Lesson Video: Graphs of Trigonometric Functions Mathematics

In this video, we will learn how to graph trigonometric functions, such as sine, cosine and tangent, and deduce their properties.

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Video Transcript

In this video, we’ll learn how to graph trigonometric functions such as sine, cosine, and tangent and deduce their properties. We’ll learn how to apply simple transformations to graph our functions in this form and recognize the relationship between each graph and the unit circle. Here is a unit circle. It’s a circle with a radius of one unit, which we’ve plotted with its center at the origin of a pair of 𝑥𝑦-axes. Let’s add a point 𝑃, which can move around circumference of the circle. We could call its coordinates 𝑥, 𝑦. But what if we then add a right-angle triangle to our diagram and define an angle here. This is called the included angle, and we’re going to call it 𝜃.

At the moment, we see that the length of the base of the triangle must be 𝑥 units, and its height must be 𝑦 units. We could, however, use trigonometry to find expressions for 𝑥 and 𝑦 in terms of 𝜃. We use the standard convention for labeling right-angled triangles. The side opposite the included angle is the opposite side. The longest side, that’s the one that sits opposite the right angle, is the hypotenuse. And the other side, the side between the right angle and the included angle, is the adjacent. We recall the acronym SOHCAHTOA. And we see that we can use the cosine ratio to find a link between 𝑥 and 𝜃. This is cos of 𝜃 is adjacent over hypotenuse.

Well, the adjacent side in our triangle is 𝑥 units and the hypotenuse is one unit, so cos 𝜃 is 𝑥 over one or simply cos 𝜃 is equal to 𝑥. Similarly, we’ll use the sine ratio to link 𝑦 and 𝜃. This time, the opposite is 𝑦 and the hypotenuse is one. So, we get sin 𝜃 is 𝑦 divided by one, or simply sin 𝜃 is equal to 𝑦. We can now say that the point 𝑃 must have coordinates cos 𝜃, sin 𝜃. Now, as we move 𝑃 around the circumference of the circle in an anticlockwise direction, the size of 𝑥 and 𝑦, and therefore the size of cos 𝜃 and sin 𝜃, will change. In fact, they do something really interesting. Let’s see what that is.

Let’s imagine our point lies on the positive 𝑥-axis here. At this point, 𝜃 is zero. And the point has coordinates one, zero. When 𝜃 is zero, then, we see that cos of 𝜃, which represents 𝑥, is equal to one. sin of 𝜃, which represents 𝑦, is equal to zero. We’ll now repeat this process here. This time, 𝜃 is equal to 45 degrees. And we can add a right-angled triangle and we see the hypotenuse is equal to one unit. However, since this is a right-angled triangle with one angle of 45 degrees, we know in fact it’s an isosceles triangle. And so, we can say that its other two sides are equal to 𝑎 units.

The Pythagorean theorem tells us that the sum of the squares of the two shorter sides is equal to the square of the longer. That is, 𝑎 squared plus 𝑎 squared equals one squared or two 𝑎 squared is equal to one. We divide through by two, and we find 𝑎 squared is equal to one-half. And then we find the square root of both sides. Now, 𝑎 is a length, so we’re going to say that it’s the positive square root of one-half, which we can write as root two over two. And so, we see when 𝜃 is equal to 45 degrees, our point has coordinates root two over two, root two over two. This tells us that sin of 𝜃 when 𝜃 is 45 degrees is root two over two, as is cos 𝜃.

And what about up here? Well, this time 𝜃 is 90 degrees. And of course, our circle has a radius of one. So, this point has coordinates zero, one. And we see then when 𝜃 is 90 degrees, cos of 𝜃, which is the 𝑥-value, is zero and sin of 𝜃, which is the 𝑦-value, is one. We can repeat this process for when 𝜃 is equal to 180 degrees. At this point, we have a point with coordinates negative one, zero. So, when 𝜃 is 180 degrees, cos of 𝜃 is negative one and sin of 𝜃 is zero. Then, when 𝜃 is 270 degrees, our point has coordinates zero, negative one. So, sin of 𝜃 is negative one, remember that’s the 𝑦-coordinate, and cos of 𝜃 is zero.

And then, we continue around the circle and we see we get back to the start. So, when 𝜃 is equal to 360 degrees, we have the same values for sin 𝜃 and cos 𝜃 as we did when 𝜃 was equal to zero. sin 𝜃 is zero, and cos 𝜃 is one. Let’s fill in these gaps. At this point, 𝜃 is 135 degrees. If we add in a right-angled triangle and use the fact that angles on a straight line sum to 180 degrees, we see we have a replica of the earlier triangle we looked at. It has a hypotenuse of one unit and an included angle of 45 degrees. This means the lengths of its other two sides must be root two over two units. In coordinate form then, this point must have coordinates negative root two over two, root two over two. And so, when 𝜃 is 135 degrees, cos of 𝜃, which is our 𝑥-coordinate, is negative root two over two, and sin 𝜃 is root two over two.

