bit now that's just going to be 1 over the square root of 2 sin 30 is just going to be 1 over 2. and now simplify each part as far as i can now i know that there are students who struggle with this so remember when you're multiplying two fractions you can just multiply the top and the bottom parts together so the 1 times square root 3 is just square root 3 and at the bottom it's just 2 root 2. over here we're going to have 1 minus 2 root 2. now we are plusing or we are minusing so either one of those we so in this case we're minusing two fractions so when you subtract two fractions your denominators have to be the same in which in this case they are so we know that the answer is going to be written over 2 root 2 and at the top you're just going to have square root 3 minus 1. and square root 3 minus 1 we can't simplify that any further so this over here is your answer want to see if it's correct well what you do is the following you type the original question in on your calculator and that's cos 75 and chances are it's not going to look the same as what we have in fact i know it won't be okay so then people often panic but guys what you've got to remember is that a calculator does not include a square root at the bottom okay but what we can do to take advantage of this is we push the sd button and we have a look at what the decimal is worth and that's 0.258 now what we do is we type this part in on the calculator and we see if we get the same decimal value okay let's push equals push sd and there the decimal value is the same so then we have a 99.9 confirmation that it is correct but remember it won't look the same and that is why it is so important that you do this without a calculator i'm talking about this part over here because teachers know what to look out for and if you're just going to use your calculator they'll pick it up straight away and you won't get as many marks as you could have