Lesson video

Hello, I'm Mrs. Lashley, and I'm gonna be talking you through the lesson today.

I really hope you're willing to try your best, and even if it gets challenging, remember I'm here to support you.

So our learning outcome today is to be able to draw the graphs for the trigonometric functions of sine and cosine.

On the screen, there are some key words that we'll be using during the lesson.

Let's read through them together now.

So trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle for a given angle.

The sine of the angle is the y coordinate of point P on the triangle formed inside the unit circle.

The cosine of an angle is the x coordinate of point P on the triangle formed inside the unit circle.

And both of those definitions about sine and cosine are referring to the diagram on the slide.

Hopefully, you can see that point P.

So I would suggest you pause the video and look at that diagram carefully and make sure you remember where sine theta and cosine theta are defined from.

Press play when you're ready to move on.

So in this lesson, we are going to break it into two learning cycles.

Our first learning cycle is going to focus on drawing the sine graph, and then when we move on to the second learning cycle, we'll look at drawing the cosine graph.

So let's make a start of thinking about the sine graph a little bit closer.

So in this first learning cycle where we're looking at drawing the sine graph, we're gonna make use of the unit circle.

So the unit circle is a circle centred at the origin with a radius of 1, and you can see a diagram of it there.

And it allows us to work out values for sine theta and cosine theta, so the sine of an angle, sine theta, is the y coordinate of the point on the triangle formed inside the unit circle.

So the point on the circumference is the point at which we're talking about, the x and the y coordinate of it.

That point comes from the triangle with a hypotenuse of 1.

And the y coordinate is the length of the opposite to that angle theta.

So we can generalise that point as x and then sine theta because the y value or the y coordinate gives us the value of sine of an angle, and that angle is the angle of rotation from the positive x-axis up to the radius.

So we can create a table of these values, either by using our calculator or by reading from the unit circle.

So in this table of values, we've got our angle of rotation.

So we're gonna look at 0, 30, 45, 60 and 90.

And then we're going to look at what the value of sine of 0 degrees, sine of 30 degrees.

So if we use our calculator, making sure we're in the degree mode because that's our angle of rotations unit, then sine of 0 is equal to 0, Sine of 30 is equal to 1/2.

0.

5 is equivalent to 1/2 as we all know.

Sine of 45, its exact value is root 2 over 2.

I'm gonna write it as 0.

7 on here.

Sine of 60 is root 3 over 2.

Again an exact value, but I'm gonna write it as 0.

9 in my table of values.

And lastly, sine of 90 degrees is equal to 1.

So remember this is the y coordinate of the points along the circumference of the unit circle.

And we've rounded all of these to one decimal place as this is gonna be useful later on.

Sine 45 and sine 60 have a more exact value than that, but we've rounded these to one decimal place.

So we can create the table by using our calculator, as we just saw.

But we also can read from the unit circle.

There is a link to an interactive GeoGebra file with the unit circle, which you can have a look at this a little bit closer yourself if you should wish.

But I'm going to go through these five particular points.

So on our unit circle, when the angle of rotation is 0, then that point where we're taking the y coordinate is on the 0.

10.

So it's y coordinate is 0.

When it's sine of 30 degrees, so that angle of rotation, then we can see our right-angled triangle and that point P on the circumference has a y coordinate of 0.

5.

When we up the angle of rotation to 45 degrees, then this has a point of 0.

7.

Again we're rounding to one decimal place.

60 degree angle, then it's 0.

9 to one decimal place.

Here we can see that when the angle of rotation is 90 degrees, then the radius is now part of the y-axis, and that point at the end of it on the circumference is 0.

1.

So it has a y coordinate of 1.

So here is a check for you.

Select the value of sine of 15 degrees when rounded to one decimal place.

So pause the video and then when you're ready to check press play.

0.

3 is the value of sine of 15 degrees rounded to one decimal place.

Option A was the exact value, so it's not rounded.

So if you put sine of 15 degrees onto your calculator, this is the value it would show you 'cause it's the exact value, and then you would need to change its form to decimal to be able to round it.

0.

7 is the value of sine of 15 when the calculator is in radians and not degrees.

So making sure that you have got that D symbol at the top line of your calculator display.

So it's quite hard to see any trends when the values are presented in the table because you may have thought, "Oh, look, it's increasing by 0.

2 each time." That's true in those middle three values, but it's not true from 0 to 0.

5 and it's not true from 0.

9 to 1.

So the gap between our theta values are also different.

So 0 to 30 is 30, and then 30 to 45, 45 to 60 are all increased by 15 degrees.

So in the table of values, it's very hard to see if there's any trend or pattern that's going on.

Andeep suggests, "Could we treat these as coordinate pairs and plot them?" So can we treat them? We know that the sine theta value is our y, so can we treat them as coordinate pairs? So absolutely, you can.

