Lesson video

Hi there and welcome to another math lesson with me Dr.

Saada.

This is our last lesson for this unit.

In today's lesson, we would be looking at sine and cosine graphs.

For this lesson, you need a pen and paper, a pencil, and a ruler.

So pause the video, go grab these, and when you're ready, let's begin.

Let's just start today's lesson by drawing a graph.

The table to the right show sine theta for 10 degrees to 80 degrees angles.

Use the table to plot sine graph.

You've been given the angles and you've been given the corresponding sine ratio for each angle.

You have been given a grid the has on the x-axis angle, so 10, 20, 30, and on the y-axis the sine of theta, the value for sine at that angle.

So all you have to do is plot the points and join them together.

Make sure that you are using your pencil for this.

If you do not have this printed, you will need to draw your own axes.

So use this one as a guide to help you draw yours.

If you have the worksheet printed, this task will only take you five minutes to complete.

If you have to draw it on axis it may take you a little bit longer.

Please pause the video and complete the task to the best of your ability, resume once you're finished.

How did you go on with this task? So this is what I have done with this task.

So I went to the table and I said, okay, my first point is going to be at 10 degrees and at 10 degrees, sine of 10 and 0.

17, so I went on the x-axis, I found where 10 is, I went up, found where 0.

17 and I marked my first point.

Then I went to 20 degrees.

So 20 on the x-axis, went up to 0.

34, and marked my second point and kept doing that until 80 degrees.

Once I had all the points, I joined the points together and it looked like a bit over curve, part of my curve, so I drew a line connecting them.

Your line should be even smoother than the line I have here using the pencil.

Now, how can I use this graph? What do I need it for? Well, one of the reasons why you can use it is if I asked you to find out, or to estimate that value of sine 45 or 65 or 35, what would you have done? So let's say, I asked you about sine 35, you need to go to the x-axis to 35, and you draw a line until you reach the graph and then from there a line to the y-axis, so you can read that and if you read this one, it's about 0.

57.

If you use your calculator, you will find that it's very close to 0.

57.

What about 65 degrees? You do exactly the same thing with a 65, draw a line until you hit the curve, and then read it off that is roughly 0.

9.

And this is part of the sine graph and we'll look at the whole graph when we get to the independent task.

Now let's look at that cosine graph.

Let's do exactly the same thing, but for cosine instead of sine, I have a table, again, I have 10 degrees to 80 degrees.

Cos of 10 degrees is 0.

98, cos of 20 is 0.

94, cos of 30 is 0.

87 and you should know this one.

I'm rounding everything to two decimal places.

Cos of 40 is 0.

77, cos of 50 is 0.

64, what is cos of 60? We should know this, excellent, 0.

5.

Cos of 70 is 0.

34 and cos of 80 is 0.

17.

Now I can plot those numbers.

So I go to 10 and opt to 0.

98, and I make a point, 20 and I go to 0.

94, and I plot the point.

So once I've plotted the points, I should have something that looks like this, then I can use a pencil to join the points together.

Now I can use the cosine graph to find out the value of cosine of angles 'cause I'm not getting them in the table.

For example, if I want to know what cos of 45 is, I go to 45 on the x-axis and I draw a line until it hits the curve and then go and read the value of the y-axis.

And therefore, I know now that cos 45 is 0.

7.

Let's say, I want to find cos of 65, I do the same thing.

So I go to 65 and I draw a line until it hits the curve and then we read off the y-axis and that's roughly 0.

43.

Now, we've looked at the sine and the cosine graphs between 10 degrees to 80 degrees.

Hopefully we can already spot some similarities between the values, how does the graph look like between zero degrees, all the way to 360 degrees? And this is what I want us to look at.

So if I want to draw a graph of y equal sinx.

So I'm going to show a sketch of how the graph looks like.

I need obviously my x and y-axis, the graph looks something like this for sinx.

On this graph there are really, really important points that I want you to know, and these are the five points that I'm marking here.

So these are five main points on the sine graph.

You can see that the sine graph looks like almost like a wave.

It goes up and then down, curves again and back up.

In fact, it repeats the same thing after 360.

So it does this and it repeats it and it repeats it and it repeats it forever.

It repeats it also on the negative side as well.

The main points that you need to know from your sine graph is sine zero is equal to zero, and that's the first point.

