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Further sketching of the trigonometic functions

We explore sketching the graphs of the sine, cosine and tangent functions.
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In this video, we will look at some further discussion of the trigonometric functions with Dr. Ria Symonds.
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The aims and objectives for this video to be able to sketch the graphs of cosine, sine, and tangent functions, and we’ll be referring to the amplitude and period of those functions, and also know some exact values of the sine, cosine, and tangent angles in radians as first refer back to our graphs of the sine and the cosine functions from our previous video.
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So recall the cosine and sine graphs create this wave. And we create this wave by looking at the unit circle. So for the cosine graph, we can think about starting at when x is 0. So here we start out when y is 1. And we get this wave function or wave sketch, and it repeats itself every so often. It’s exactly the same for the sine function. This time when we start at 0 though, we start at when y is 0. So you get the exact same curve, but it’s slightly translated along the x-axis. Now when we look at these curves, they have a very similar shape. What I can say is that they’re bounded.
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I’ve got a very top value and a very top bottom value of where the curve sits. This top value is 1, and this bottom value is minus 1 and exactly the same for the sine function. What I can also see is that every so often it repeats itself. So if I maybe start looking here– so when x is 0– and I follow my curve along, when I get over here, which is 2 pi, the curve starts again. So I start at the top. Da, da, da, da– I get back to the top and it repeats itself. Now these descriptions of how we describe this curve are called particular things. We’ve got the amplitude.
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The amplitude is half of the vertical height. So here, since we’re looking between 1 and minus 1, that’s a height of 2 If we look from top to bottom. So half of that will be 1. So the amplitude of the cosine and sine functions is 1. And then the period describes the horizontal width of one wave section. And we’ve already seen that one width is here, repeats itself every 2 pi. So the period for cosine and sine is 2 pi. Now one of my functions look a little bit different, so let’s say I’ve got y equals a times cos of bx. What happens in terms of the amplitude and the period now?
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Well we can take that value of a to describe the amplitude. So the amplitude is given as a, but, in fact, I’m going to take the modulus of a. Because if this was a negative value, then to describe the vertical height as a negative number, would it really make any sense? So we take the modulus to describe the height as a positive value. So the amplitude always has to be positive. We then use this value b to work out the period. So for the period, I know that one period of cosine is 2 pi, so then I divide it by modulus b. So whatever that value is divide by, again, the positive value of b.
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And I can do exactly the same for sine. So let’s say I had y equals a sine bx. Again, the amplitude is described as modulus a, and the period is 2 pi over the modulus of b. Now as I look along these graphs, I can see that there might be particular angles that are exact. I.e. I can take particular values along the x-axis. So we’re writing this in terms of radians, so 0 pi, 2 pi. They give me exact values when I put it into either the cosine or the sine function. So some of them are very useful to know, especially when we’re doing further calculations with those functions.
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So some of the important ones that we should look at if we think about moving along the x-axis. So let’s start at 0 and choose some particular points. Well cos of 0, I can see, is 1, because at 0, I’m at the top of the graph, which is 1. If I move along the x-axis, cos of pi by 2 is here, and that is 0. Cos of pi, which is here, gives me a bottom value of minus 1. And then cos of 3 pi by 2 gives me 0. So that’s here. We can do the same thing for the sine function. So we’re going to start at 0. So sine of 0 is 0. Sine of pi by 2.
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So pi by 2 is about here. I’m at the top of my curve. So that’s 1, sine of pi would be 0. That’s here and sine of 3 pi by 2– oh, that should be a sine, not a cos, sorry– is minus 1 because it’s down here. So this is 3 pi by 2. So if you remember how to sketch the graphs, you can kind of read these exact values off. But it’s nice to remember them just so you can easily refer to them if you’ve got further calculations. Now we’ve not yet looked at the tangent function, so let’s have a look at the sketch of the tangent function.
