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.

An outcome today is to be able to draw the graph for the tangent trigonometric function.

On the screen, there are keywords that I'll be using during the lesson.

So I'm gonna talk us through them now.

You have met them before in your learning, but it's important that we feel familiar and comfortable before we make a start.

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

The sine of an angle is the y-coordinate of point P on the triangle formed inside the unit circle, and the cosine of an angle is the x-coordinate of point P.

A tangent to a circle is a line that intersects the circle exactly once, and the tangent of an angle is the y-coordinate of point Q on the triangle, which extends from the unit circle.

The definitions about sine, cosine and tangent are all referring to the diagram on the slide.

So I would suggest you pause the video and really refer to that diagram, seeing where the sine theta value is, the cosine theta and the tangent theta values are located.

Press play when you're ready to move on.

We do have a new word for this lesson.

So the word we are looking at is period, and in this context, it's for a repeating function.

So for a repeating function, the period is the distance of one repetition of the entire function.

So we will be using that during the lesson when we're looking at the tangent graph as well as others.

Our lesson on drawing the tangent graph is going to be split into two learning cycles.

In the first learning cycle, we are gonna focus on drawing the tangent graph, whereas when we get to the second learning cycle, we're going to look at the sine, cosine, and tangent graphs together.

So let's make a start at first of all finding out how to draw the tangent graph.

So the unit circle allows us to work out values for tan of theta.

So the tangent of an angle is the y-coordinate of the point on the triangle which extends from the unit circle.

So we can see this on the diagram here.

We've got our tangent to the circle.

Remember a tangent meets a circle at exactly one point, and at this case, it is where the coordinate of one zero.

This tangent is a vertical line of x equals one.

And then we have a line that passes through the origin.

Where that intersects the tangent is where it forms this right angle triangle, and that right angle triangle is what we're going to make use of to get our values for tan, and it's that y-value of the point of intersection.

And you can see the generalised form of that point.

So the point is one and then tan theta.

The reason is one, the x-coordinate is one because that's what the value of the tangent line is.

So your check is to create the table of values by either using your calculator, so type in tan of zero, tan of 30, or by reading those values from the unit circle, and you can use the link there to a GeoGebra file to look at those values.

So I'd like you to pause the video and complete those five values for tan, and you can give your answers to one decimal place.

So tan of zero is zero.

Tan of 30 is 0.

6.

Tan of 45 is 1.

Tan of 60 is 1.

7.

And tan of 90 is undefined.

So when you did tan of 90 on your calculator, it will have come up with math error to indicate that this just isn't possible mathematically, so we say it's undefined.

If you looked at the unit circle file, then when you increase that rotation to 90 degrees, then there wouldn't have been a point of intersection, and that's because the line that passes through the origin is now parallel to the tangent.

So when we look at these tan values, it's quite hard to see a trend when it's presented in the table.

We've got things that make that more difficult and that's that the angle of theta is not increasing by the same amount each time.

We go from 0 to 30, that's an increment of 30 degrees, and then we go from 30 to 45, which is 15 degrees.

But also with our values, there doesn't seem to be any kind of pattern going on there, so it's quite hard to look at shape and trend from a table of values.

So Andeep says, "I'm not sure I can treat these as coordinate pairs." Why do you think Andeep thinks this? Pause the video and think about that, and then press play when you're ready to look at why.

So tan of 90 degrees is undefined, so you cannot plot that as a pair of coordinates.

You can't plot 90 degrees with undefined.

But what we can do is take a value very close to 90 degrees.

So if we take 89 degrees, that does have a value for tan, it's 57.

3.

You might be shocked at that value, you might be thinking that surely that's incorrect, but you'll see in a moment that it's not.

So if we look at this as an x and y-axis.

Now, x is our theta value and our y is our tan of theta, then we can see the purple crosses are our pairs of coordinates.

So we've got zero, zero, then we've got 30, 0.

6, 45, 1, 60, 1.

7, and lastly 89 and 57.

3.

So because that value for 89 degrees is so much larger, it's quite hard to distinguish the difference of the first four.

And Alex recognises that this graph curves upwards rapidly, and that difference between the value for 60, which was 1.

7, and the value for 89, 57 point something.

So if we plot that with a smooth curve between those five points, you can see that this graph is curving upwards rapidly, as Alex said.

So Alex says, "We have seen graphs with undefined values before." So your check is give an example of the type of graph that Alex could be thinking of.

So what graphs have you met before in your learning where there were undefined values? So pause the video.

