Fill in the gaps: In  a right-angled triangle, the sine of an acute angle is equal to the length of the side that is opposite to the angle, divided by the length of the …………………...

What is the length of the unknown side in the triangle below?

Which image shows the hypotenuse correctly labelled?

Which triangle shows the adjacent side to angle a?

Which triangle shows the opposite side to angle a?

The sine ratio

Key Stage 3

Maths

Surds and trigonometry

a is half the size of the hypotenuse when 𝑥 = ?

﻿Hello, and welcome to another maths lesson with me Dr. Saada. In today's lesson, we're going to look at the sine ratio. It's nothing completely new. It follows on from what we have been learning in the previous lesson, and it introduces the correct language or the name for the ratio that we were looking at in the previous lesson. You need a pen, paper and a ruler for this lesson. So, go grab these and when you're ready, come back and let's make a start. The "Try this" for today's lesson is the following. You have been given two triangles. What's the same and what's different about the two triangles? Construct the two triangles that you have in front of you, measure the side lengths. What do you notice? Please pause the video to complete the "Try this" task. It should take you roughly five minutes. Resume once you're finished. Welcome back. How did you go on with the Try task? Okay. Now this is what you may have noticed about the two triangles. They are both right-angled triangles, the longest side, i.e the hypotenuse, and both of them is equal. They're both five centimetres. If you constructed the triangles correctly, you should have this side being roughly 2.5 centimetres. It may be slightly different to my measurements. And this side should be roughly 4.4 centimetres. Did you get that? Well done. Now the second triangle, you should have had roughly this side being 1.3 centimetres, and this one roughly 4.8 centimetres. Did you get this? Really good. Now I wonder if any of you managed to think about the sides and what they should be without actually measuring them. Or before and then checked when you measured them. In particular, this first triangle. Cause it's something that we looked at in yesterday's lesson. So, with this side here, did any of you notice that I have a 30-degree angle? Therefore, this side, opposite to the 30-degree angle should be 1/2 of the hypotenuse. And just knew that it will be roughly 2.5 centimetres. It's really good to know this information as well, because if you make a mistake when you are constructing the triangle, and you end up with with a side length here, that is a lot less than 1/2 or a lot more, then you know that you made a mistake and you can go back and correct it. So, I told you earlier on today that we are going to be looking at the sine ratio. What is the sine ratio? Let's read this together. In a right-angled triangle, the sine of an acute angle is equal to the length of the side that is opposite to the angle, divided by the length of the hypotenuse. Okay, so we say that for the marked angle theta. And theta is just a symbol for showing the angle. So, here, this we read it as theta. We say that sine of that angle... So, sine theta is equal to the opposite divided by the hypotenuse. Basically the ratio of the opposite to the hypotenuse. What does the whole of this mean? Let's look at a right-angled triangle. So, if I have a right angled triangle, and I have this angle here being marked theta, what this tells me is, the sine. Sine is just the ratio, the ratio that we have been looking at. The sine of it. The sine of this angle here, is equal to the opposite divided by the hypotenuse. That is the length of the opposite divided by that length of the hypotenuse. How do I know which side is which? Let's have a look at that. Okay, so we've got three triangles here that we want to label. The easiest so to label is always which one? Really good. So, the hypotenuse is always the easiest. If I look at the right-angled triangle, the side that is opposite to the right angle. So, this is the right angle. Opposite to the right angle and it's the longer side. This one here is the hypotenuse. That's one is the easiest. Now we mark the other two sides based on the marked angle or the angle that we are interested in. So, in this case, this is your marked angle here. Now if I draw an arrow from that to that, I'm looking now at the opposite side to that angle. So now side AB here, is the opposite. So, I can label it opposite. This side here is the adjacent. So, I can write adjacent. It is adjacent because it's adjacent to the angle. It's the side that is between the right angle and that marked angle. So, we have the hypotenuse. The hypotenuse is the longest side in the triangle, it's opposite the right angle. We have the opposite and that is opposite the marked angle. The third side is obviously the adjacent. It is next to the actual marked angle and it is between the angle and the right angle. The marked angle and angle. So, let's look at the second triangle here. Which side is the hypotenuse? Tell me the name. Excellent, really good. So, the hypotenuse here is AB. So, here is the hypotenuse. Which one is the opposite? Really good. AC is the opposite. So, this side here is the opposite and the adjacent is BC. Okay, next one. Look at the triangle. Which one is the hypotenuse? Opposite the right angle, it's the longest side. It's EF, well done. And which one is the opposite? Really good. It's opposite, this one. It's EF, it's opposite that marked angle. And this means that DE is the adjacent and it is the adjacent because it is the one between or connecting the right angle to the marked angle. So, this one here is the adjacent. And for the Independent task, you have two questions. The first one, you have to label the opposite, adjacent and the hypotenuse in relation to the marked angle A. So, I've marked angle A for you. So, you do what we have just covered together. In the second question, I have labelled the sides for you. And I want you to mark the angle theta in relation to the labelled sides. Now it's your turn to have a go at that Independent