In this explainer, we will learn how to graph trigonometric functions, such as sine and cosine, and deduce their properties.
Let us begin by reviewing the special angles on the unit circle.
We know that the -coordinates of these points represent sine values of the corresponding angles. Using degrees, we can construct the input–output table for the function .
| 0 | 1 | 0 |
A key feature of , which is demonstrated in its graph, is that this function begins with the value 0 when , and it increases to the maximum value 1 when . By plotting the points from the input–output table above, we can approximate the graph of .
As noted earlier, the graph of begins at zero when , and it increases to the maximum value 1 when .
Since represents the angle on the unit circle diagram, we know that each of these values repeats every , or radians. This leads to the fact that is periodic, with a period of , or radians. The graph of can be extended beyond the interval by creating copies of the graph of this interval. For example, the graph of over is shown below.
From this graph, we can see that has roots at every starting from . We also note that the sine function is odd, which means that its graph is rotationally symmetric about the origin. This property is algebraically expressed by for any real number .
Properties: The Sine Function and Its Graph
The graph of the sine function demonstrates the following characteristics:
- The -intercept of is 0, and it increases to the maximum value 1.
- The roots of are , or , for any .
- The maximum value of the function is 1 and the minimum value is .
- The function is periodic, with a period of , or radians.
- is an odd function; that is, .
We can determine the graph of the cosine function using a similar process. We know that the -coordinates of points on the unit circle represent cosine values of the corresponding angles. Then, we can obtain the input–output table.
| 1 | 0 |
By plotting these points, we can approximate the graph of the cosine function.
Unlike the graph of sine, cosine begins at the maximum value 1 at and decreases to the minimum value at . Like sine, cosine is a periodic function with a period of , or radians, and we can extend this graph to a larger interval by making copies of the graph over . The graph of over is shown below.
The roots of begin at and repeat every . Unlike sine, we can see that cosine is an even function, which means that it has reflective symmetry with respect to the . Algebraically, this means
Properties: The Cosine Function and Its Graph
The graph of the cosine function demonstrates the following characteristics:
- The -intercept of the function is 1, and it decreases to the minimum value .
- The roots of are , or , for any .
- The maximum value of the function is 1 and the minimum value is .
- The function is periodic, with a period of , or radians.
- is an even function; that is, .
For our first example, let us practice identifying the graph of the cosine function by recalling the properties that it has.
Example 1: Identifying the Graph of the Cosine Function
Which of the following is the graph of ?
Answer
Let us attempt to answer this question without referring to a graph of but instead using our knowledge of the properties of the cosine function. Let us begin by focusing on the property:
- The -intercept of the cosine function is 1, and it decreases to the minimum value .
In particular, this property means that
This property is very important for distinguishing cosine graphs from sine graphs, since , meaning the -intercept is different. If we examine the five given graphs, we can see that only options A and C have their -intercepts at 1 (we also note that for these graphs). This means that options B, D, and E cannot be correct.
Let us now consider another useful property:
- The cosine function is periodic, with a period of , or radians.
Recall that a periodic function repeats itself in cycles, and the period of a function is how long it takes to complete one full cycle and return to its original position. We can determine this by considering the point and seeing how long it takes the graph to return to , since it only reaches once per cycle. If we do this for option C, we have
We can see that the period of this graph is less than (specifically, it is ), so it cannot be correct.
On the other hand, if we consider the period of option A, we find
Here, we can see that the period is ; therefore, option A is the only remaining option, and so it must be correct.
Although not strictly necessary, we can compare this graph to the other properties of cosine functions to see whether they all hold true. Recall the following:
- The roots of are , or , for any .
- The maximum value of the function is 1 and the minimum value is .
- is an even function; that is, .
For the first property here, we need to look at the roots of the graph (i.e., where it intersects the ). The first positive root is at , the second is at , which is more, and the first negative root is at , which is less, indicating that the roots are indeed at for .
