in this video we're going to look at how to solve trig equations all right so at this point of the year the idea of an equation makes sense to you it's got an equal sign you solve it a trigonometric equation just has synchros and tan involved so there's a nice structure to follow in solving these things the first thing you typically want to do when you've received an equation is you would like to simplify so you want to simplify the equation as far as possible sometimes that means that you will eventually have to factorize so after you've simplified you might need to factorize many times you won't you're then going to get a reference angle okay now each school might sorry different teachers or different schools might call that something else the reference angle is the thing that you get when you use the calculator typically on the casio calculators you would say shift cause i mean not shift cause shift and then you would type in the angle for like sin or cause or tan then you would choose your quadrants choose quadrants and then solve okay so we'll remember those five steps going into the next slide which has a couple of examples so here we have seven examples over the years i have seen all the different types of general solutions that they can ask and i've included most of them over here and so each of them is going to be slightly different and they'll be testing various areas that you have to encounter or they'll be you will encounter all the different types of ways that they can ask the question something i'd just like to quickly mention is that some of them have a interval given whereas some of them don't okay so this one has an interval this one doesn't this one has an interval this one doesn't and this one also doesn't so the ones that don't have an interval we call that the general solution whereas the ones that have an interval are called specific solution they both get solved in the exact same way it's just that with the specific solution which is the one with the interval you have to do one small step at the end but up until that point both methods are actually the same thing all right what i also want to do is just quickly write down that five-step procedure that we wrote down on the previous slide so there on the right hand side we've written down the five different steps so let's start with number one they tell us that sin x is equal to 0.4 now the first step says that we should simplify okay well this one and with practice you'll see this one already is simplified you can't you can't move anything to the left to the right you don't need to divide number two says factorize if possible well everything is factorized answer number three says get the reference angle and so it says sin x equals to 0.4 so what it's saying it's saying sin of what angle gives you 0.4 so you type on your calculator on the casio calculators this is where you would say shift and on the sharp calculators you might have to say second function you'll say shift then you'll say sin and you'll type in 0.4 and if you do that on the calculator you're going to get a reference angle of 23.58 degrees just remember your calculator needs to be on degrees and not radians so sometimes our calculators do slip into radian mode maybe you push something by mistake it's usually not that easy to switch it to radian mode but i have heard of students who have gone into an exam and got everything incorrect because their calculator was in the wrong setting all right so there we've got our reference angle then the quadrants okay so it's saying that sin is equal to 0.4 they're telling us that sin is something positive now remember from your caste diagram we know that sin is positive in quadrants number one in quadrants number two and so we'll answer this question in quadrants number one and quadrants number two okay so we've chosen our quadrants now it's time to solve so we start off by looking at what's inside here okay so it's just a normal x sometimes it says other things there like x minus 5 or 2a or 3c or whatever so you just need to look what's there so it's x equals and x equals then for quadrant number one it doesn't say 180 minus 180 plus or 360 minus so you just start off by saying the reference angle you then say plus k times 360 and then you should say k is an element of z now i've used k some schools use the letter n it doesn't really matter all right so that's for quadrant number one for quadrant number two you would say 180 minus and then 180 minus what well that's your reference angle so 23.58 and then also plus k times 360 k is an element of z then you would just simplify that part and that's going to give you 156.42 plus k times 360 k is an element of z all right so our two main answers for this question would be that over there and that over there now remember this question is within a specific interval okay so we have to give specific answers not general answers general answers have the k360 at the end so what we now do is we can start off by looking at this one over here and if k was equal to zero so i'm going to give a few numbers for k and i'm going to give a few numbers for x so if k is zero then if you plug k is zero over here you're just going to end up with 23 23.58 if k is now remember k can be numbers such as 1 2 3 0 minus 1 minus 2 any type of integer that's why we say integer over there so let's try minus 1. if you plugged minus 1 as k you would get 23.58 minus 360 which is -336.42 now that isn't going to be correct i'll explain why just now if k was 1 then it would be 23.58 plus 360 which gives you 383.58 but if we scroll up to the original question the interval only goes from 0 to 360. so both of these are invalid okay so for that one there that's the only answer for that and then we're just going to look at this one over here so we'll choose a few k values they can be totally different k values to what you've already chosen and then let's choose x so if k is 0 then you're just going to get 156.42 if k was -1 you would end up with negative 203.58 and if k was 1 you'd end up with 516.42 both of which are invalid and so the only answers for this