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Now we can’t quite easily sketch it using the unit circle, but what I can do is use the relationship of tan x– i.e. the relationship between sine and cosine– to graph my tangent function. So tan is described as sine of x over cosine of x. So if you try putting some values in, what you’ll find is you get the shape of the graph over here where you kind of get like s shapes that go up through the x-axis at certain points. So for example, it crosses at 0, pi and 2 pi. So every pi it crosses. You’ll get that shape. So notice again it repeats itself every so often.
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But what you also see are some kind of gaps in between the graphs, and these are asymptotes. So it will never ever reach that point. And the reason why it’ll never ever reach that point– so this is pi over 2, the next one will be 3 pi over 2– is because we’re dividing by cos of x. And at certain points, cos of x give you 0. So cos of pi by 2 is 0. We can’t divide by 0, so therefore there is no value for the tan function at pi by 2 or 3 pi by 2. Now we can kind of describe tan in terms of its amplitude and its period, kind of.
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In fact, there is no amplitude because if you think about the amplitude, it’s half the horizontal height, but this graph just keeps getting bigger and bigger and bigger or smaller and smaller and smaller. So there’s no amplitude.
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Let’s just say it doesn’t exist. Well, I mean, we could say it’s infinite or it’s infinite. There is a period, though. So this will repeat itself every so often. And I’ve seen that it repeats itself every pi. So the period of tan is pi. And if I’ve got the function a tan bx, I take pi and divide it by b. And the same kind of way that those exact values of some of the angles for cosine and sine, there is for the tangent as well. So some of those– oh, sorry let’s just delete that bit out. So tan of 0 is 0. Tan of pi is 0.
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And then we’ve seen that at those half values of pi– so pi by 2, 3 by 2, and so on– it’s undefined. We get these asymptotes. OK, let’s finish up this video, let’s just see if we can sketch then a couple of functions knowing what we’ve just done. So I’ve got two functions here– 2 cos of x over 3 and y equals 1/5 sine of 4x. Let’s try and sketch both of these now. So we should be able to write down what the amplitude is and what the period is for both of those functions. Let’s start my first function. So cos looks like this as the graph.
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Now, I know that this 2 in front means that my amplitude is 2. So half the vertical high is now going to be 2, not 1. So that means I’m going to be stretching my graph. Here at my b values, it’s like I’ve got 1/3. So the period of this function is going to be 2 pi divided by 1/3, which is 6 pi. So that means it’s going to repeat itself now every 6 pi rather than 2 pi. So I’m kind of stretching it out that way. So I’m stretching it this way and this way. What that’s going to give me is a graph that looks like this. So you can see it’s kind of bigger and fatter.
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What about the second function? So this is a sine function. Well, the sign graph looks a bit like this. Again, if we think about the amplitude– so the amplitude is now 1/5. Now these are both positive values for these particular examples. But if I had a negative value there, I would use a positive value, modulus. So if the amplitude is 1/5, I’m going to be squishing it down. And say our equivalent value is 4– so I’m going to work out the period. I take 2 pi and divide it by 4. So that’s going to give me pi over 2. So that means the curve is going to repeat itself every pi over 2.
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So I’m going to be pushing it in. So that means when I sketch this graph, it looks like this. So we’ve condensed it down. But the curves happen more frequently. They repeat themselves every pi over 2. So knowing those things about the amplitude and the period is really useful for us to help sketch these graphs. OK, so let’s check that we’ve hit our aims and objectives. So we’re able to sketch now the graphs of the cosine, sine, and tangent functions. And we know how to change the amplitude on the period. And we’ve also seen some exact values of some of the cosine and tangent angles in radians. So move on to our next section.

Watch this video for a more detailed explanation of how to sketch graphs of the sine, cosine and tangent functions.

Try it yourself:

On paper, have a go at sketching the graph of y = 3tan(x) then comment below to share:
1. What is its period?
2. What is its amplitude?

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