Lesson Video: Graphs of Trigonometric Functions Mathematics

Video Transcript

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

We’ll begin by looking at special angles on the unit circle. This is a circle of radius one unit whose center lies on the origin of the Cartesian plane. We might recall that the 𝑦-coordinates of the various points that lie on this circle correspond to the sine of the various angles. For instance, this point here tells us that sin of 30 degrees is one-half, and this point tells us that sin of 45 degrees is root two over two.

Now, we can draw the given table to relate the input, 𝑥, in degrees to the output for sin of 𝑥. Of course, since we can continue moving around the circle infinitely and in either direction, we could extend this table either way. This means that sine is a periodic function. In particular, sin of 𝑥 has a period of 360 degrees, or two 𝜋 radians. A key feature of sin of 𝑥, which is demonstrated in its graph, is that the function has a value of zero when 𝑥 equals zero degrees, and it increases to the maximum value one when 𝑥 equals 90 degrees.

By plotting the points from the table above, we can approximate the graph of sin of 𝑥. As we saw, since the sine function is periodic, we can extend it in either direction by repeating the graph over each 360-degree interval as shown here. We can now also see that the sine function has roots; that is, it intersects the 𝑥-axis at every 180 degrees, starting at zero. We can also see that there’s rotational symmetry about the origin; this means it’s an odd function. Let’s briefly summarize all of this.

The graph of the sine function has the following features. It has a 𝑦-intercept of zero and has a maximum value of one and a minimum value of negative one. It has roots at every 180 degrees, starting at zero. It is periodic with a period of 360 degrees or two 𝜋 radians. Finally, it’s also an odd function. Formally, that means that sin of negative 𝑥 is equal to negative sin of 𝑥. Informally, it means it has rotational symmetry about the origin.

Now, we can perform a similar process to find coordinates on the graph of 𝑦 equals cos of 𝑥. This time, the 𝑥-coordinates of the various points that lie on the unit circle correspond to the cosine of the various angles. In fact, the input–output table looks like this. This gives us the graph of the cosine function between zero and 360 degrees. Unlike the graph of sine, cosine begins at the maximum value one when 𝑥 is zero degrees and decreases to the minimum value negative one when 𝑥 is 180 degrees. Like sine, cosine is a periodic function with a period of 360 degrees. And so, we can extend this graph over a larger interval by making copies of the graph over the interval between zero and 360 degrees.

Let’s summarize the key features. The 𝑦-intercept of the cosine function is one, which is its maximum, and it decreases to a minimum value of negative one. It has roots every 180 degrees, starting at 90. It has a period of 360 degrees or two 𝜋 radians. Finally, it is an even function. In other words, cos of negative 𝑥 is equal to cos of 𝑥, and the graph is symmetric over the 𝑦-axis.

We will now consider our first example in which we will look at how to use these properties to help us recognize the graphs of these functions.

Which of the following is the graph of 𝑦 equals cos of 𝑥?

Remember, one of the key features of the cosine graph is that it has a 𝑦-intercept of one. This means we can immediately disregard any graphs that don’t pass through the point zero, one. So, we can get rid of options (B), (D), and (E) straightaway. Next, we know that it’s a periodic function and that it repeats every 360 degrees. Option (C) appears to have a much smaller period; in fact, it repeats every 120 degrees, so it can’t be this one.

This leaves only option (A). Let’s check by looking at the other properties. Some of the roots of 𝑦 equals cos of 𝑥 are 90 degrees, 270 degrees, and negative 90 degrees. We can see that option (A) passes through all of these values on the 𝑥-axis. The curve has maxima at one and minima at negative one. It is also an even function, so it’s symmetric over the 𝑦-axis. Hence, we have confirmed that option (A) is indeed the graph of 𝑦 equals cos of 𝑥.

In this example, we saw that it is possible to identify the graph of cos of 𝑥 centered at the origin from its features such as its 𝑦-intercept and its periodicity. The same principles apply to graphs of cosine and sine when viewed at values of 𝑥 shifted away from the origin. In particular, we can use the periodic nature of these functions to help us determine the location of key features of the graph. Let’s demonstrate this in our next example.

Consider the following figures. Part (1) Which function does the plot in the graph, figure (a), represent? Is it (a) cosine or (b) sine? Part (2) Assign each region of the plot in figure (a) to the corresponding quadrant of the unit circle in figure (b).

