Lesson video

Hi, I'm Mr. Bond.

In this lesson, we're going to learn how to sketch and recognise the graphs of y is equal to cosine x, y is equal to sine x, and y is equal to the tangent of x.

Let's start by recapping how to find the sine of an angle using your calculator.

We're going to work out the value of sine 32.

The first thing that I'd like you to do is make sure that your calculator is in degrees.

It should say D there and not R or G.

Once you've done this, you need to hit the following buttons to type in sine 32.

First, sin, that will bring up sine on your display like this, then three, then two, and finally, hit Equals.

And this will give you this number as a result.

So this is the value of sine 32.

Double-check that your number is the same as the one here.

So sine of 32 degrees is equal to 0.

530 to three decimal places.

Now we're going to start plotting the graph of y is equal to sine x.

We'll use a table of values to help us.

Here we have some angles from zero to 360 and we're going to find sine x for each of those angles.

If we do that using our calculators, we'll get these values, and we can plot these on a graph.

There they are.

But you can see we also have values of x on our graph from zero to negative 360, so let's find those as well using our calculator.

That would give us these values, and if we plot those onto our graph, it would look like this.

We need to connect to each of these points.

But this doesn't look quite right, does it? Normally, if the points aren't in a straight line, we'd connect them with a smooth curve.

So we'll need to find some more points and plot those on our graph as well.

But before we do, what symmetry do you notice? Will the graph change or stay the same if more information is plotted? Let's find out.

Let's find the value of sine x for these angles, 30, 45, and 60.

That will give us the following values.

And if we plot those on our graph, it would look like this.

We could plot some more values.

And in fact, we could choose any angles we wanted.

I've chosen 120, 135, and 150 degrees.

Sine x would them be this for those three values.

And if I plot those on my graph, it would look like this.

So that would change my line between those points to something like this.

What do you notice? Could you predict what the graph will look like between negative 360 and zero and negative 180 and 360? Here's what it would look like if we plotted some more points.

What about if even more points were plotted? Then we'd get an absolutely smooth curve.

Here it is.

The graph of y is equal to sine x is a smooth curve that repeats itself every 360 degrees.

Repeating graphs are known as periodic functions, so this is a periodic function.

These arrows now show the turning points.

The turning points are simply where the gradient turns from positive to negative, or from negative to positive.

Here's a question for you to try.

Pause the video to have a go and resume the video when you're finished.

Here are the answers.

In part a, you needed to sketch the graph of y is equal to sine x.

It didn't need to be perfect, because it's only a sketch, but it did need to have some key features, one of which, for example, it needed to go through the origin.

And then in part b, you needed to label the coordinates of each of the turning points.

Here's another question for you to try.

Pause the video to have a go and resume the video when you're finished.

Here are the answers.

In this question, you simply needed to write down the points where the graph cross the x-axis.

Now we're going to plot the graph of y is equal to cosine x.

Again, we'll use a table of values with some angles and then we'll find the cosine of those angles using our calculators.

That will give us these values.

And again, we could plot them on a graph like this.

Once again, we could choose negative values for x.

And if we find those on our calculators, that will give us these values.

And again, plotting those will look like this.

If we connect those with a line, like we did for sine, again, this doesn't look quite right.

We're expecting this to be a smooth curve again, but we can identify the turning points again.

What symmetry do you notice this time? Will the graph change or stay the same if more information is plotted? Well, of course, just like the sine graph, it's going to change.

Let's look at some more values.

If we use our calculators to find the values of cosine x for 30, 45, and 60 degrees, we'll get these values.

And if we plot those on our graph, it will look like this.

So that will mean we'll start to see a curve when we join these points.

If we find some more values, we'll find cosine x is equal to these values for these other angles.

And if we plot those on our graph, they'll look like this.

Now if we join each of those points, we start to see a smoother curve appearing.

Can you predict what the graph will look like between negative 360 degrees and zero degrees and negative 180 degrees and 360 degrees? It'll look like this.

What about if more points were plotted? Again, if more points were plotted, we can start to see a smooth curve again.

So the graph of y is equal to cosine x is a smooth curve that repeats itself every 360 degrees, just like y is equal to sine x, which means it's also a periodic function.

Here's another question for you to try.

Pause the video to have a go and resume the video when you're finished.

Here are the answers.

You needed to identify the coordinates for each of the points A to G.

Hopefully you got all of those right and it's also worth pointing out that B, D, and F are the turning points shown on this graph and A, C, E, and G are where the graph intersects the x-axis.

Finally, we're going to investigate the graph of y is equal to the tangent of x.

Again, we'll use a table of values.

This time, when you use your calculator to find values for the tangent of x for each of these, something unusual happens.

For zero, 180, and 360 degrees, we get zero, but for 90 and 270 degrees, this is undefined.

You might also have seen the phrase Math error on your calculator when you try to do this.

And this also appears when we try to divide by zero on our calculators.

We say that this is undefined for those values.

We're going to try and plot this on a graph again.

We're also going to need to find values for zero to negative 360, and that will look like this.

Again, we have a couple of values that are zero and two values that are undefined.

If we plot this information onto our graph, we'll get this.

Does that help us plot this graph? No, a lot more information is needed.

Let's have a think about these values then, 30, 45, and 60 degrees.

If we find the tangent of these angles, we'll get these values, so that gives us three more values that we can plot onto our graph.

We could also choose some more values between 90 and 180, 120, 135, and 150.

If we use our calculators to find the tangent of these angles, we'll get these values.

And we can plot those onto our graph.

We still need more information though.

And what will undefined look like? The fact that the tangent of x is undefined for negative 270, negative 90, 90, 270, and so on means that we won't be able to plot any points on our graph along these dashed lines.

These dashed lines are known as asymptotes.

But what happens as we get very close to these asymptotes? Let's have a look at these three values, 89, 89.

9, and 89.

99.

If we find the values for the tangent of x for these three angles, we'll get these values.

So as we get closer to 90, the value of the tangent of x increases rapidly.

What about on the other side of the asymptote, so 91, 90.

1, and 90.

01? This time, the values decrease rapidly as we get closer to the asymptote.

This happens for all of the other asymptotes as well.

If we plotted lots of points, it would look like this.

So we can see that once again the graph of y is equal to the tangent of x is a periodic function.

But rather than repeating itself every 360 degrees, like y is equal to the sine of x and y is equal to the cosine of x, this one repeats itself every 180 degrees and it has asymptotes at 90 degrees and every 180 degrees above 90 degrees, and also 180 degrees below 90 degrees.

Here's a final question for you to try.

Pause the video to have a go and resume the video when you're finished.

Here are the answers.

First, you just needed to sketch the graph of y is equal to the tangent of x and then you needed to identify the equations of the asymptotes.

So the asymptotes are vertical, so they're going to be of the form x is equal to a constant.

And you can see that this is x is equal to 90, then every 180 degrees above 90 and every 180 degrees below 90.

That's all for this lesson.

Thanks for watching.