Trigonometry - Using SOHCAHTOA to Find Missing Angles or Sides Part 1 _ GCSE Maths 9-1 Revision

Okay, so in this video we look at how to find missing sides or angles using sine, cosine, tan and SOH CAH TOA. This comes up every year in some way or another, so it is a very good idea to really understand it. I remember seeing this in year 8 thinking, you know, wow, this is so confusing, but in reality it is actually quite easy. So let's begin. We do these problems in two parts. I think people go wrong in the first part and that is why they get confused. The first part is simply labelling the triangle. SOH CAH TOA problems are for right angle triangles only and the three sides are always hypotenuse, adjacent and opposite. We use shorthand when we label them, so, you know, I've written them like so, you know, with height, edge and up. The hypotenuse is the easiest to label, it is always the longest side and it is always opposite the right angle. So, you know, let me, you know, let me label the hypotenuse like so. So we know there are three angles to right angle triangles. One of them is obviously a right angle. So with the SOH CAH TOA problems, you always focus or zone in on one of the other two angles, so this one or that one, and sometimes they will only give you one of these angles. I've used the Greek letter theta here, this just means some angle, you know, it could be 60, 90 or whatever. OK, so this is the part people find confusing. Labelling the adjacent and the opposite sides really depend on where the angle is. So if the angle is here, the adjacent side simply means, you know, adjacent to the angle. So I will just write adjacent there and then, you know, the opposite just means opposite the angle, you know, it's quite obvious. So I can just label it there. So this is the confusing part. Really, it depends on where your angle is. If your angle is here, then obviously these two, you know, swap about or they can swap about. So, you know, consider this triangle here, looks like the same triangle. However, the angle of interest is now up here. So hypotenuse, as before, is the easiest to label because it's opposite the right angle and it's always the longest side. However, the adjacent is the side adjacent to the angle, which would be over here, and that would mean the opposite side is over here, which is opposite to the angle. So that's the first part done. Labeling is pretty important and once you label the triangle, I think, you know, the next part becomes easier. Okay, so the second part is where we use the Sokotoa formulas. So if I come down here, you know, I've written this out and you would have had this memorized and you will also know the formulas that they correspond to. So if you look at sine of theta over here, it is for the SOH, it's opposite over hypotenuse, and obviously we have the cos of theta is equal to adjacent over hypotenuse, KAH, and TOA is, you know, tan of theta is equal to opposite over adjacent. You know, we use this Sokotoa as an easy way to remember these formulas. And I've noticed some people tend to use sine of x, you know, you can use that, it's not a problem, but as you'll come to see, you know, sometimes we use x to denote sides and therefore theta is, you know, it can be a more appropriate variable to use. So I'm going to very quickly go over a worked example first so you can see how these formulas work. So let me put my theta over here. I've already put my theta here and so we know here theta is 60 degrees. So let's now label the triangle as we've done before. I know the hypotenuse must be here, it's the longest side and it's opposite the right angle. Our theta, so this is where you have to be careful, you've got angle there and angle there. However, our theta, the angle that we are focusing in on is over here, which is the 60 degrees. So therefore, the adjacent side is adjacent to the angle that we are interested in is will be over here and this is the opposite to the angle. So this is the opposite side. So I'm going to, I'm going to try out the sine formula. So the sine formula says that sine of theta is equal to opposite over hypotenuse. So we have our theta here and we have obviously the opposite and hypotenuse that we've just labeled. So let's just see what happens if we, if we go ahead and do this. So we've got theta is equal to 60 degrees. So we say sine of 60, because that's what theta is. And the opposite is, is 1.732 over the hypotenuse, which is 2. So I'm literally going to just chuck these values into my calculator just to see if it actually works or not. So I've got 60 is equal to, okay, this is in a simplified form. So in decimal would be 0.866. Let's just say approximately. So 0.866. What about here, we've got, we've got 1.71732. That's divided by 2, that is 0.866. And wow, as you can see, I think, you know, these, these, these values are correct, they work out. So this formula is definitely correct. And what you can do here is you can pause the video and you can actually check whether the cos formula works with these values by doing the adjacent over hypotenuse and putting in cos 60 and just check if they work. And you can also do it for, for tan as well. And furthermore, what you can do is you can move the theta up here, and then you can relabel the triangle, because obviously, as you know, the adjacent would become there if that was theta and the opposite would become there. And you can again just check if all three of the formulas work. And that would really improve your understanding. Okay, so moving on to sort of the main part of this video, which is finding missing angles or missing sides. So here we've got, you know, a typical problem from