Lesson video

Hello, and welcome to another lesson with me, Dr.

Saada.

In today's lesson, we're going to be looking at the sine and the cosine ratios.

For today's lesson, you need a pen, a paper, a ruler, protractor, and a pencil.

So please pause the video and go and grab these.

And when you're ready, click resume and we'll make a start.

I would like us to start today's lesson, by recapping what we have learnt in our previous lesson.

So can you try this? Can you find the length of sides A to D? You've been given two triangles and some of the sides are missing.

I want you to find the length of these sides.

If you're feeling super confident about this, please pause the video.

And if not, I will be giving some support.

So pause the video now, if you're feeling confident.

I'll be giving support in three, two, one.

It's a good idea to start by labelling the sides of the triangle.

If you look at the first triangle, I know that this is the hypotenuse.

It's the longest side of the triangle.

It's opposite the right angle.

If I look at the marked angle, angle 30 and draw an arrow from there, it helps me identify where the opposite side is.

So I know that this is the opposite.

And now I know that this is the adjacent.

Now I need to start thinking about the ratios of the sides of a triangle that has a 30 degree angle as the marked angle.

What did we say about the relationship between the opposite and the hypotenuse? There we go.

I think you can make a start on your own now.

Off you go.

Please pause the video to complete the try this task.

Resume once you're finished.

Welcome back.

How did you find it? Really good.

Okay, so these are my answers.

To start with, the hypotenuse is double the length of the opposite.

So the hypotenuse is 10 centimetres.

The opposite must be five centimetres.

And to find the adjacent I use Pythagoras.

So 10 squared, minus five squared, is equal to A squared and 75, is equal to A squared.

And if I just want A, I can square root it and that is square root of 75 centimetres.

I left it in surd form.

For the second to triangle, I started by labelling it.

So I've got my hypotenuse here.

I have the opposite, opposite and marked angle.

So D is opposite side and C is the adjacent.

And I started by thinking about the relationship when I have a marked angle of 60 degrees, in a right angled triangle, the adjacent is half of the hypotenuse.

So therefore the adjacent must be five centimetres.

And now I can again, use Pythagoras to find that the opposite is the square root of 75.

There's something really really important about those two triangles.

And this is why I chose them.

The second one is the same as the first one, but it's just been rotated around.

And we can see that because look at the sides that we have, they both have 10 centimetre hypotenuse.

They both have the shorter sides as squared or two 75 and five centimetre.

And the angles are identical.

Both of them have right angles, we've got right angle here, we've got right angle here.

We have a 30 degree angle, but what is this angle? Interior angles in a triangle add up to 180.

So this angle must be? Excellent.

It must be 60.

So this angle here is 60 and we have a 60 degree angle here.

So we have a, that's in common.

And then here I have 30 and this angle here, we have 60 plus the 90.

If we subtract that from 180, that gives us 30.

So this angle here must be 30 degrees.

And there's something really special, about the 60 degree and the 30 degrees angles.

And this is what we're going to look at in today's lesson.

Thank you.

So in our last lesson, we explored the sine ratio, which is a relationship between two of the three sides in a right angled triangle, the opposite divide by the hypotenuse, is what we know as the sine ratio.

But are there any other ratios that we can find? Thinking about the three sides.

Really good.

So I can have so many ratios.

I can think about, the opposite divided by the adjacent, opposite divided by the hypotenuse, is what we have for the sine ratio, hypotenuse divided by the opposite, the hypotenuse divided by the adjacent, adjacent divided by the hypotenuse, or adjacent divided by the opposite.

So really because I put the three sides, if I want to relate one to the other, I can end up with six ratios.

We looked at this one and we called it the sine ratio.

Now I want you to look at this one here.

It's very similar, isn't it? So with the first one, we have the opposite divided by the hypotenuse.

The second one we have, the hypotenuse divided by the opposite.

Okay.

So in order for everyone to be doing the same thing, mathematicians decided that we have to stick to one of these.

So do we look at the ratio? Do we, when we're looking at the ratio, do we divide the opposite by hypotenuse or hypotenuse by opposite? And then mathematicians decided that we're going to go with the opposite divided by the hypotenuse.

So it's just a convention for why we use this one when we're talking about the sine ratio.

Okay.

So at the beginning of the lesson, I told you, we are going to look at the sine and the cosine ratio.

We know what the sine ratio is.

So let's look at what is the cosine ratio.

In a right angle triangle, the ratio of the length of the side, adjacent to angle X, or angle theta, the given angle, the marked angle to the length of the hypotenuse, is called the cosine ratio.