The x coordinate will be the value of theta, and the y coordinate will be the value of sine of theta degrees.

So plotting the coordinate pairs in that way, so if you look at this table of values, the first column is our x coordinate, and our second column is our y coordinate, results in this.

So here you can see that our x-axis has been labelled as theta degrees 'cause that is the value.

That's what it's representing.

Our y-axis is labelled as sine of theta because that's what it's representing.

So we've got our coordinate 0,0, then we've got a coordinate of 30, 0.

5, then we've got 45, 0.

7, 60, 0.

9, and 90, 1.

Alex says, "How do I join these points?" So it's not a linear graph.

They're not increasing by the same amount each time.

So it's not a straight line graph.

So that's not one way to.

Can you think about how you're gonna join this up? So we're gonna join it up with a smooth curve in the same way that we would join up other non-linear graphs.

So a smooth curve between those points looks like this.

So Alex says, "I wonder what happens when we choose values for theta greater than 90 degrees." So in my table of values, I did 0, 30, 45, 60, and 90.

So Alex is, and I'm hoping some of you are thinking about what happens past 90.

Izzy says, "Is that even possible?" So what do you think? Pause the video.

If you're sitting next to somebody, you might wanna discuss this one and think about whether Alex can consider points past 90 degrees, or if Izzy's right that actually this is not possible.

Press play when you're ready to look at this.

The unit circle shows that we can generate values for sine theta when theta is greater than 90.

Remember, theta on this unit circle is thinking about the angle of rotation of the radius from the positive x-axis.

So it can continue to rotate past the y-axis.

It can continue to go past 90 degrees.

So for the first task of the lesson, you are gonna investigate that, exactly that.

So using your calculator to come up with tables of values in a similar way that we did previously or the link to the unit circle.

Draw the sine graph between 0 degrees and 360 degrees.

So 360 degrees, we know is a full turn.

So if you're thinking about your unit circle, the angle of rotation for the radius, it started at 0, which is where it is horizontal on the positive x-axis, and it will have rotated completely around the unit circle one full turn.

And what values, what Y values, does it take as you go through that rotation? And then question two, write down any key features of the graph that you observe.

So once you've drawn your sine graph, look at it and what key features can you see? Press pause whilst you're doing those two questions.

And when you press play, we'll move to the next question.

So question three, I'd like you to go to the Desmos website, and there's a hyperlink there which takes you straight to the relevant place, and draw the graph of y equals sine of X.

So you need to type that in so that the Desmos will plot that for you.

You'll need to change a few settings.

So you need to go to the spanner, which will go onto the settings, change it to degrees.

Also change your x-axis so you can input some values there from minus 360 up to 360.

So that's your range of x values.

And set the y-axis from negative 2 to 2.

And then you need to comment on what do you notice and draw a sketch of what you think the graph will look like if your x-axis goes from minus 720 to 720.

So once you've drawn your sketch, you can obviously change the values of your x-axis in the settings and have a look to see if you were correct in your prediction.

Press pause whilst you're doing all of that, and when you press play, we'll go for our answers to task A.

So using the calculator or the unit circle link, you needed to draw the sine graph between 0 degrees and 360 degrees.

So here I've used Desmos to plot that for us.

And then you needed to write down any key features, and things that I was hoping you were going to mention, so it looks like a wave.

The maximum value is 1.

So the highest y value that it takes is 1, and the minimum value it takes is at negative 1.

And it touches or crosses the x-axis three times, so at 0, at 180, and at 360.

And if you think about the unit circle, 0 is when the angle of rotation is 0.

Then 180, it's now on the negative x-axis, the radius.

And then at 360, it's done its full turn, so it's back on that positive x-axis, the radius that I'm talking about.

On question three, you needed to go to the website, change the settings, and then comment on what you notice and draw a sketch of what you think the graph would look like between minus 720 and 720.

So the top graph is the one you will have seen when you've changed all the settings, put minus 360 to 360, and the bottom one is from minus 720 to positive 720.

So what did you notice? Well, that sine wave that we saw in question one is repeated in the negative part.

Negative 360 through to 0 has exactly the same wave as 0 to 360.

And if we then extended that interval from minus 720 to positive 720, then it repeats itself two more times.

And so we can see that the sine curve has this repetitive nature to it.

However, it still only goes to 1 and down to minus 1.

Its range is 1 to minus 1, no values greater than 1 and no values less than minus 1.

So we've now finished looking at the sine graph.

So let's now move on to the second learning cycle, where we'll be focusing on the cosine graph.

So we're looking at the cosine graph in this learning cycle.

So in a similar way that we use the unit circle to get our values for sine theta, we're now gonna do the same to look at the values for cosine theta, but this time, the cosine theta value is the x coordinate of the point.