To answer part (1), let’s recap on the values of some of the coordinates of the graphs of the sine and cosine functions. The coordinates of points on the unit circle are given by cos 𝜃, sin 𝜃, where 𝜃 is the counterclockwise angle of the radius to that point measured from the positive 𝑥-axis. Specifically, in the given graph, we can see that the value of the function is zero when the angle is two 𝜋 radians; that’s a full revolution. The coordinate of the point on the unit circle that corresponds to an angle of two 𝜋 is one, zero, which means that cos of two 𝜋 is one and sin of two 𝜋 is zero. The given graph indicates that this function takes the value of zero at two 𝜋, so this agrees with the sine function. This is option (B).

To answer part (2), we can use the values of the 𝑦-coordinates on the unit circle. Let’s look at region A which lies between three 𝜋 over two and two 𝜋, where two 𝜋 is a full revolution and three 𝜋 over two is three-quarters of this. That means that region A must lie here, in the fourth quadrant. In a similar way, five 𝜋 over two must be a quarter turn further than a full turn. So, region B is here, in the first quadrant. Region C is between five 𝜋 over two and three 𝜋, one and a half turns, so it’s the second quadrant, meaning that region D must be in the third quadrant. So, we assign A to quadrant IV, B to quadrant I, C to quadrant II, and D to quadrant III.

So far, we have only looked at the graphs of 𝑦 equals sin of 𝑥 and 𝑦 equals cos of 𝑥. So, what happens if we multiply one of these by some constant real number? For instance, take the function 𝑦 equals two cos of 𝑥. This means we input the value of 𝑥 into the function 𝑦 equals cos of 𝑥 then multiply that result by two. This means that all of the values of 𝑦 would double. So, the maxima would now be two, and the minima would now be negative two. So, multiplying the entire cosine function by two gives us a vertical stretch by that scale factor. In fact, we will achieve a similar result when multiplying the sine and cosine functions by a single constant value. Let’s look in particular at what happens if we multiply by a different constant.

Consider the graph below. Which function does the plot in the graph represent? Is it option (A) 𝑦 equals cos of 𝑥, (B) 𝑦 equals two cos of 𝑥, (C) 𝑦 equals negative sin 𝑥, (D) 𝑦 equals sin 𝑥, or (E) 𝑦 equals negative cos 𝑥?

In this example, we have been given a graph and need to decide which of the options represents it. Since all of the options include sine or cos or constant multiples of these, we should begin by looking at the features that these functions have. Let’s first recall the 𝑦-intercepts of sine and cosine. The sine graph passes through the 𝑦-axis at zero, whilst the cosine graph passes through at one. Comparing this to the graph given, we see that the 𝑦-intercept is at zero, meaning that 𝑦 equals cos of 𝑥 cannot be an option. In fact, we can see that this also applies to the other options that are multiples of cos of 𝑥.

In particular, the graph of 𝑦 equals two cos 𝑥 is found by multiplying the values of the output of cos of 𝑥 by two. This means the 𝑦-intercept would be one times two, which is two. Similarly, the 𝑦-intercept of 𝑦 equals negative cos of 𝑥 is one times negative one, which is negative one. So, we can disregard options (B) and (E) too. Let’s look at the remaining two options.

Both of the graphs will pass through the 𝑦-axis at zero, since negative one times zero is zero. We also know some of the key values for the sine function. For the equation 𝑦 equals sin 𝑥, when 𝑥 is equal to 𝜋 by six, 𝑦 is equal to 0.5. And when 𝑥 is equal to 𝜋 by two, 𝑦 is equal to one. Our graph passes through the points 𝜋 by six, negative 0.5 and 𝜋 by two, negative one, which means the output values are being multiplied by negative one. To achieve this, we need to multiply sin of 𝑥 by negative one. So, the graph has equation 𝑦 equals negative sin of 𝑥. That’s option (C).

Let’s now recap the key points from this lesson. The graphs of the sine and cosine functions are periodic, and each function has a period of 360 degrees or two 𝜋 radians. We saw that both have maxima and minima at one and negative one, respectively. The 𝑦-intercept of the sine curve is zero, whilst for the cosine curve it’s one. Finally, we learned that sine is an odd function and cosine is even.