Graphs of trigonometric functions

We have defined the sine function and the cosine function by a definition mode that used the unit circle. And we’ve derived several properties of these functions. But you know, I don’t really feel, myself, that I know a function until I have an idea of what its graph looks like. So what are the graphs of these trigonometric functions we’ve defined? We’re going to examine that now. Let’s start with the sine function. As you know, the sine function is the y-coordinate of the point on the unit circle corresponding to the angle theta measured out in the standard way. Now, as theta increases from 0 to pi over 2, then the sine will increase from 0 to 1.

Remember pi over 2 means 90 degrees, quarter circle. And as theta, it continues to increase from pi over 2 to pi, 90 degrees to 180. Then the sine of theta will decrease from 1 to 0, the y-coordinate. And as theta increases from pi to 2 pi– so it comes back where it started– then the sine of theta goes down to minus 1, and then rises up again to 0. When you put all this and get the graph of the function, what you’ve shown is that the sine of theta has a graph whose shape is like this. It’s a sinusoidal, literally. And we’ve graphed it from 0 to 2pi.

So here’s our graph from 0 to 2 pi. What about the rest? What about other values of theta that are not between 0 and 2 pi? Well, here the key is to recall periodicity. The sine of x plus any multiple of 2 pi, positive or negative, will be the same as the sine of x. Because you always come back to the same point on the unit circle. So therefore, to get the rest of the graph, all you have to do is take this piece, which is of length 2 pi, and continue the same graph to the right. And then continue it to the left. And the graph will look the same everywhere, as this piece just repeated.

Now this is helpful in constructing the graph of certain functions, which you need to do from time to time, as in this example. We want to sketch the graph of the function 2 sine 2 x. We know the graph of the function sine. There it is. What is the graph of f going to look like? Well, it’s going to look like this for the reason that follows. The 2 in front of the sine means that, instead of alternating between minus 1 and plus 1, as the sine does, the extremes are going to be minus 2 and plus 2. So it’s going to be a bigger function with a higher amplitude, as we will say.

And the fact that the x is multiplied by 2 means that you’re going to have a double oscillation. It’s going to jump around faster. It’s going to oscillate faster. That’s the graph of the function 2 sine 2x. And we should have expected it to look like that– well, with a little experience, of course. Now here’s a fact about cosine and sine. If you start with the cosine of x, it’s equal the sine of pi over 2 minus x, because that’s a complementarity formula. Now that’s minus the sine of x minus pi over 2, because sine is an odd function. And then if you add pi, you get the sine of x plus pi over 2 by the antipodality formula.

So if you look at the extremes of these equations, you see that the cosine of x is always the same as the sine of x plus 2. This tells you that, if you look at the graphs of cosine and sine, the graph of the function cosine is just the graph of the function sine but translated to the left by pi over 2. So here’s the graph of sine. You translate it to the left by pi over 2. You reach the graph of cosine. So it’s pretty much the same curve, but shifted. These curves, by the way, the curves generated by sine and cosine, are called sinusoidal curves. So the canonical sinusoidal curve is the curve given by sine theta.

As you see, it’s an odd function. And when is it equal 0? Sine theta is 0 exactly when you’re at a multiple of pi, so 0 pi, 2 pi, 3 pi, and so forth, or minus pi, minus 2 pi to the left, and so forth. As for the other kind of sinusoidal curve from cosines, well, it gives you an even function. And that function is equal 0 if and only if you differ from pi over 2 by a multiple of pi– sinusoidal curves. Now what about the graph of tangent? We’ve ignored tangent so far. Well, tangent is defined in our new definition mode by being sine over cosine.

Evidently in this definition, you cannot consider the thetas for which the cosine is 0, because you’d have 0 in the denominator. In other words, the tangent is not defined when x is pi over 2 or differs from pi over 2 by a multiple of pi. At those points, the cosine will approach 0 as you approach those points. And the sine in the numerator will be either plus or minus 1 in the limit. And the tangent will blow up either to plus infinity or minus infinity. Now the tangent of x plus pi, it turns out by certain facts about antipodality, it turns out to be the same as the tangent of x.

So the tangent function is not only periodic of period 2 pi– which it is– but it has a smaller– the real period of the tangent function is actually pi. Now let’s get all these facts together. And we get the graph of the tangent function. We wind up with something like this. For example, between minus pi over 2 and pi over 2, the function tangent is strictly increasing. It’s 0 at 0. And as you can see, it has vertical asymptotes. In this case, as you approach pi over 2 from the left, the graph goes up to minus the plus infinity. And as you approach minus pi over 2 from the right, the graph goes down to minus infinity.

Now here’s a length, a pi interval, where we know the graph of tangent. But tangent is periodic of period pi. Therefore, to get the rest of the graph, all we need to do is repeat this block to the right and to the left, like so. And here is the graph of the function tangent x. Next, we’re going to be studying the subject of the dreaded trigonometric identities.