This is usually abbreviated to cos.

The formal way of writing it down, is cosine of theta is equal to the adjacent divided by the hypotenuse.

What does this mean? That if I have a right angle triangle with an angle, marked theta, we mark and label this, label the size as we usually do.

So, the longest side is in the hypotenuse.

The opposite is the one opposite to the marked angle.

The adjacent is that adjacent.

And what we are saying, is that the ratio of the adjacent, let me just scrap this.

So we're saying that the ratio of the adjacent to the hypotenuse, is given a more formal name and that's cosine.

So we are just saying the ratio of this to this, we call it cosine.

Okay, although the last couple of lessons, I have been asking you to construct the triangles, to measure the sides, to calculate different ratios.

And in particular, we were focusing and concentrating on 30 degree angles and 60 degree angles.

So when we did that, when we were constructing those triangles, we always found that that, that opposite divided by the hypotenuse, in a 30 degree angle triangle, where we are labelling the opposite, being the angle opposite, the 30 degree angle, we always found out that the opposite divided by the hypotenuse was 0.

5.

And we also found out that in a 60 degree angle triangle, so where we are labelling in reference to that 60 degree angle that the adjacent divided by the hypotenuse, was always again 0.

5.

So even though there were two different angles and we were using different sides.

So this is what we are going to do today.

We're going to make it a little bit more formal and think about it as sine of 30, and cosine of 60.

And this is how we are going to deal with it.

If we look at the first triangle, we have a right angle triangle with a 30 degree angle.

So I'm going to start by, labelling the sides of the triangle.

So I can label the five centimetre as the? Excellent.

The opposite, and the 10 centimetre as the?.

Really good.

So as the hypotenuse.

Now, this is the formal way of writing it.

It's not different what we have been doing before, but it's just driving it, a bit more like mathematicians.

Okay, so we write down sine 30, is equal to, opposite divided by the hypotenuse.

We write it as a fraction because that by means division.

Now, I know what the opposite is.

I know what the hypotenuse is.

So I can say that sine 30, is equal to five out of 10, and I can simplify this and write sine 30 is equal to a half.

Okay.

And now I want to, before we move on to the next to triangle I want you to continue looking at that same triangle here.

So look at the first triangle.

Do we know the size of this angle here? Well this is 90 degrees plus the 30, 120, interior angles in a triangle add up to 180.

So this must be? 60 degrees.

Excellent.

So you should all be happy with me labelling this as 60 degrees.

Now I literally just took this, into this, into here, to the second triangle.

So I still have that this triangle, is exactly the same as the first one.

but this time I didn't mark the 30 degree, I marked the 60.

So I'm going to label everything in relation to the, 60.

So I already have labelled.

I have one of the sides, 10 centimetres is still the hypotenuse.

And I'm going to label this one as the adjacent.

Yeah? Very good.

So now I'm not using sine, I'm using cosine because I am looking and I'm using the adjacent and the hypoytenuse.

And the ratio that relates to adjacent to the hypotenuse, is not sine is cosine.

So this time I'm going to write cos of 60, remember I told you, we abbreviate it to cos, so cos 60 is equal to, adjacent divided by the hypotenuse.

That cos 60 is equal to five out of 10 and of course 60 is equal to a half.

So we really, what we can gather from this, from this task is, I have exactly the same triangle.

It doesn't matter which angle I use, as long as I'm naming the size, according to that marked angle and calculating the ratio, I end up with the same thing.

So here I had sine 30 is half and here I have cos 60 is half.

Sine 30 and cos 60 are always going to be equal.

No matter how much we enlarged that triangle, make it bigger, make it smaller, it will always give us the same thing.

And it is time now for you to have a go at the independent task.

So for this task, I would like you to construct and explore the sine and the cosine ratios in right angled to triangles.

I want you to pick your own side lengths, but I want you to choose the following angles, 10, 15, 30, 45, 60 and 75 degrees.

So I want you to start by drawing a triangle, a right angled triangle.

The other angle has to be one of the other acute angles.

Has to be 10 degrees, the lengths of the sides is entirely up to you.

Once you've constructed that triangle.

I want you to measure all the three sides, put it in the table, and I want you to calculate the sine ratio and the cosine ratio.

So by doing the opposite divided by the hypotenuse, you get your sine ratio and the adjacent divided by the hypotenuse, will give you the cosine ratio.

Then I want you to repeat the same process for 15 degrees, 30 degrees and so on.

Now, as I said, the lengths are entirely up to you.