whole thing would be 23.58 and 156.42 and so you would typically show your answer like this you would say x is equal to and then some teachers do this funny bracket it doesn't really matter 23.58 and the other answer is 156.42 and that would be your final answer okay so that's the specific solution and then i just want to mention something with that last part we didn't choose any further k values because choosing any other k values we're just going to go further and further out of the given interval and so we stopped all right so now we're going to do number two it starts off with sin x minus 20 equals to minus 0.3 so the first step we should do is to simplify that is as simplified as possible there's no numbers that can move across there's no numbers that can be divided so that is simplified it's also factorized there's no factorizing possible there so we can go straight to the reference angle so remember when doing reference angle you completely ignore this part and you look at this part here so some schools include the negative on the calculator but most schools like 99 of all schools do not include the negative on the calculator so if your teacher does then it's absolutely fine they will just use a slightly different technique when solving okay so just look out for that but 90 99 of schools don't put the negative so you would just say shift sun of 0.3 so this is now for the reference angle so i'm just going to call it our angle okay so if we say shift sin of 0.3 you get an answer of 17 comma 4 6 degrees let me just write it here reference 17 comma 4 6 degrees okay and then we go to the quadrants now this is where the negative comes in it's telling us that sin is in a negative quadrant okay now sin is positive remember from your cause diagram sin is positive in those two so it's obviously negative in quadrants three and quadrants four okay so now remember how we start this part you write whatever you see there so you say x minus 20 and x minus 20 equals equals quadrant 3 says 180 plus quadrant 4 says 360 minus then you just put your reference angle so 17.46 17.46 and then we should also say plus k times 360 k is an element of z and on the other side plus k times 360 k is an element of z now what you want to do is solve okay so you want to get x parts off so we're going to take this 20 over and combine it with the 180 and the 17 and so you're going to end up with 217 0.46 plus k times 360. then if we go solve this one you move the 20 over do all of that and you're going to end up with 362.54 plus k times 360. k is an element of z now we don't have to go any further because they haven't given us an interval and so we can just finish off like that that's called the general solution let's take a look at number three so with number three we should always start our first step as number one sorry the first thing we should do is simplify okay so with number three the first thing we should do is simplify so now we can simplify because this 2 needs to end up on the other side okay so we get it there by dividing by two and so we can end up with cos of two x minus ten equals a half and now remember because we just have syn causal tan and some type of angle on the inside there you ignore that part and now we go to step three which is to get the reference angle so you would say shift cos of a half and so your reference angle is going to be 60 degrees okay then you choose your quadrants it says that cos is something positive so that's going to be quadrant number one and quadrant number four so in and then what we do is we we start off by writing whatever we see over here so it's going to say 2x minus 10 equals and then on the same for quadrant 4 then for quadrant one it doesn't say 180 minus 180 plus or 360 minus so we just go straight to the reference angle plus k times 360. i'm not going to put k as an element of z that you should but i've just run out of space then for quadrant 4 we know that quadrant 4 starts off by saying 360 minus and then we put our reference angle which is 60 and then we can say plus k times 360. then what we need to do is go and solve all right so we get x by itself so we do that by moving the 10 to the other side where it will be plus with the 60 and so we will end up with 70 we then divide by 2 so we end up with 35 plus k times 180 many times i see people forget to put k 360 in the beginning then at the end they remember to do so but then they forget that maybe that 360 has now been divided by two okay so you want to get into a habit of putting that k360 in the very beginning okay so i'm now going to solve the quadrant 4 so it's going to give us 2x is equal to 360 minus 60 is 300 plus 10 is 310 plus k times 360. we can then divide everything by 2 and we'll end up with x is equal to 155 plus k times 180 but then please keep in mind this one is a specific solution and so we have to try a few values of k and so we'll take these two solutions which is 35 plus k times 180 and the other one is x equals to 155 plus k times 180 and now we will try different values of k we'll try values such as zero one minus one and then maybe we would be able to squeeze in two and minus two depending on if we have gone out of the given interval by that time or not and then we'll see what the x value is for each of those and then we'll do the same on the other side okay so i'm going to quickly go fill in all of those values by testing different values of k and then i'll fill in all the values and then just make sure that you get the same values and so there i have gone and filled in all the different values and so the ones that fit into the given interval which is minus 360 to 360 would be 35 215 minus 145 minus 325 155 335 minus 25 and minus 205 all the others are with they fall out of the given interval and so the answer will be and this doesn't have to be listed from smallest to big but i'm going to do that so it will be minus 325 minus 205 minus 145 okay now make sure which ones i've chosen so i've chosen that one i've chosen minus 205 minus 145 minus 25 35 and there we go so we've got eight different solutions for this question over here