Sine is 90 is one, that's when the graph reaches and when you're at maximum point, then it drops on sine 80, it crosses the x-axis, it's at zero, and it goes underneath the x-axis, so sin 270 is negative one, and then it goes back up at sin 360 and hits zero and then it repeats the same process.

Now, the cosine graph looks similar.

If I drew the cos graph, that's how it looks.

So the difference is that this time, the cos graph crosses or dissects the y-axis at one, so it starts at one and then it goes down and then under then under the x-axis and then back up, and then it reaches the same starting point and it would be the same thing.

And the points that are important here are the following.

So the first one cos of zero is one whereas sine zero is zero and we are going to look why is that in our explore task.

The next point is cos of 90 degrees and that is zero.

Next point is cos of 180, and that's a negative one, and that's when cos graph reaches the minimum point of negative one, and then next one, 270, cos 270 is zero and cos 360 is at one, and that's when it reaches that maximum again.

And then the cycle repeats itself again, so you will see that the graph repeats itself.

Now, if you look at the two graphs, there are similarities in the sense that they both look like waves and it almost feels like the sine graph, I can show you this here, it feels like the sine graph is bit, isn't it? The cosine graph, it starts from here, and it goes there, up and then obviously up there.

But almost feels like this red bit that I drew has been shifted to the y-axis, closer to the y-axis, or actually it starts here and it does that, so they're really, really similar.

Again, now let's do exactly the same thing, but for cosine instead of sine.

I have a table, again, I have 10 degrees to 80 degrees.

Cos of 10 degrees is 0.

98, cos of 20 is 0.

94, cos of 30 is 0.

87, and you should know this one.

I'm rounding everything to two decimal places.

Cos of 40 is 0.

77, cos of 50 is 0.

64, what is cos of 60? We should know this, excellent, 0.

5.

Cos of 70 is 0.

34 and cos of 80 is 0.

17.

Now I can plot those numbers.

So I go to 10 and up to 0.

98, and I make a point, 20 and I go to 0.

94, and I plot the point.

So once I've plotted the points, I should have something that looks like this, then I can use a pencil to join the points together.

Now I can use the cosine graph to find out the value of cosine of angles, if I want to know what cos 45 of is, we go to 45 on the x-axis and I draw a line until it hits the curve and then go and read that value of the y-axis, and therefore, I know now that cos 45 is 0.

7.

Let's say, I want to find cos of 65, I do the same thing, so I'll go to 65 and I draw a line until it hits the curve and then read off the y-axis and that's roughly 0.

43.

Now we've looked at the sine and the cosine graphs between 10 degrees to 80 degrees.

Hopefully we can already spot some similarities between the values, but how does the graph look like between zero degrees, all the way to 360 degrees, and this is what I want us to look at.

So if I want to draw a graph of y equal sinx.

I'm going to just show you a sketch of how the graph looks like.

I need obviously my x and y-axis, the graph looks something like this for sinx.

On this graph there are really, really important points that I want you to know, and these are the five points that I'm marking here.

So these are five main points on the sine graph.

You can see that the sine graph looks like, almost like a wave, it goes up and then down, curves again and back up.

In fact, it repeats the same thing after 360.

So it does this and it repeats it and it repeats it and it repeats it forever.

It repeats it also on the negative side as well.

The main points that you need to know from your sine graph is sine zero is equal to zero, and that's the first point, sine is 90 is one, that's when the graph reaches and when you're at maximum point, then it drops sin 80, it across the x-axis which is at zero, and it goes underneath the x-axis, sin 270 is negative one, and then it goes back up at sin 360 and hits zero, and then it repeats the same process.

So, the cosine graph looks similar.

So if I draw the cos graph, that's how it looks.

So the difference is that this time, the cos graph crosses or intersects the y-axis at one, it starts at one, and then it goes down and then under the x-axis, and that went back up and then it reaches the same starting point and it repeats the same thing.

And the points that are important here are the following.

So first one, cos of zero is one whereas sin zero is zero, and we're going to look why is that in our explore task? The next point is cos of 90 degrees and that is zero.

The next point is cos of 180, and that's at negative one.

And that's when cos graph reaches the minimum point of negative one.

And next one, 270, cos 270 is zero.

And cos 360 is at one and that's when it reaches that maximum again.

And then the cycle itself again.

So you will see that the graph repeats itself.

Now, if you look at the two graphs, there are similarities in the sense that they both look like waves.

