hey guys hope you're well this is the very first lesson of grade 12 trigonometry so compound angles your teacher's been showing you these weird looking uh formulas in class they look pretty intimidating don't they I remember the first time I saw them I was like okay math let be looking good right now anyway so I'm going to be showing you exactly how to use these formulas they're actually very straightforward they always going to give them to you in your um tests and exams so you just need to know how to use them okay and in this lesson I'm going to show you how to use them going from from when it looks like this and I'm going to show you how to make it look like that but then I'm also going to show you how to make it look like how to take it from this and then make it look like that and then we're going to do a bunch of stuff with um special angles like 30s and 45s and 60s okay so we're going to be doing things like expanding then we're going to do things like simplifying and within all of those examples that I've just shown you there's going to be some interesting things popping up all right so let's have a look all I want you to do is just look at this over here and try to tell me or you can't tell me but you can just like in your head try to figure out which one is it does it look like number one does it look like number two does it look like number three or does it look like number four well it's number four because look it's got a C and then it's got some angle minus another angle here we've got cos some angle minus another angle all right so then we ignore all of this stuff okay we ignore that one ignore that one ignore that one and then I want you to just focus your attention on this one over here and I want you to do your best to take it from what it looks like over here and now rewrite it okay so there's a equal sign then it says we must put cos okay so we're going to put cos then it says that we must use this angle which is this one over here okay so that's going to be um for us it's going to be X okay easy so far then the next thing is it says we must put another cause you can put the little dot there but that's not necessary we must put another cause Okay and then we must put the other angle so what is the other angle it's that one over there so for us that's going to be a 10 so I put a 10 then what comes next then it says there's a plus okay so I put a plus and then it says s all right and then it says um this angle again which is this one which for us was X cuz that was our first angle and then there's another sin and then it's the other angle which for us was a 10 and that's it okay so yeah might still be feeling a little bit uncomfortable a little bit weird don't worry that's normal uh we're going to do a bunch of these now so let's go do another one so once again all I want you to do is look at that over here and then try to identify where do you see it over here and then just copy so if you see it's got a sin okay so it's definitely not going to be this one and it's not going to be that one so it's got a sin then it's got a plus in between ah look at that so it's not going to be that one so we're going to just copy this now so we're going to say equals then says you put the you put the angle sin not the angle you put s and then the first angle is this one so that's going to be a 20 for us then you put the word c and then it's it's this angle beta which is this one so for us that's a y so we put a COS y okay then the formula says put a plus so we put a plus then we put cos why because okay that was lame okay so um this one over here so now is this angle over here which for us is the 20 can you see this is like our Alpha and this is like our beta so we're just going to put um 20 and then the next part says sin okay so we put sin and then it says the Sin of beta which is a y for us and there we go that's all that you need to be comfortable with right now now as I said as we move along these questions are going to start getting a bit more interesting like this one over here it's going to get quite weird but also quite fun to be honest uh okay Kevin that's lame trying to tell people that are learning something for the first time and it looks terrifying and you're telling them it's fun don't be cringey please woo what happened there okay so let's move on so sin of 50 minus W so all you're going to do is you're going to look for so it's definitely one with sin okay so it's not this one and it's not this one and then it's got a negative so it's not going to be this one cuz this one has a positive so this is our um the one we need to use over here so so all we do is our Alpha is that angle there and our beta is that one over there and now you're just going to go copy this over here so you're just going to say sin and then what are the I don't know why I keep using a capital letter sin um then you're going to use Alpha which in our scenario was 50 and then COS of um B which in our scenario was a w and then we're going to put a minus and then it says C then it's Alpha again which for us was a 50 and then sin of um the B angle which for us was W all right I'm hoping by now some of you are starting to feel a little bit more comfortable and those thoughts of math literacy are slowly but surely fading away all right so let's go to this one now so um this one has a cause so it's definitely not going to be that one not going to be that one and it's got a plus so it's not going to be this one so it's going to be this one down here so the alpha angle you can see over there so I'm just going to say that that's the same as