task. Please pause the video and complete that. Resume once you're finished. Welcome back. How did you find this Independent task? Okay, really good. I have labelled the first three triangles for you. And I'm going to go through the last two in Question one together. So, I like to do it this way. Now you can start from any side, it really doesn't matter. I prefer to look at the hypotenuse first. So, I always look at the triangle and I say, "Where is that side that is opposite the right angle? "Which one is the longest side?" And I label it. So, I start with this one I label it hypotenuse. Then I like to go next label the opposite side. Again, it's entirely up to you. You may choose to label the adjacent first. The opposite is actually opposite that angle. And I do use an arrow, cause I find that helps me avoid making any silly errors. So, I draw an arrow from that angle and I say, "Okay, opposite to that angle is the opposite." Sorry. Now, the side that is the remaining must be the adjacent. I label it as the adjacent. But I also know that because it is the side that is connecting the right angle with that marked angle. Next one, again, I start with the hypotenuse all the time. I say to myself, it's the easiest. Opposite I draw an arrow from the marked angle, and I'm labelling it as opposite. And now the last side is that adjacent. How did you do on Question one? Really good if you have this correct. For Question two, it's among the angle theta in relation to the labelled sides. So, this time the sides were labelled. Let's look at the first one. The three sides we're given and labelled for you. So, you want to know what theta is. Again, you can use more than one side. I like to look at the opposite sign and say what, "If the opposite is here, "where is that arrow originating from? "Where is it coming from?" It's from here. So, this angle must be theta and opposite to where is the opposite side. Next one. Again, I start with an arrow and I say, "Where is it coming from?" So, the angle here must be opposite to that. And next one, I've only labelled one side. Which one is the hypotenuse? Really good. So, this is the hypotenuse here. And I can go and say, well, this here is the opposite side. That's only when I have left. So, that angle must be theta. So, I can label it as a theta. Now, the important point of today's lesson is one, to be able to label these triangles. So, you need to be able to label the sides: Adjacent, hypotenuse and the opposite, properly. And you also need to learn one important thing. If is this theta here... So, if you look at this triangle in the middle in Question two, the second one, if this theta is 30 degrees, what does that tell us? Okay? It tells us that the opposite is 1/2 the hypotenuse. If I tell you that the opposite is five, the hypotenuse must be 10. If I tell you that the hypotenuse is 12, then the opposite must be six. So, it must be 1/2 of it. So, that's one of the most important ratios that I want you to learn from today's lesson. For today's Explore task, I would like you to construct right-angled triangles with the angles shown in the table below. Fill in the table by finding the ratio of the opposite to the hypotenuse. So, opposite divided by the hypotenuse for the given angles. And I want you to have a look and think about what do you notice. So, the angles I want you to look at for today's lesson are: 15 degrees, 30 degrees, 45 degrees and 60 degrees. So, you construct this, fill in the table and see what's happening to the ratio. For an extension, you can try and draw different triangles or enlarge the triangles that you already have. So, the triangle that you have, whether 15-degree angle, enlarge it, see what happens. Does the ratio stay the same? Do the same for 30, for 45 and for 60. Please pause the video and have a go at the Explore task. Resume once you're finished. Welcome back. How did you get on with the Explore task? Really good. These are some of the triangles that I constructed. And I filled in the table with my answers. Did you do the same? Do we have any numbers in our tables that are similar? I guess that the length opposite and the length of the hypotenuse for you is slightly different from mine. But our ratio should be very very similar. Because if I now try and draw another right-angled triangle with a 15 degree, and I take the length of the opposite to the length of hypotenuse, even if I try a slightly different triangle in terms of the length, I'm going to end up with a similar ratio of 0.26. Did you get that? Well done. And for the 30-degree triangle, whenever you have a right-angled triangle, if you have a 30 degree, then the hypotenuse is always going to be double the opposite. And that's why we end up with a ratio of 0.5. For 45 degrees, did we even need to do the calculation for this one? If we have 45 degree, we know that we have an isosceles triangle. And if you have an isosceles triangle, then the two sides are equal. So, the length of the opposite is going to be equal to the length of the hypotenuse. And therefore, the ratio is going to be one to one. For the 60-degrees triangle, did you end up with something close to 0.87? Really good. Okay, so that was the point of today's lesson. For us to look at the ratio in right-angled triangles. When we change that angle, is that ratio going to change? Well, if we have a 30-degree angle, the hypotenuse is always going to be doubled the opposite. If we have a 45 degree, the two sides are always going to be equal. And if we have a 60-degree angle, the hypotenuse is always obviously going to be the longest. And it's going to be the ratio between that from the opposite to the hypotenuse, is always going to be 0.87. You have done lots of mathematical thinking in today's lesson. So, well done. You should be really really proud of yourself. This brings us to the end of today's lesson. Can I please remind you to complete the Exit quiz. That's it from me for today. Please enjoy the rest of your learning for the day and I'll see you next lesson. Bye.

The sine ratio

The sine ratio

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UnitMaths / Surds and trigonometry

Maths