It feels like the sine graph is this bit, isn't it? The cosine graph, it starts from here, it goes there up and then obviously up there, but almost feels like this red bit that I drew has been shifted to the y-axis, closer to the y-axis, or actually it starts here and it does that.

So they're really, really similar.

And for your independent task, you have been given two graphs sine and cosine graph.

I want you to look at the two graphs and tell me, how are these graphs similar? How are they different? And then I want you to think about two angles, which have the same sine value using the graph and two angles that have the same cosine value.

My hint for you is to try and think about drawing lines on this graph to find out angles that have the same sine value and same cosine value.

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

If you're not, I'm going to give you a hint in three, in two and in one So for the first part, when it says, how are these graphs similar and how are they different? I want you to look at the graphs.

One of the things that you can look at is where do these graphs cross the y-axis? Where do they cross the x-axis? Are these points similar? Are they different? You can use any words that I have used in my previous description to describe these graphs.

So with this hint, you should be able to make a start, off you go.

Please pause the video for roughly 10 minutes to complete the independent task, resume the video once you're finished.

Welcome back, how did you get on with this task? What did you write down about the graphs? And how are they similar and how are they different? Some of you may have written that they have similar shape or the same shape.

Some of you may have written that they repeat, because I mentioned that earlier, some of you may have mentioned that they have turning points.

Some of you may have mentioned that they looked like waves.

And some of you may have mentioned that it feels like we took that sin graph and moved it a little bit across the y-axis in order for us to obtain the cos graph, and these are all correct.

Some of you may have mentioned that cos graph starts at zero and the cycle ends at 360.

So it takes 360 degrees to complete a cycle.

And the same thing applies to the sine graph, so they're similar.

So you can see that they complete one cycle in 360 and after that, they're going to repeat again.

So some of you may have written this.

Some of you may have said that cos starts at one or crosses the y-axis at one, which is more accurate and sine starts at zero or crosses the y-axis at zero.

Some of you may have mentioned where the graphs cross the x-axis and that sine graph crosses the x-axis at three points, between zero degrees and 360 degrees, whereas cos only crosses the x-axis twice, between zero degrees and 360 degrees.

You may have mentioned that the maximum point for both graphs is one and the minimum point or the minimum value for sine and cosine is negative one.

Really good if you've done this.

Now, write down two angles which have the same sine value.

In order for us to do that, we can go to the sine graph and draw a line anywhere in y-axis, across the y-axis like this.

And now I can just read this value, this is 150, this is 30 degrees.

So this tells me that 30 degrees and 150 degrees, both of them have the same sine value.

In fact, if I read it, it's about 0.

45, so I can write that down.

Now, if I want another two points, I can draw this line here.

This line tells me that 250 degrees and 290 degrees have the same sine value.

And now for two angles that have the same cosine values.

I already have here drawn a line.

This one there if you can see it, this pink one here.

I guess I have a line there and it shows me that cos at 90 degrees is equal to zero and cos at 270 degrees is equal to zero.

So I can say that cos 90 and cos 270 have the same value.

Now I draw another line above that, at about just above 0.

6.

And if I draw a line from it to read the values from the x-axis, I can see that cos of 50 and cos of 310 have the same value.

Cos 50 is just above of 0.

6, so maybe 0.

61 and cos 310 is the same.

So I can write these down for sine and cos.

But there are so many different answers, I wonder which ones you wrote done.

And did you check that is in your calculator? I gave a new explore task for today's lesson.

I would like you to think about everything that we have done so far in this unit about the sine ratio and the cosine ratio.

I want you to explore those statements here.

I want you to think, why is cos 45 is equal to sine 45? You can test on your calculator if you want.

Why is cos zero equal to one? So we just looked at the cosine graphs, and we said that the cos graph crosses the y-axis at one.

But why? We said that the sine zero is zero, why is that the case? Why is sine 90 degrees is equal to one? And why is cos of 90 is equal to zero.

Lots of questions here for you to go and explore and think about.

If you're feeling super confident about making a start, please pause the video now and off you go.

If not, I will be giving you support in three, in two and in one.

And for support, I want you to think about actually drawing some triangles and testing those statements and seeing, well, how can I get to this statement? What do I need to do to the triangle? What's going to happen to the sides? The first one is probably the easiest.

So it will be a good idea for you to start with cos 45 degrees equal sine 45.