um the alpha angle and then the beta angle is usually over there now you're just going to go um copy this down literally like that so it's just going to be cos then they're saying it's the alpha angle but for us that is a t then they're going to then they put a COS again so we put cos and then they put um this which for us is a five all right all right fromberg okay and then and then um you're going to say um minus sin then it's the alpha angle so the alpha angle is a t and then it says sin again and then you're going to say um the beta angle which for us is a five and there we go so now let's keep moving on now it's going to start getting interesting so all I want you to do when you get something like the this I want you to only think about special numbers what are the special numbers that we know 30 45 and 60 all I want you to do is can you make this number using these numbers now Kevin what are you talking about dude are you okay well what I mean is the following I want you to take the number 75 and break it up into any of those numbers so for example I could say 45 + 30 why because that does give me 75 ah now all of a sudden what I can do is realize that I've got this one over here can we agree it's a cause and it's got a plus so here has a cause with a plus so now what I can do is I can go and expand it into this so remember that this is Alpha this is beta so here's our Alpha and here's our beta so now I'm just going to go expand it so it starts off with um cos over here so cos then it's the alpha angle which for us is a 45 then it's cos again and then it's the beta angle so for us that is 30° so we just do that then we're going to say minus sin now Alpha angle again which is 45 sin again and then the beta angle which for us is 30 now we know that each of these from um those are special those are special numbers now back in the day um they used to be very when I started teaching and all of that they were quite strict with this thing of special triangles some of your teachers maybe still use special triangles however um and you must just check what your teacher prefers but what I've started noticing on the official memos for like your metric exams now is that they don't require special angles anymore what they want you to now do what it looks like you can do is you can just go type cos 45 on your calculator so you just go type cos 45 on the calculator so I'm going to do that quickly and that's going to give us < tk22 okay then go type in cos 30 on your calculator so what is cos 30 just type it in on your calculator quickly and that'll give you < TK 3/2 and then so as long as you just do each one individually don't what you don't want to do is don't go write this whole expression on your calculator they're not going to give you marks for that go do each one by itself okay so what is sin 45 that will give youare < TK 2 if you're using special triangles it will your this will look different for that I think you would actually get 1/ < tk2 but mathematically these two things are actually the same so some of you are just going to be using the calculator which I think most of you are going to be doing and then some of you are going to be using special triangles and the answers are going to look or this part here is going to look different but mathematically it's still the same okay and um so now what you do is you're just going to go handle this part here so mathematically you multiply top and top together not together together luckily I don't teach English and so we multiply these two together and that's going to give us square otk of 6 then you multiply these two together and that's going to give you four then you're going to multiply these two together that'll be sare < tk2 then you're going to multiply these two together and that'll give you four now because these two have the same denominator you don't cancel them we're not busy with an equation you can just say that the answer will be over it's almost like saying 1 3 plus or no let's say rather say 1 6 + 3 over 6 what does that become 4 over 6 so you just keep it like that so what is this going to be well don't say < TK 4 um it's just going to literally be < TK 6 - < tk2 you can't really simplify that and so there's your answer and what's really cool is you can go type this on your calculator and you'll see that your calculator is literally going to give you that awesome so let's move on okay so here has another one so when they just give you the single little angle over here now this one's pretty cool the reason is I said that you must only use these now this one you could say 60 - 45 but some of you are like Kevin you could also say 45 minus 30 yes you can so which one should you choose well you have to take there's a special coin that they're going to give you in grade 12 um it's called a special angle coin it's it's literally a five round that your teacher just puts like a little tipic dot on it and what that does is um the tipic just allows the coin to flip nicely and so in your exam you'll take that coin out and you'll flip it and then whatever your coin lands on I'm talking such nonsense whatever your coin Lads are that will help you to decide if you should choose 60 minus 45 or if you should choose this one okay guys I'm talking nonsense choose whichever one you like mathematically it is going to to give you the same answer okay sorry if I'm being annoying so you're just going to go choose any one of them so I feel like choose okay I'm just going to quickly take my coin out of my pocket has it got its tipic yes it does okay flip the coin boom okay 45 - 30 45 - 30 all right and so um but you could have chosen you could have chosen 60 - 45 it really doesn't matter okay so um so it's definitely not this one and it's definitely not this one and remember to stick around for the rest of this lesson cuz we're also going to learn how to simplify you see how we've been expanding we also need to learn how to simplify so how do you take this and then make it smaller and we're going to do it like that we're also going to do ones with special angles so yeah still a few more exciting things coming up stick with me so um that's going to be uh which one did we say the one with the minus it won't be that one okay so this is going to be Alpha this is going to be beta now you're just going to go copy this all right and so that's going to be um I really think I come from job I keep saying all right BR Ben you for a warmup okay so uh sin 45 sin 45 and then cos 30 I'm literally just copying this partt all right and then minus um cos 45 and then the next one is sin so I say sin and then it's the beta angle which is 30 there we go now remember each of these are special um triangles so go ahead use your calculator and sin 45 will give you that cos 30 would give you that and then cos 45 would give you that and then sin 30 would give you that okay once again multiply the tops multiply the bottoms then multiply the tops swe we're going to get the same answer as the previous one what was the previous question cos 45 oh yes cos 45 and sin 15 are the same because of co- functions Kevin shut up dude no one cares minus < TK over 4 and then you're going to end up with um just a four at the bottom and then < TK 6 - < tk2 when I said the whole co- functions thing some of you are like nerd okay so there's the answer for that one let's move on now we're going to start learning how to um simplify so now I'm giving you this partt or this partt or this partt or this part whereas the first partt I was giving you in the first examples I was giving you that see the difference Okay so we've definitely got a C cause so it cannot be this one and it cannot be that one we need one that has a cause cause so here has a cause cause and here has a cause cause all right then there's a plus okay so we want the one that has a plus or there's the plus now don't get confused don't say say yeah but Kevin there's the plus no look here we're looking for the plus over there in between them so it's not so it's not going to be this one cuz this one has a minus over there so we're not going to use you um we're going to use this one so we're going to take this and we're going to convert it back to there so it says that the answer is going to be a cause you see there's a cause here then it says I must open a bracket okay and then it says that I should take um this first angle so I'll just take that angle there which in our situation was an 80 and then it says that I'm must put a minus now okay so I'll put a minus now and then I'm must take a um the next angle which was this one over here and so that's a w done okay not too bad we're just doing it the other way around now let's try this one so it's got a c c again okay so it won't be that one and it won won't be that one um it could be any one of these bottom two so it's got a COS C but then there's a minus in between so there's a minus in between so we're now going to use this one okay it won't be this one because this one has a plus we don't want that all right so now we're going to say um so the answer definitely becomes C there we go open a bracket they say and then they say take the alpha angle which is this first one so that's going to be a 50 for us then they say put a plus okay so we'll put a plus and then you put the other angle which for us is a 10 and look at this this will give you what is 50 + 10 um let's quickly whip out the calculator 50 + 10 ah my calculator says it's running out of memory it can't hold that many values uh 50 + 10 guys I'm just joking 60 hey 60 um and then cos 60 is a special angle go ahead type it on the calculator and it'll give you a half off let's move on let's keep practicing these we've still got another two more this one and this one okay so this one has a sin and a cause Okay so it can't be this one because the only one and and it can't be this one because this one has a cause cause but we looking for a sin cause so here we see a sin cause here we see a sin cause so which one do we choose well have a look here there's a minus ah so we look at that there's a minus okay so we can sare this one or take that one away so what does the answer become well if you look here it's got a sin then they tell us please open a bracket so we open a bracket so what goes in the bracket well it starts with Alpha which is this part here which in our scenario is a t see there's a little T angle there so we put a t okay then there's a minus so we put a minus then the next part is beta which is this part here so for us that'll be 20 and there's is the answer let's do one more and well done if you've made it this far you awesome okay so here we have a sin cause Okay so that means it won't be this one and it won't be this one and then it's got a plus in the middle here we've got a plus in the middle so it won't be this one okay so it's going to become sin and then it says that see cuz we have this right we have that ours is just a 10 and a 20 and a 10 and a 20 so you see how um it says that we must put Alpha so that's going to be a 10 plus and then you must put a beta which is a 20 and then that becomes sin 30 30 is a special angle type it on your calculator and you get a half