It might be that you need to go and look that up, do a bit of research, have a go on Desmos and type some graphs in to see which ones have undefined features.

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

So Izzy said, "Yes, reciprocal graphs of the form y = a/x are undefined when x or y equals zero." And that's because we can't divide by zero.

So Izzy has just given one example.

So there are other examples you could have chosen that would also be valid, but the reciprocal graph is one of the most common ones that you will be familiar with where there are values that are undefined.

So on this task you're gonna use your calculator or the unit circle, that link there, to draw what you think the tangent graph will look like between zero degrees and 360 degrees.

And then on question two, you're gonna write down any key features of the graph that you observe from your drawing.

So pause the video whilst you do that.

Don't be surprised if you get some very large values for tan theta.

Press play when you're ready to go onto the next part of the task.

So here is question three, which is the final question of task A.

And here you're gonna use the Desmos link and draw the graph of y equals tan of x.

You will need to go onto the settings, which you can find under the spanner, to change the settings to degrees rather than radians, and set the x-axis from minus 360 to 360.

So that's gonna be our interval for the x values.

What do you notice? And then I would like you to draw a sketch of what you think that graph will look like if you change the x-axis, so that interval is from minus 720 degrees up to positive 720 degrees.

So pause the video, access that link, change the settings, and then look at what you notice.

Press play when you're ready to go through our answers.

So on question one, you should have got a graph that looked like this when you plotted it.

We were looking at the interval of our x values from zero degrees to 360 degrees on question one.

We'd already seen that first part between 0 and 90.

Remember that at 90 was undefined.

Then from 90 up to 270, we've got, we've had this shape here, and then from 270 to 360, we've got sort of part of that shape.

So what did you notice? Key things that you may have written down.

So the graph looks like it jumps over some of the values on the x-axis, and at that point, we're discussing the 90 degrees and the 270 degrees.

You may have also said that it touches or crosses the x-axis three times, so that's at zero degrees, 180 degrees and 360 degrees.

You may have written things about it rapidly increasing and rapidly decreasing, and you can see that at those 90 and the 270 sort of extreme values.

Your question three was to go onto the Desmos website and use that to graph tan over a large interval for x values, and then to think about the sketch for the minus 720 to positive 720.

So the top one is our range from minus 360 to positive 360.

And hopefully you discussed in terms of what did you notice, there were still some of these sort of areas that it jumped and then it repeated, that shape we can see has repeated three times, and you could also think about the fact that there is a fourth one there, but it's in two parts, one right on the minus end and one at the positive end.

And then when we extend the interval for the x values from minus 720 to 720, it just repeats some more.

We still have these places where our values seem to jump, go from really, really positive values to really, really negative values.

So now we know what the tangent graph looks like, we're going to look at the sine, cosine, and tangent graphs together in this learning cycle.

So the graphs of y equals sine theta and y equals cosine theta are very similar.

We've got them both on the screen here.

We can describe them both as waves.

They both have a maximum value of one and a minimum value of minus one, so they are very similar as functions.

The graph of y equals tan theta is not similar to the other two graphs, and you can see that here, and that's what we were working with in the first learning cycle.

So now Alex is going to quiz his classmates on the trigonometric graphs.

See how many questions you can get right.

So here we've got Alex, he's gonna be our quiz master.

He's looking very dapper there.

And he's gonna ask us some questions and you need to see how you get on.

So Alex's first question, the sine graph is the only trigonometric graph that looks like a wave.

So is that true, y equals sine theta produces a wave? Is it B, false, y of cosine theta also produces a wave? Or is it C, false, y equals tan of theta also produces a wave? So pause the video and then when you're ready to move on with Alex's quiz, press play.

So B is what you should have gone for.

Sine and cosine can both be described as a wave.

The tangent graph does have that sort of curve to it that there isn't, it's discontinuous.

It's a non-continuous function.

Question two, for which functions can y only take values between negative one and one? So pause the video, have a think about that.

When you're ready to check your answers with Alex, then press play.

So, sine and cosine have ranges between negative one and one.

The maximum value is one, the minimum value is negative one, whereas the tangent function goes to positive infinity and down to negative infinity.

Question three, select the graph of y equals tan of theta.

So pause the video, and then when you're ready to look at the answer for that, press play.

So the answer was C.

A is a reciprocal curve, so it does have a feature like the tangent curve.

B is our sine curve, and C is our tangent.

Question four of Alex's quiz, part of the graph of y equals sine theta is shown.