So that point along the circumference, which is coming from a rotation of the radius, it's the x coordinate that gives us our cosine value.

And that's because that is the length of the adjacent of that right-angled triangle.

And we know that cosine is the relationship between adjacent and hypotonus, and this hypotonus is 1.

So cosine of theta is equal to adjacent when the hypotenuse is equal to 1.

So the x coordinate is our values.

So we can, again, generalise that point to say that's cosine theta.

We do know that that y value is the sine theta value, but for the moment, we're just gonna focus on cosine.

So for the check, just like we did for sine theta, we can create a table of value for cosine theta by using the calculator or reading from our unit circle.

So your check is to complete the table of values.

Pause the video, fill in those values, and then when you're ready to move on, press play.

So cosine of 0 is 1.

Cosine of 30 is 0.

9, rounded to one decimal place.

Cosine of 45 is 0.

7, rounded to one decimal place.

Cosine 60 is 0.

5, and cosine 90 is 0.

So hopefully, you have got those values and would've recognised that we've seen those values before.

It is quite hard to see any trend when the values are in that table for the same reason we discussed with sine.

So Andeep says, "Can we also treat these as coordinate pairs and plot them?" What do you think about that? Just have a moment to think about that or discuss it if you're sat with somebody before we continue.

So, absolutely.

This time, the x coordinate will be your value of theta, but the y coordinate will be the value of cosine.

So plotting the coordinate pairs results in this.

So we've plotted 0 to 1, 30 and 0.

9, 45 and 0.

7, 60 and 0.

5 and 90 and 0.

So these points are also gonna be joined with a smooth curve.

Alex recognises that the way we're going to draw the graph here is by joining them up with a smooth curve.

So here is that smooth curve, passing through those five points.

Alex says, "I think the graph will also be a wave." Izzy said, "I think it'll also go between minus 1 and 1 on the y-axis." So these were the properties of the sine curve.

We said that the sine curve looked like a wave, the sine wave, and we also noted that it had a maximum value of 1 and a minimum value of negative 1.

So what do you think? Pause the video.

Think about that.

Do you think the cosine graph will be very similar to the sine graph? Discuss it if you can, and then when you're ready to look at it a bit further, press play.

So let's have a look.

So in your final task of the lesson, you're going to use your calculator or the GeoGebra profile that's linked to draw the cosine graph between 0 degrees and 360 degrees.

And then for question two, you're gonna write down any key features of the graph that you observe and maybe you are going to see what they suggested.

So pause video whilst you do that, and then when you press play, we'll move to question three.

So question three is very similar to what you did in task A.

Go to the Desmos website.

Ask it to plot y equals cosine of x.

Go to the settings, make sure that it's in degrees.

Put your interval for the x values as minus 360 to 360 and your y-axis from negative 2 to 2.

Think about what you notice and then draw a sketch of what it will look like from negative 720 up to positive 720.

Pause the video whilst you do that and then when you're ready to go through the answers, press play.

So question one, draw the cosine graph.

Well, the cosine graph starts at 0,1 and then we saw it all the way down to 90, 0.

And then as we go past that into values greater than 90, we can see this sort of bucket shape.

So what did you notice? Well, it does look like a wave.

Its maximum value is 1, and its minimum value is negative 1.

It touches/crosses the x-axis twice.

So there is the difference to sine.

Sine, we said, touched and crossed the x-axis three times between 0 and 360 degrees.

Whereas on the cosine graph, we're only crossing the x-axis at 90 and 270.

On question three, you needed to go and use the Desmos to really look at the graph rather than drawing over and over again using the graphing feature to look at it over a larger interval.

So what did we notice? Well, once again, we can see the cosine graph between minus 360 and positive 360 at the top, and then from minus 720 to positive 720 is the second graph.

So what do you notice? Well, it's still got maximum values of 1 and minimum values of negative 1.

So that is the range of the cosine function.

It repeats in a similar way to that sine did.

The shape that we see between 0 and 360 is repeated between 360 and 0 and is the same on from negative 720 all the way up to positive 720.

So this repetitive nature of the cosine curve, and that's why we might think of it like a wave.

So to summarise today's lesson on drawing the sine and cosine graphs, values for sine theta and cosine theta can be generated using the calculator or the unit circle, and we can plot the coordinate pairs of theta to sine theta in order to draw the graph of y equals sine theta.

And we can plot the coordinate pairs of theta and cosine theta in order to draw the graph of y equals cosine theta.

Both graphs look like a wave.

Both have a maximum value of 1 and a minimum value of negative 1.

So there are some really key similarities between sine and cosine, and that shouldn't be too surprising when you just think about how they are named.

One is sine, and one is cosine.

So there is a really strong relationship between the two of them.

Really well done today, and I look forward to working with you again in the future.