And that's the beauty of this.

Is that even though you're doing it now, and I've done it while I'm recording this video for you, you and I should have really similar, sine and cosine ratios if we do this accurately.

Okay, and when you're done, I would like you to use your calculator, to check the accuracy of your answers.

You do that if you have a scientific calculator, by pressing the sine buttons, you've got sine button here, you press it.

And then you press the number for the angle that you have.

So depending on which angle you chose.

So for example, if I choose 45 and then you go to equal sign and you press equal and then it should tell you the answer.

Okay? So that's what you do to check using a calculator.

Once you have the ratios, the answer that displays on your calculator, should be very similar to the sine ratio that you have.

You press the cosine function, or button.

If you want to check your cosine ratio.

So if you're feeling super confident about this, you can pause the video now and make a start.

If not, I'm going to give you a bit more support.

In three, two, one.

Because I'm going to talk you through the steps of the first triangle that I drew.

I drew a triangle that looks like this.

If you cannot remember how to construct triangles, please watch, go back and watch the lessons, about constructing triangles.

So I started with just the base of four centimetre and marked the 10 degree angle, and then I joined, joined the points to make a triangle because I was not given any constraints about the sides.

Then I measured the sides.

The length of the opposite, opposite side was only 0.

8 centimetres.

The hypotenuse was roughly 4.

1 and the adjacent was four centimetres.

So I filled that in the table.

Then to find the sine ratio, I did the opposite, which is 0.

8 and divided that by the hope to use 4.

1, and I wrote down my answer, which was 0.

17 to two decimal places.

I did the same for the cosine, but this time I divided the adjacent, which is four by 4.

1.

And the answer was 0.

57.

And I filled that in the table.

Now it is your time to have a go at this.

Please draw another triangle that has 10 degree angles, use different sides to me, and you should be able to get the same or very, very similar sine and cosine ratios.

Okay.

Please pause the video and have a go at the independent task.

Remember to try and draw your triangles as accurately as possible and measure them to the nearest millimetre.

When you're rounding your sine and cosine ratios, do round them to two decimal places.

And remember to check your answers using a calculator.

This task should take you about 15 minutes.

So pause the video and resume once you're finished.

Welcome back.

How did he get on with this task? Okay, this is what I have done.

I drew the triangles, so I drew various right angled triangles with 10, 15, 30, 45, 60 and 70 degree angles.

And then I measured the three sides and I started by completing the table first.

When I finished that, I calculated the sine ratio and cosine ratio for each triangle.

And I did that for the same ratio, by dividing the opposite by the hypotenuse and for the cosine, I did it by dividing the, adjacent by the hypotenuse.

Then I dialled up the calculator to compare the accuracy or to check the accuracy of my answers and compare that with the display on the calculator.

Did you do that too? Okay.

Really good.

So this is what I did.

I grabbed the calculator, then I pressed sine 10 equal, and I looked at the number that was displayed.

And I found that, that it was really, really similar to mine.

In fact, mine was almost rounded to two decimal places.

And the reason for that is when we're drawing, we're only measuring the side links to one, the closest millimetre.

So it's not as accurate as if like I'm getting a programme or a computer to do it.

And then I did the same for sine 15 on the calculator compared it, sine 30 and so on.

Then I did the same with cos.

So I went to my calculator and I pressed cos button, cos 10 equal.

And I compared it to the ratio that I calculated.

I found out that my results were pretty accurate.

How about yours? Okay.

Really good.

Now, there are couple of ratios that I really want to highlight for you today.

And these were the main things that we have been exploring.

So we have really been exploring, the sine ratio at 30 and the cos at 60.

And they are both equal to 0.

5.

They are both equal to a half because if we have a 30 degree angle and we labelled the triangle based on that marked angle of 30, then the opposite is always half hypotenuse.

If we have the 60 degree, if we look at the 60 degree, and we mark out or we mark the sides of the triangle based on the 60 degree angle, we always have the adjacent being, half of the hypotenuse.

So this is, the ratio at, the sine ratio 30 is always equal to cos 60.

And you really need to know that.

Now the other one I want to highlight to you is this one here.

The sine ratio and the cosine ratio at 45 degrees, they are equal because the two angles are 45 degrees.

So it really doesn't matter which way you do it.

You will end up with the same ratio.

And the other one that we looked at, okay, is sine 30 and cos 60.

So we've looked at these two, okay? And we found out that they are equal.

So cos of 30 is equal 0.

87 and sine of 60 is equal to 0.

87.