When 𝜃 is 225 degrees, we have a similar situation. We have another isosceles triangle with a hypotenuse of one unit and two other sides of root two over two units. This means when 𝜃 is 225 degrees, our point must have coordinates negative root two over two, negative root two over two, meaning sin of 𝜃 and cos of 𝜃 are both negative root two over two. Finally, when 𝜃 is 315 degrees, we have another isosceles triangle. This time, the coordinates of our point are root two over two and negative root two over two, meaning that when 𝜃 is 315 degrees, sin 𝜃 is negative root two over two and cos 𝜃 is root two over two.

Now, it should be quite clear that if we were to continue moving around this circle in an anticlockwise direction, we would meet all of these points once again. So, when 𝜃 is 405 degrees, sin of 𝜃 and cos of 𝜃 are the same when 𝜃 is equal to 45 degrees and so on and this leads us to a definition. We say that sin of 𝜃 and cos of 𝜃 are periodic functions; that is, they repeat. They do so every 360 degrees. So, we say their period is 360 degrees. Let’s sketch the graphs. We’ve seen that both graphs oscillate; that is, they move between the values of one and negative one. Those are their maximums and minimums.

So, we’ll start with the graph of sin of 𝜃 or, in this case, 𝑦 is equal to sin of 𝑥. It passes through the origin; that is, when 𝜃 is zero, sin of 𝜃 is zero. So, when 𝑥 is zero, sin of 𝑥 is zero. Then, when 𝑥 is 45, we get sin of 𝑥 to be root two over two. That’s roughly here. When 𝑥 is 90, sin of 𝑥 is one. Then, when 𝑥 is a 135, sin of 𝑥 is root two over two. And when 𝑥 is 180, sin of 𝑥 is zero. We could continue in this manner. Let’s join these up with a smooth curve. And when we do, we find the graph of 𝑦 equals sin of 𝑥 between the values of zero and 360 looks a little bit like this. Of course, we said that these graphs are periodic. They repeat, so we could continue in this manner, repeating this exact graph every 360 degrees.

Similarly, the graph of 𝑦 equals cos 𝑥 in the interval from zero to 360 looks like this. Since it is also periodic, we could continue either side in the same manner. And so, to recap a few features of our graphs, they’re periodic; they have a period of 360 degrees. They both have maxima at one and minima at negative one. Both of them, in fact, have a little bit of symmetry. The graph of 𝑦 equals cos of 𝑥 has reflectional symmetry about the line 𝑥 equals 180 or the line 𝑥 equals zero. The graph of 𝑦 equals sin of 𝑥 has rotational symmetry about the origin. But if we narrow it down and just look at, say, a part of the graph, that is, the interval from zero to 180, we see that 𝑦 equals sin of 𝑥 does have a little bit of reflectional symmetry about the line 𝑥 equals 90 degrees.

These features can help us to solve problems involving cos and sin of 𝑥. The graph of 𝑦 equals tan of 𝑥 is a little bit stranger than this. And rather than using the unit circle method, we’re going to plot a table of values using our calculator. For values of 𝜃 of zero, 45, 90, 135, and so on, we get the following table of values. Notice we have an error at 𝜃 equals 90 degrees and 𝜃 equals 270 degrees. This will continue every 180 degrees. But what’s actually happening here? Well, we know tan 𝜃 is opposite over adjacent, but we can’t actually draw a right-angled triangle at 𝜃 equals 90 degrees. And so, we say that as 𝜃 approaches 90 degrees, tan of 𝜃 approaches ∞. We can’t evaluate it, and so we represent this fact using vertical asymptotes on our graph.

The graph of 𝑦 equals tan 𝜃 gets closer and closer to these asymptotes but will never quite touch them. And so, the graph of 𝑦 equals tan of 𝑥 looks a little something like this. Once again, we see that the function tan of 𝜃 is periodic. But this time, its period is 180 degrees. It has rotational symmetry about the origin. And we can’t define its maximums or minimums because we saw that as 𝜃 approaches 90 and then multiples of 180 degrees, tan of 𝜃 approaches ∞.