GCSE maths. And so we're given an angle, we are given a side length, and we are asked to find another side. Yes, so the first step is always to label the triangle, as I said before. So we can start by the the hypotenuse, which is always obviously the longest side, and the opposite to the right angle, I must have said this so many times by now. The next we label the adjacent and the opposite, well, I can see the angle that they've given us is over here, which is 60 degrees. So I'd say this is the adjacent side. And this leaves us with the opposite side here, because that is opposite the angle that they've given us. Second step is we write down SOH CAH TOA. And then we tick off the sides that we are interested in, or that we have given to us. So and we can only solve this if we have two out of the three unknowns. So we always choose a formula with two ticks, I'll show you what I mean. So here we are trying to work out x, we are given the opposite side. So I'm going to tick that off, I'm going to tick that off. And we are interested in the hypotenuse. So I'm going to tick that off, I'm going to tick that off. And we always choose the formula with with two ticks, because that's two out of three unknowns. And we can actually use this to work out the formula, or the answer, I should say. So in this case, we are obviously going to be using the sine formula, the SOH, the SOH part, because we've got two ticks. So literally, I'm just going to write out sine of theta is equal to its opposite over hypotenuse. And as you can see, memorising this SOH CAH TOA makes it, you know, really, really easy, you know, to write up these formulas in your exam. So then what we do is we take our our figures that we have. So we've got the angle theta here, we take, we say sine of theta is 60. We have that now. So we can say sine of 60 is equal to opposite over hypotenuse. That's what the sine formula says. So its opposite is 1.732. And the hypotenuse, which we don't have is, is the unknown, which is x. And as you can see, that's why we choose the one with two ticks as opposed to these one tick ones. And then, you know, it's just a case of solving for x. So I'm going to move the x. I'm going to times both sides by x. So we have x times sine of 60 is equal to 1.732. And that leaves us with x is equal to 1.732 over sine of 60. And I'll just go to my calculator. So it's 1.732 divided by sine of 60. That's equal to 1.9994. So I'm just going to round that up to two. So I'll just say x is equal to two. And I'm just going to write approximately, normally, they would have given you, you know, either put it to three significant figures or two decimal places, you know, whatever I haven't, I haven't specified anything in just for the purposes of demonstration here. So but anyway, you know, we can see our answer is x is equal to two. And quite interestingly, I think we've seen this triangle somewhere before. If we go back up to our example here, we've got 60 degrees and 1.732 as our sides, and we were asked to find out x. And we've just found out that x is equal to two. And that is what this triangle had. So you know, we know that our answer is correct, and it works out. Okay, next one more example. We've got a triangle here. And again, we are asked to find x. But more interestingly, we are asked to find an angle this time, as opposed to a side. So let's just go through the steps that you know, we've just done. So the first thing that we do is we label the triangle. So the hypotenuse is over here, I'm sure you know, it's because it's opposite the right angle, and it's the largest side, then we label the adjacent and the opposite. Well, the angle is over here. So this must be the adjacent because this side is adjacent to the angle. And then we label this which is the the opposite. So we write down our SOH CAH TOA tick off the sides that we have or and that we are interested in. So we've got the adjacent certainly. And we've got the hypotenuse that's involved as well. So two ticks for the CAH or the cos formula. So we are going to be using that. So it's cos of theta is equal to adjacent over hypotenuse. And then once we have our formula, it's just a case of substituting in what we have from our triangle. So theta is the unknown. So that's x. So the right cos of x is equal to the adjacent is 1.732 divided by two. Okay, so let's, let's try and work solve for x. And I'm sure you know what to do here. But you know, just in case you don't, you can refer to what is this 1.732 divided by two, and that's 0.866. So we're saying that cos of x is equal to 0.866. And I'm sure your teachers have explained to you how you get the x from that part. All you do is with your calculator, you do shift and inverse cos. And you put the answer in obviously, whatever answer this is, or you can type it in one. So you get here, you know, and the reason why you do the answer is because sometimes you might get a massive long decimal. But regardless, you do inverse cos, whatever answer you get, and we are getting 30.002, which I'll just say it's 30 degrees approximately, I'm just rounding it up, I haven't again stated how many significant figures or decimal points. So we're saying 30 degrees. And, you know, so we can say, yeah, we know x is equal to 30. So again, we've seen this triangle before, we've actually done it right at the beginning, 1.732 and two, and we know that it's 30 degrees. So as you can see, sine, cosine, tan is a very powerful method of finding missing sites and missing angles. In the next video, we will go through several more examples and work through more interesting problems. Anyway, remember to like and subscribe and I'll see you in the next video.