What are the coordinates of point A? So is it 90, 1, 180, 1, or 360, 1? So pause the video and then when you're ready to check, press play.

So the coordinate of A is 90, 1.

So that is our maximum value there.

So they've all ended with one, which is the y value equals one.

But our graph, our sine curve over a 360 degree period, 90 is at that local maximum.

Question one, you're gonna investigate the graphs of sine, cosine and tangent, either by using the link to GeoGebra files or through Desmos or both.

And what you need to do is make a sketch of each graph and annotate the graph of its key features.

Those key features are the coordinates for points of intersection with the axes, coordinates for any local maximums or minimums, and equations of any asymptotes.

So pause this video whilst you do that.

That might take you a little while, but that's absolutely fine.

Really important that you think about those key features and get them annotated.

When you press play, we'll have a look at them together.

So, this is the sine curve.

So our axis is theta and sine theta.

And so you should have noticed that the graph repeats itself every 360 degrees, and this is the period of the function.

For example, between the 0 degrees and 0 to the 360 degrees and 0, that's one complete cycle.

So on this graph, you can see three complete cycles, from negative 360 to zero is a period of one, then from zero to 360 is the second period, and then from 360 to 720 is our third period.

So it's true for being negative values and as well as positive values that we can see this periodic nature.

So I did ask you to annotate the coordinates of whether it crosses the x-axis.

So for the sine curve, it's minus 360, 0, minus 180, 0, 0, 0, 180, 0, 360, 0, 540, 0, and 720, 0.

I also asked you to label up the local maximums or the local minimums. The reason we say local is in that area of the graph because there is more than one maximum and more than one minimum, and it is a continuous function.

So we've got minus 270, 1 is one of our maximums. 90, 1 is another one of our maximums. 450, 1 is the third one on this particular part of the sine curve.

And hopefully you can see that they are 360 degrees apart from each other because of the period being 360.

And then if we look at our local minimum, so that's the minimum y value that it takes, we've got negative 90, negative 1, 270, negative 1, and 630 negative 1.

Once again, every 360 we find ourselves another minimum.

Other features that hopefully you're recognising is that there is some symmetry to the graph as well.

So if between 0 and 180 degrees, that is symmetrical.

From 360 to 540, it is symmetrical.

From 180 to 360, it is symmetrical.

In a similar way to a quadratic and a parabola being symmetrical.

Cosine, we know cosine and sine are very closely linked.

They're both waves, and in the same way that they both repeat each other every 360 degrees, so that is the period.

So again, between minus 360 and positive 720, there are three periods, so we can see the wave repeating itself three times.

The coordinate axes, where did it meet the coordinate axes? Well, it met along the x-axis at minus 270 degrees 0, minus 90, 0, positive 90, 0, 270, 0, 450, 0, and 630, 0.

And it also meets the y-axis at zero, one.

Zero, one is one of our maximums. We also have one at minus 360, 1, positive 360, 1, and 720, 1.

So again, every 360 degrees, we find a new maximum, a new local maximum.

And then when we get our minimums, we've got minus 180 minus 1, positive 180 minus 1, and 540 minus 1, again every 360.

In terms of symmetry, this time the cosine graph is symmetrical across the whole period of 0 to 360.

There is a line of symmetry at the 180 degrees, and that's important as we use this further in mathematics.

So the tangent graph has got asymptotes, and those asymptotes is where the value of theta is undefined.

So we know that it's undefined at 90 degrees, but it's also undefined at 270 degrees, and undefined at 450 degrees, et cetera, et cetera.

And that's because the tangent graph repeats every 180 degrees.

So it crosses the x-axis at 0, 0, 180, 0, 360, 0, 540, 0, and 720, 0.

So every 180, it crosses back through the x-axis, and that does the same going negative.

It only crosses the y-axis at one point and that is zero, zero.

But we have got these lines that are the asymptotes, the part of the function where it's undefined.

So the period for the tangent function is 180 degrees, which is different to the sine and cosine.

They were both 360.

And it also has the asymptotes for the undefined parts.

That's where the graph we spoke about earlier looks like it jumps across.

So for this lesson, I'm drawing the tangent graph.

We have also looked a little bit at sine and cosine.

So the unit circle can help you predict what the tangent graph will look like.

The tangent graph has asymptotes, and the trigonometric functions have different periods.

So the sine and cosine have a period of 360, that means it repeats every 360 degrees, whereas the tangent function repeats every 180 degrees.

Its period is 180 degrees.