The second property can be verified by drawing horizontal lines at and and showing that the graph touches both lines but does not cross them. We show this is the case below.
Finally, we can verify that the graph is even by checking whether it is symmetric in the . As this is indeed the case, we can conclude that the graph is of an even function.
In conclusion, the solution is option A.
As we have just seen, it is possible to identify the graph of centered at the origin from its features such as its -intercept, its periodicity, and its maximum and minimum values, and the same applies to .
These same principles apply to graphs of cosine and sine when viewed at values of shifted away from the origin. In particular, we can use the properties of periodicity of these functions (i.e., that they repeat every or radians) to help us determine the location of key features of the graph.
In our next couple of examples, we will determine which of the trigonometric functions correspond to the given graph, and consider which portion of the graph of a trigonometric function results from each quadrant in the unit circle diagram.
Example 2: Recognizing Trigonometric Functions from Their Graphs
Consider the following figures.
- Which function does the plot in the graph, figure (a), represent?
- Cosine
- Sine
- Assign each region of the plot in figure (a) to the corresponding quadrant of the unit circle in figure (b).
Answer
Part 1
Let us compare the values indicated by the graph with the values of sine and cosine functions.
The coordinates of points on the unit circle are given by , where is the counterclockwise angle of the radius to the point with respect to the positive . In the given graph, we can see that the function value is equal to 0 when the angle is at radians. We know that is the angle of a full revolution, which brings the point back to the positive .
The coordinate of the point on the unit circle corresponding to the angle is , which tells us
The given graph indicates that this function takes the value 0 at , so this agrees with the sine function.
This is option B.
Part 2
We know that the values of the sine function are given by the -coordinates of the points on the unit circle. To find the region of the unit circle corresponding to each part of the given graph, we draw these angles on the unit circle. Region A includes the angles between and . We know that is a full counterclockwise revolution. In the unit circle, these angles can be drawn as below.
Hence, the angles between these two values lie in the fourth quadrant, meaning that region A is assigned to quadrant IV.
Similarly, we can draw the angles in region B.
So, these angles lie on the first quadrant. Hence, region B is assigned to quadrant I.
Let us look at the remaining regions.
We can see that region C corresponds to quadrant II, and region D corresponds to quadrant III. In conclusion, we have the assignments
Let us consider another example where we will determine the trigonometric function represented by a given graph and associate regions of the graph with parts of the unit circle.
Example 3: Recognizing Trigonometric Functions from Their Graphs
Consider the following figures.
- Which function does the plot in the graph, figure (a), represent?
- Cosine
- Sine
- Assign each region of the plot in figure (a) to the corresponding quadrant of the unit circle in figure (b).
Answer
Part 1
Let us compare the values indicated by the graph with the values of sine and cosine functions.
The coordinates of points on the unit circle are given by , where is the counterclockwise angle of the radius to the point with respect to the positive . In the given graph, we can see that the function value is equal to 0 when the angle is at radians. We know that negative angles are clockwise angles from the positive . Since , this angle is obtained by rotating clockwise from the positive by (one revolution and a half) and going an additional quarter revolution clockwise.
The coordinates of the point on the unit circle corresponding to the angle are , which tells us
The given graph indicates that this function takes the value 0 at , so this agrees with the cosine function.
This is option A.
Part 2
We know that the values of the cosine function are given by the -coordinates of the points on the unit circle. To find the region of the unit circle corresponding to each part of the given graph, we draw these angles on the unit circle. Region A includes the angles between and . We know that negative angles are measured clockwise from the positive and that is a clockwise full revolution. We can write
So, this angle represents two and a half clockwise revolutions followed by an additional quarter turn. Also so this angle represents two and a half clockwise revolutions. These angles are drawn below.
Hence, the angles between these two values lie in the second quadrant, meaning that region A is assigned to quadrant II.
Similarly, we can draw the angles in other regions.
