The three trigonometric ratios; sine, cosine and tangent are used to calculate angles and lengths in right-angled triangles. The sine and cosine rules calculate lengths and angles in any triangle.
In this lesson we are going to be working through the basic trigonometry section which is using trigonometric ratios to find lengths and angles in write right angle triangles. So we are going to learn how to label a right angle triangle so that you can use these trigonometric ratios to find missing lengths and angles. So let's make a start on labelling your triangle. So here is a right angle triangle and they've given you an angle x and we've also got the 90 degree angle there as well. So the first side that you should label is the Hypotenuse which is the longest side on your triangle.
It is always opposite the square symbol, the 90 degree symbol, it's always this slant. Now the next side that we should look at is the side directly opposite the angle and that's called the opposite. Now sometimes the given angle can be at the top of your triangle in which case we would be looking at this base side as your opposite. So the location of the given angle determines where the opposite side is. The final side is called the adjacent which is just basically next to the angle or between the 90 degree angle and the given angle.
It's always on that side where both angles lie. So again, if x was here then this would be your adjacent. So it's really important that you can label your triangles accurately before you try to use the trig ratios. So I would suggest you pause the video here, quickly draw these triangles out and label the three sides on each diagram with hypotenuse or Hype for short, opposite or Op for short and adjacent or add for short. So pause the video and when you're ready to cheque, replace play.
So here are my labelled sides. We've got the slant which is the Hypotenuse. The angle is up here so the opposite side would be down here and the adjacent is the last side. And you apply that same sort of process to these two triangles here. So if you have struggled with the labelling, feel free to again try these again.
Redraw your triangles, put x in a certain place, certain angle and relax. Okay? It's really important that you can do this before we move on.
Okay? So the actual trigonometric ratios. So one of the easier ways to work with these trigger ratios is using a process called Socartoa which is essentially linking your three sides together opposite hypotenuse and adjacent. Now that's what the O, the H and the A stand for. But we've got three other letters here, s, C and T and they stand for sign, cos and tan which you will see are on your calculator next to each other as well.
Okay, now this is what all of these mean. Now what's important to understand is the S, the C and T. They are essentially linked to angles. Okay? So sine angle, cos angle and tan angle.
And this will make a lot more sense once we apply it to a question. But essentially what you would work with is as follows. You would have two of the three bits of information from one of these triangles and you would use these triangles accordingly. So for example, if I labelled a triangle and it had the hype and you needed to look for the opposite, this triangle is, is the only triangle you could use because it's the only one that contains on h. Because for the second triangle I've got the height but I don't have opposite.
That's not going to help me. And here I've got the opposite. This time I don't have the height. So again, talking at you isn't going to make as much sense. Let me show you an example to work with.
Okay, so here we have a triangle with a given angle of 36, a hypotenuse of twelve. And we've also got an additional side and that's what we're trying to find, the A. The first thing you need to do is label the three sides. So we know this is the hypotenuse, that's the hype. We know that the side opposite, the angle is called the opposite and the final side would be called the adjacent.
So those are my labels. Now what I would suggest to do is write down the diagram, the information that we have in the diagram, because we don't need to use all three of these sites. So I've got an angle of 36, I've got the hypotenuse at twelve and the opposite is what I'm looking for and they've labelled that A. So the ratios that we need, we've got opposite and height. So which one of these three triangles, these three trig ratio triangles contains O and h.
It's the first one. So that's the triangle we're going to look at and we are looking for the opposite. If you sort of put your finger over what we're looking for, which is the O. So I'm going to put my cursor here. We're looking for the opposite.
So if I cover that, the formula is telling me to do S times h. So when they're next to each other we times and then one on top of the other we divide. So because we're looking for the O, I'm going to cover the O. And we are essentially working out S times h. So remember, O equals the S which stands for sine x times the height.
And we've got all the information here. The angle was 36. I'm going to substitute that in there and the Hypotenuse was twelve.
Plug that into your calculator. And what's important is that you know that there is actually a bracket around this 36. Please remember to put that in, otherwise it will calculate 36 times twelve first and then work out the sign of that which is incorrect. So sign 36. I think the bracket automatically pops open after sign.
So just remember to close it times twelve gives you an answer of that to two decimal places. Now note that the question didn't actually tell me what to round two, which is why I've stated that I have rounded it to two DP. What you don't want to do is round it and then not state your rounding because that's inaccurate. Okay, so just to reiterate what we've done, we labelled all three sides of our triangle. We wrote down which trig ratios we have and by doing so, we worked out which two letters we're working with which was O and H.
And this is the only triangle that contains both O and H. So that's the triangle we're looking at. Cover the letter that we're looking for, which in this case is O. And so we're working out sine angle times the height and that's how you find length.
Now, finding angles using soccer Torah, same kind of process as before. So we need to label the three sides. So we've got the hypotenuse here, we've got our angle up here. So this is the opposite side and this is the adjacent side. So let's label our triangle, let's write down what we have.
So this time we don't have the hypotenuse, so we're not going to be using that side. What we do have is the opposite, we have the adjacent and we're looking for the angle. The three bits of information you should be writing is the two bits of information they've given you. And the one thing we're trying to find, which in this case is X. Okay, now looking at the lettering, ignore the angle for now.
We've got the two sides, A and O. So which one of these triangles, these trig ratios contains A and O? Not the first one, it's not the second one, but the third triangle contains O and A. So that's the triangle that I'm going to use. We are looking for the angle.
Now remember, the angle is always this bottom left. So if I cover that with my mouth or with your finger, the formula is odd by A. So tan angle equals O divided by A. So write that down. Let's plug in all our giving information.
So the angle was X, the opposite is nine and the adjacent is twelve. So at the moment we have ten. X equals nine over twelve, which is a quarter. But I don't want Tan. X.
I want X. And this is where finding angles is slightly different to finding lengths to undo tan. To get rid of tan, you do a special thing on your calculator. So if you get your calculators out and in the top left, press Shift and then press Tan shift and then Tan on your calculator. And what should appear on your calculator is tan of minus one.
You need to put that bracket in, make sure the brackets in if it's not there, and then plug in nine over twelve. So that's what you need to put into your calculator. That's what it will look like on your screen. Make sure you close the bracket and that will give you the answer. So if you press equals, that's your answer.
And again, I've rounded to one DP and I've stated my rounding. So again, to reiterate what we've done, we labelled our three sides, we wrote down the information that we have. So the two sides were given and what we're looking for. We decided which trig ratio is most appropriate to use, which was the tan triangle. And we wrote the formula down, we wrote the ratio down.
We then plugged in all the values that we have. And the final step is to get rid of tan on your calculator. Instead of just pressing Tan, make sure you press Shift first. So shift Tan and then your fraction. Close your bracket equals, and then that's your answer.
So that's using Soccer Tower to find angles. So, I've put together a few of these questions for you. What I would suggest is pausing the video, jotting these down, trying them yourself, and feel free to post your solutions onto my profile, and I'm more than happy to go through those. So, overall, we have figured out how to label a right angle triangle, so it is ready to be used with the trigonometric ratios. We've used the Soccer Toa triangles to find missing lengths in a right angle triangle.
And we've used Soccer Tower to find missing angles in right angle triangle.
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Sanaa is currently Deputy Head of Maths at a secondary school and has been tutoring GCSE Maths students for over 10 years. She tailors her approach so that no student is left behind.