We're talking about 45 degrees, so we may as well draw a right angle isosceles triangle that has base angles as 45 degrees.

It's an isosceles triangle, so we know that the sides are equal.

So you can assume if this is one, then this must be one.

Now we'll label that sides according to one angle, and you choose which angle you want to choose other marked angle.

You may choose that this is your marked angle.

If this is your marked angle, and this is your opposite, find the height and find the adjacent and write down the equations for these and see how you get on with it.

Now for a cos zero, sine zero and sine 90 and cos 90, I would suggest that you draw another triangle, just so you can have a bit more space and think about the angles, well, mark one of the angles and then say, well, if this here is your marked angle.

So if this year is your marked angle, think, if I want to find cos of zero.

Now this marked angle needs to get smaller and smaller and smaller and smaller and smaller and smaller and smaller.

'cause I want it to be zero.

So what happens to the other sides? Or if I want it to become, again, this is my marked angle.

If I'm looking at the ratio of a 90 degree angle, well, this angle is going to be bigger.

It's going to be an acute angle anymore, it's going to be bigger and bigger and bigger and bigger and bigger and bigger and bigger and bigger and it's going to be like this.

So what effect does it have on this side and on the third side? So it would require a lot of imagination here for you.

I think with this hint, you shouldn't be able to make a start, off you go.

This explore task is the longest explore task that you've had in this unit and the longest task of today's lesson, it should take you roughly 25 minutes.

So about five minutes to spend on each of those statements.

You may want to pause this video and actually do some research while you're doing the explore task.

And you might be able to see some demos online that would help you with this.

So spend enough time on the explore task, looking at all of these statements and seeing why aren't they true? And what makes sine zero, or makes cos 0 and sine 90 and cos 90, and just look at the triangles and how the angle or changing the angle, what effect it has on the sides of a triangle.

So please pause the video and complete your task to the best of your ability.

Resume once you're finished.

Welcome back, how did you go on with the explore task? Really good.

Well, this is our opportunity to really think about the sine and the cosine ratios at a greater depth in our last lesson.

So let's make a start on the first one.

Sine 45 is equal to cos 45.

I told you to draw the triangle.

If you mark the triangle, this is the hypotenuse and this is the opposite side.

Again, we are marking this in relation to this here being the marked angle.

So this is the opposite.

And because it's an isosceles triangle, I can say that the two sides are equal.

So if one of them is one centimetres, the other is going to be one centimetre.

I could have used any numbers, any adjustment to this triangle it would not make a difference.

Now let's write the sine ratio in relation to that marked angle, while sine theta is equal to the opposite divided by the hypotenuse, do we know what the opposite is? Yes we do, it's one.

Do we know what the hypotenuse is? Not really at the moment.

We can calculate it using Pythagoras, but we don't really need to.

So I'm going to write down to that.

Sin 45, sin theta and theta in this case is 45, is equal to one out of the hypotenuse.

Using the same marked angle, let's write down cosine ratio.

Well, the cosine ratio is cos theta, so cos 45 will be equal to that adjacent divided by the hypotenuse.

So now I need to know where the adjacent is and it's there.

So I can write down cos of 45 is equal to one out of the hypotenuse, because the adjacent is also one.

Now sin 45 is one divided by hypotenuse, cos 45 is one divided by the hypotenuse for this triangle.

Therefore, I can say that they're equal.

Because that hypotenuse is the same one for both triangles.

This will still be the case if I had a bigger isosceles triangle, if this was enlarged, it will still be the same.

And this is why it cos 45 and sin 45 are equal because the two shorter sides of the triangle are equal.

And next one, cos of 0 degrees, why is that equal to one? So I will need you to have a bit of imagination here when we are looking at this one.

Look at this triangle here for me.

I have marked an angle with theta, and this is the right angle on here.

Now, according to, if we label this triangle using this theta here, this is the opposite.

And this here, the pink one is the adjacent.

So the pink line is the adjacent and the blue one is the hypotenuse, and we've got the angle in there between them.

Now, if I want cos zero, what am I going to do? I have to make this here, smaller, smaller, even smaller, even smaller, even smaller.

Can you see now that by moving, I'm using my arms, of course, but this is the blue one.

So as I move it down, as I'm making the angle smaller, I'm moving it down, it's becoming really, really close to that adjacent.