It’s really important that we can identify and sketch the graphs of 𝑦 equals sin of 𝑥, 𝑦 equals cos of 𝑥, and 𝑦 equals tan of 𝑥. And we also need to be able to transform them such that for the graph of 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑥 plus 𝑎 is a translation by the vector negative 𝑎, zero, whereas 𝑦 equals 𝑓 of 𝑥 plus 𝑏 is a translation by the vector zero, 𝑏. 𝑦 equals 𝑓 of 𝑎 times 𝑥 is a horizontal stretch with a scale factor one over 𝑎. And 𝑦 equals 𝑏 times 𝑓 of 𝑥 is a vertical stretch by a scale factor of 𝑏. Then, 𝑦 equals negative 𝑓 of 𝑥 is a reflection in the 𝑥-axis, and 𝑦 equals 𝑓 of negative 𝑥 is a reflection in the 𝑦-axis.

Finally, sometimes we measure angles in radians such that two 𝜋 radians is equal to 360 degrees and 𝜋 radians is 180 and so on. Now, don’t worry if you’ve not encountered these yet, they’re just another way of representing an angle. So, let’s have a look at some questions on trigonometric graphs.

Assign each plot shown in the graph below to the function it represents.

Here, we see we have two very similar-looking graphs. These graphs are clearly periodic. They appear to repeat every two 𝜋 radians. Remember, that’s repeating every 360 degrees. We see they have maximums at one and minimums at negative one, respectively. In fact, we know that we can describe the graphs of the sine and cosine functions in the same way.

The main difference is where these graphs intersect the 𝑦-axis. 𝑦 equals sin of 𝑥 passes through the 𝑦-axis at zero, whereas 𝑦 equals cos of 𝑥 passes through at one. And we said, of course, that both have a period of 360 degrees or two 𝜋 radians, maxima at one, and minima at negative one. This means that the red plot which intersects the 𝑦-axis at one must be the cosine curve, whereas the blue plot must be the sine curve.

Now, this actually shows a really interesting feature of these functions. Let’s define 𝑓 of 𝑥 to be equal to sin of 𝑥. Then, we see that the function 𝑓 of 𝑥 plus 𝜋 over two or 𝑓 of 𝑥 plus 90, which represents a translation by the vector negative 90, zero, maps the sine curve onto the cosine curve. And of course, the converse is also true. So, we’ve seen that there’s a relationship between the sine and cosine graphs by a horizontal translation. Now, let’s look at a stretch.

Find the maximum value of the function 𝑓 of 𝜃 is equal to 11 sin 𝜃.

Firstly, we’re going to recall what the graph of 𝑓 of 𝜃 equals sin of 𝜃 looks like. It has maxima and minima at one and negative one, respectively. We know that it passes through the origin and that it’s periodic and it has a period that repeats every 360 degrees. So, 𝑓 of 𝜃 equals sin of 𝜃 has a graph that looks a little something like this. But of course, we were actually interested in the graph of the function 𝑓 of 𝜃 is 11 sin 𝜃.

And so, we recall that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑎 times 𝑓 of 𝑥 represents a vertical stretch by a scale factor of 𝑎. In this case, we can see our scale factor is 11. And so, 𝑓 of 𝜃 equals 11 sin 𝜃 looks something like this. It still intersects the 𝑥- and 𝑦-axes at the same places, but now it travels as high as 11 and as low as negative 11. And so, the maximum value of the function 𝑓 of 𝜃 equals 11 sin 𝜃 is 11.

We’ll now have a look at a reflection.

Which of the following is the graph of 𝑦 equals negative tan of 𝑥?

Let’s just begin by recalling what the graph of 𝑦 equals tan of 𝑥 looks like. It’s periodic, and it repeats every 180 degrees. It passes through the origin, the point zero, zero. It has vertical asymptote at 𝑥 equals 90 degrees but also every 180 degrees either side of this, in other words, 𝑥 equals negative 90, 𝑥 equals 270, and so on. In fact, the graph of 𝑦 equals tan of 𝑥 is this one. It’s (A).

Notice that the graph approaches the asymptotes but never actually quite touches them. But of course, we were interested in the graph of 𝑦 equals negative tan of 𝑥. So, we recall that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals negative 𝑓 of 𝑥 is a reflection in the 𝑥-axis. And we see that the only graph that matches this is (D). (D) is the graph of 𝑦 equals negative tan of 𝑥.

In this video, we’ve learned what the graphs of 𝑦 equals sin of 𝑥, 𝑦 equals cos of 𝑥, and 𝑦 equals tan of 𝑥 looks like. We saw that the graphs of 𝑦 equals sin of 𝑥 and 𝑦 equals cos of 𝑥 are periodic; they have a period of 360 degrees, whereas the graph of 𝑦 equals tan of 𝑥 repeats every 180 degrees. Finally, we saw that 𝑦 equals sin of 𝑥 and cos of 𝑥 have maxima and minima at one and negative one, respectively, whereas the graph of 𝑦 equals tan of 𝑥 has asymptotes at 𝑥 equals 90 and then multiples of 180 degrees.

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