And I really want you to become really familiar with these numbers and kind of know them by heart if possible.

'Cause then your exit quiz, I'm going to be asking you about those numbers.

So I want you to be able to answer them.

So if you want to make a note of this, you can pause the video and just make a quick note that sine of 30 is equal to cos of 60, is equal to 0.

5 At 45 degrees, the sine ratio is equal to the cosine ratio.

And cause of 30 is equal to sine of 60.

And they're both equal to 0.

87 to two decimal places.

This brings us to our explore task where I have given you two right angled triangles.

One of them has a 30 degree angle marked and one of them has a 60 degree angle marked.

In both cases, I gave you the length of the hypotenuse.

Can you find the length of the missing sides? And if yes, how many methods can you use to arrive to the same answer? So I don't want you to just find that using one method.

I want you to really try so many different methods, as many methods as possible.

What is the exact value of sine 60 and cos 30? Can you perhaps use these to help you? This student here says, I could use Pythagoras theorem with right angle triangles, as well as sine 30 and cos 60.

Can you combine both Pythagoras and what we have been learning, about ratios and triangles.

Can you perhaps redraw your own triangles? When you're draw them, only measure one of the sides at the beginning and try and calculate the length of the other two sides and then measure them to see you were accurate or not.

You will need about 10 to 15 minutes to complete the explore task.

Maybe a bit longer if you want to draw a lot more triangles and really explore it.

Please pause the video and complete the task.

Resume once you're finished.

Welcome back.

How did you get on with this task? Okay.

Now there are so many ways that you can approach this question.

And I'm going to share my thoughts on this with you.

So I looked at this question, I thought, let me start by labelling the triangle and the first triangle in relation to the marked angle 30 degrees.

So I labelled this side here as the opposite because it's opposite in the 30 degree.

And this is the adjacent.

I know that the opposite is half of the hypotenuse, If we have a 30 degree angle.

So I knew that this side here must be 10 centimetres.

And then I thought I can use Pythagoras now to find the adjacent side.

So if you've done that, that's amazing, brilliant.

But I could also use the ratio of the sides that we've been looking at.

Now, the relationship between the adjacent and the hypotenuse is, this cosine ratio.

So I said, well, the cos of 30, will be the adjacent divided by the hypotenuse.

I know what cos 30 is because we've discussed that just few minutes ago.

I know that cos 30 is roughly 0.

87, isn't it? Or you could have used the calculator to find the exact value.

So I said, well, 0.

87 is equal to the adjacent, which I don't know the length of, but I know that the hypotenuse is 20.

Can I use this now to help me find the adjacent? I can.

If I rearrange the equation and multiply both sides by 20, I get the adjacent, which is 17.

4 centimetres.

If you use Pythagoras theorem, you should have ended up with exactly the same answer.

Now, one of the other methods, or ways to look at this question, which is also really interesting because you can use it as a method to check that your answer is correct, is to look at this angle here.

We know that it must be 60 degrees, okay? Cause we have 90 degrees, 30 degrees, and we know that we add up 60 to make the 180 degrees inside the triangle.

Now I can label this site as the opposite because it's opposite that 60 degrees.

I'm looking at the 60 degrees as my marked angle.

This side is the adjacent, which I know now that adjacent is half of the hypotenuse.

And it makes sense in a 60 degree.

If we're looking at 60 degree and we have that side, we already labelled it as 10.

So I know that that's correct.

Now let's look at finding the side that we have just labelled as opposite.

So looking at this side here.

Let's find the length of that side, by using that 60 degree angle.

Instead of the 30 degree angle.

One, that side is the opposite, and we know that sine 30 is, opposite divided by the hypotenuse.

Sorry sine 60 is opposite divided by the hypotenuse.

And I know that sine 60 is 0.

87.

So I can write that down equal the opposite, I don't know it, divide by 20, rearrange the formula by multiplying both sides by 20.

And that would give me the opposite is equal to 17.

4.

And you can see here that now I got the same answer using both the sine and the cosine rules or ratios.

Now, if you've done something similar to the next triangle, you should have this side being four centimetre.

And this one has 6.

9 centimetres.

Again, you can use sine, you can use cosine.

You can use a combination of both and you can use to Pythagoras, it's really interesting to use more than one method to check our answers.

So really good.

Did you try and draw some different triangles? If you have, I would love to see these.

A huge well done on all the fantastic learning that you have done today.

Please do not forget to complete the exit quiz.

Enjoy the rest of your learning for today and I'll see you next lesson.

Bye.