And if I keep doing this eventually it will, at zero degrees when I have no angle, the hypotenuse now is equal to the adjacent, it goes right on top of it.

If I move it like this on the diagram up to here, and the blue line becomes there because that angle is going to be zero and now all of a sudden, the hypotenuse of the triangle is equal to them adjacent.

Now, how can we write cos ratio? Well, cos is, cos theta is equal to the adjacent over the hypotenuse.

Now adjacent on the hypotenuse if the angle is zero, the adjacent and the hypotenuse are the same.

If I have something divided by the same thing, it's always one.

If I divide that adjacent by the hypotenuse and they are both the same value or have the same value, then the answer is one.

And this is why cos of 0 is equal to one.

Now sin zero is equal to zero for a very similar reason.

So hopefully we're starting to get the idea of how do we imagine zero degrees in triangles.

So now I have this triangle here and this is the right angle.

Again this side here is the opposite.

The pink one is the adjacent and the blue one is the hypotenuse.

So I want to sine of theta.

So this is the marked angle, I want this angle to become zero.

Again, imagine the same thing if it becomes zero what's going to happen? The hypotenuse and adjacent are going to be equal, but I am thinking about sine not cosine.

So I'm not really interested about what's happening to the adjacent.

If the hypotenuse comes down, down, down, down like this, what's happening to the opposite? How big is the opposite side? So look at the diagram here, if this blue line just moves down and down and down, closer, closer, the close it gets to the pink line or to the adjacent, the smaller the opposite is going to be.

And at some point it's going to be flat on the adjacent, and the opposite is going to be zero.

Now, how do you calculate sin on the angle? So the opposite is becoming zero, sin theta is equal to opposite divided by the hypotenuse.

Now the opposite is going to be zero, zero divided by anything is always going to be zero.

So sin of zero is going to be equal to zero, because the opposite becomes zero.

Now, moving on to sin 90 and cos 90, we're going to do something really similar, but this time, instead of making the angle smaller and making it zero we're going to make it bigger and we're going to make it 90 degrees.

So let's make a start.

I have this triangle here, I have angle theta.

I didn't fully draw the triangle, there we go.

If I have, this is the original triangle.

Now, imagine I am going to move this line, the hypotenuse, I'm going to make it bigger, bigger, bigger, because I want this angle to be bigger.

Now, I don't want it to be smaller anymore.

I want it to reach 90, so I want it to get to this point.

So if I move it, eventually this line will become here.

I know we can not have a triangle where we have a 90 degree here and the 90 degree there, and that's why I'm saying we have to imagine stuff.

So if I move it up, up, up, then that angle now is 90 degrees, What is happening to the opposite? The opposite now it's equal to the hypotenuse.

And therefore, sin theta is equal to the opposite divided by the hypotenuse.

If they are equal to one another, then the answer must be one.

Then divide the number by itself, we've discussed that earlier on.

Next one, cos of 90 is equal to zero.

Now we're going to do the same triangle, again that line, the hypotenuse becomes bigger, bigger, bigger, bigger.

What happens to the adjacent? Well, obviously the adjacent is going to now become zero because that line is moving and the other one is getting closer and closer to it.

The opposite to the adjacent is zero now.

Therefore, I can say that cos of 90 is being to be zero.

This brings us to the end really of the explore task, which was not an easy one, wasn't it? Required a lot of mathematical thinking, deeper understanding of the ratios and how they work, sine and the cosine ratios, imagining triangles, imagining angles zeros, imagining moving the triangle to have two right angle triangles and what not.

So it was a lot going on with this task, but it was such a rich task.

You should be super proud of yourself for all the resilience that you're showing in tackling questions that are as challenging as this one.

So huge a huge well done.

This brings us to the end of the lesson and the end of the unit.

We've learned so much during the last 12 lessons.

I would like you to take a couple of minutes to think about what you've learned in this unit and reflect on that learning.

Write down the three most important things that you've learned from this unit, it can be anything, it's entirely up to you what you've learnt.

What is the most important thing that you've learned? What do you remember? So please make sure you spend some time writing that reflection down and then remember to do the exit quiz, to show what you know.

I would love to see you work.

So if you would like to share you work with OAK National, please ask your parent or carer to share your work on Twitter, tagging at OakNational and #LearnwithOak.

This is it from me for today.

Enjoy the rest of your day, and I'll see you in another lesson and another unit.

Bye.