We can see that region B corresponds to quadrant III, region C corresponds to quadrant IV, and region D corresponds to quadrant I. In conclusion, we have the assignments
So far, we have considered the behavior of the graphs of and both at the origin and away from it. In addition to the basic versions of these functions, we can also consider their behavior when multiplied by a constant value.
For instance, suppose we have the function
To understand how this affects the graph of the function, it is useful to consider how the maximum and minimum values are affected. For a standard cosine function, the maximum value is 1 (e.g., when or ) and the minimum value is (e.g., when ). If we consider , then the minimum and maximum values are doubled; we highlight this in the graphs below.
As we have indicated by the direction of the arrows shown, the maximum and minimum values have increased and decreased, respectively, leading to the shape of the graph changing. Nevertheless, many of the other features of the graph remain the same: the period is still the same, the roots are the same, and the function is still even. It is worth keeping in mind these similarities when comparing variations of sine and cosine functions.
Just as it is possible to multiply a trigonometric equation by 2, other constants including negative constants can be used. In the next example, we will identify a function where some of the possibilities feature trigonometric functions multiplied by a constant.
Example 4: Identifying the Graph of a Trigonometric Function
Consider the graph below.
Which function does the plot in the graph represent?
Answer
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 or or constant multiples of them, we should begin by reviewing the qualities that these functions have.
Let us first recall the -intercepts of sine and cosine (i.e., their values when ). We have
Comparing this to the given graph, we see that the -intercept is at 0, meaning that (option A) cannot be an option. In fact, we can see that this also applies to the other options that are multiples of . This is because for options B and E, we can see that
That is to say, neither of the -intercepts are equal to 0, and since we already know that the -intercept is at 0, neither of these options can be correct.
We now have two remaining options, and . Both options have their -intercepts at 0, so we have to consider other properties of . One thing we can consider in particular is the behavior of as increases from 0. This can be seen by considering a table of some of the first few values (in radians).
| 0 | ||||||
| 0 | 0.5 | 1 |
From this, we can see that increases from 0 to 1 as increases from 0 to . Considering the graph shown, we can see that the opposite occurs; as increases from 0 to , the value of the graph goes to . On the other hand, if we consider some of the first values of , we can see the following.
| 0 | ||||||
| 0 |
As we can see, the entries are the same as the first table, except they have been multiplied by . As it happens, this behavior indeed corresponds with what we can see in the graph. In particular, the value of the graph at is indeed . Therefore, option C, , is the solution.
In our final example, we will apply function transformations to the sine function to obtain a new graph.
Example 5: Finding the Maximum Value of a Given Sine Function
Find the maximum value of the function .
Answer
We recall that the graph of starts at 0 at , and it oscillates between the maximum value 1 and the minimum value .
We know that multiplying a positive constant by results in a vertical dilation (stretching or contracting) with the scale factor . Here, we are multiplying by 11, so the new graph of this function can be obtained by stretching the graph above by a factor of 11.
Here, the solid blue graph represents the original function, , and the dashed curve represents the function . The double-sided red arrows indicate vertical dilation. We can see from this graph that the function oscillates between and 11.
We can also come to this conclusion algebraically. We know that
Multiplying this inequality by 11, we have
This leads to the maximum value 11.
Hence, the maximum value of is 11.
Let us finish by recapping a few important concepts from this explainer.
Key Points
- The graph of the sine function demonstrates the following characteristics:
- The -intercept of is 0, and it increases to the maximum value 1.
- The roots of are , or , for any .
- The maximum value of the function is 1 and the minimum value is .
- The function is periodic, with a period of , or radians.
- is an odd function; that is, .
- The graph of the cosine function demonstrates the following characteristics:
- The -intercept of the function is 1, and it decreases to the minimum value .
- The roots of are , or , for any .
- The maximum value of the function is 1 and the minimum value is .
- The function is periodic, with a period of , or radians.
- is an even function; that is, .
