In this explainer, we will learn how to graph trigonometric functions, such as sine, cosine, and tangent, 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 -axis. 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, .
Finally, let us consider the graph of the tangent function. We know that . This means that the tangent function is not defined when is equal to zero, which is when
Using the input–output table for sine and cosine, we can construct one for tangent.
| 0 | 1 | Undefined | 0 |
By plotting these points, we can approximate the graph of the tangent function.
Again, it is worth highlighting that the function is undefined at and , since at these angles. Unlike sine and cosine, the graph of the tangent function is unbounded and includes vertical asymptotes. Similar to sine and cosine, we can extend this graph periodically to a larger interval, .
From this graph, we note that the period of the tangent function is , as opposed to the sine and cosine functions that have periods of . Also, like sine, the graph of the tangent function has rotational symmetry about the origin, meaning that the tangent function is odd; that is,
Properties: The Tangent Function and Its Graph
The graph of the tangent function demonstrates the following characteristics:
- The -intercept of the function is 0, and it increases without bound until .
- The roots of are the same as those of . These are , or , for any .
- The graph of has vertical asymptotes at the roots of , which are , or , for any .
- The graph of is unbounded.
- The function is periodic, with a period of , or radians.
- is an odd function; that is, .
In the first three 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 1: Recognizing Trigonometric Functions from Their Graphs
Consider the following figures.
- Which function does the plot in the graph, figure (a), represent?
- Cosine
- Sine
- Tangent
- Assign each region of the plot in figure (a) to the corresponding quadrant of the unit circle in figure (b).
Answer
Part 1
Recall that the graph of the tangent function has vertical asymptotes at every interval of length . Since this graph does not have any vertical asymptotes over an interval with length greater than , this cannot be the graph of the tangent function. Hence, it must be the graph of either sine or cosine. 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 -axis. 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 -axis.
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 2: Recognizing Trigonometric Functions from Their Graphs
Consider the following figures.
- Which function does the plot in the graph, figure (a), represent?
- Cosine
- Sine
- Tangent
- Assign each region of the plot in figure (a) to the corresponding quadrant of the unit circle in figure (b).
Answer
Part 1
Recall that the graph of the tangent function has vertical asymptotes at every interval of length . Since this graph does not have any vertical asymptotes over an interval with length greater than , this cannot be the graph of the tangent function. Hence, it must be the graph of either sine or cosine. 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 -axis. 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 -axis. Since , this angle is obtained by rotating clockwise from the positive -axis 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 -axis 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
Let us consider another example where the graph of a trigonometric function with vertical asymptotes is given.
Example 3: Recognizing Trigonometric Functions from Their Graphs
- Which function does the plot in the graph, figure (a), represent?
- Cosine
- Sine
- Tangent
- Assign each region of the plot in figure (a) to the corresponding quadrant of the unit circle in figure (b).
Answer
Part 1
Among sine, cosine, and tangent functions, the tangent function is the only one that grows without bound. In other words, the graph of a tangent function contains vertical asymptotes, while the graphs of the sine and cosine functions are bounded between and 1. Hence, this graph represents the tangent function.
This is option C.
Part 2
Region A contains angles between and 0. Angle 0 represents the positive -axis. We also know that negative angles are measured clockwise from the positive -axis, so angle represents a quarter revolution clockwise from the positive -axis, which places this angle on the negative -axis. This tells us that angles in region A lie between the negative -axis and the positive -axis. Hence, region A corresponds to the fourth quadrant.
Angles in region B lie between 0 and . We know that 0 represents the positive -axis, and represents the positive -axis since it is a counterclockwise quarter revolution from the positive -axis. These angles lie in quadrant I.
Region C has angles between and . Since is a counterclockwise half-revolution, it represents the negative -axis. Hence, angles in region C lie between the positive -axis and the negative -axis, which is quadrant II.
Finally, region C has angles between and . The latter angle is a three-quarter counterclockwise revolution, which lands on the negative -axis. So, the quadrant between the negative -axis and the negative -axis is quadrant III.
In conclusion, we have the assignments
In the next example, we will recognize the correct graph of a trigonometric function using its characteristics.
Example 4: Identifying the Graph of the Tangent Function
Which of the following is the graph of ?
Answer
Let us recall a few important characteristics of the tangent graph:
- is periodic, with a period of .
- has vertical asymptotes at the roots of , which are for any .
- The roots of are the same as those of , which are for any .
Let us begin by examining the first property, which tells us that the period of is . We see that option A has a period of , so we can rule this out. All other graphs are periodic, with a period of .
The second characteristic tells us that we should see vertical asymptotes at . Since we already checked the periodicity of , it would be enough to see whether we can find a vertical asymptote at . Option B does not have a vertical asymptote at , so we can rule this one out. The remaining options, C, D, and E, all have a vertical asymptote at .
Next, let us examine the roots. We know that the roots of the tangent function are . Once again, since we have already checked the periodicity, it is enough to check whether the graph demonstrates a root at . Option C does not have a root at , so we exclude this option. The remaining two options, D and E, both have a root at .
Let us compare the remaining two graphs, D and E. We note that the function in D is negative over the interval , while the function in E is positive in that interval. We know that represents acute angles, or angles in the first quadrant. Since the tangent function is positive for acute angles, we see that D is not the graph for the tangent function.
This leads to option E.
In our next example, we will identify a graph of the tangent function after a function transformation.
Example 5: Identifying the Graph of a Trigonometric Function after Transformation
Which of the following is the graph of ?
Answer
We can begin by drawing the graph of . Let us recall a few important characteristics of the tangent graph:
- is periodic, with a period of .
- has vertical asymptotes at the roots of , which are for any .
- The roots of are the same as those of , which are for any .
Additionally, we know that is positive for acute angles; that is, . This leads to the graph of the tangent function.
We know that multiplying a function by results in reflection over the -axis. Since we are multiplying by to obtain the given function, , we need to flip this graph over the -axis.
In the graph above, the solid blue graph represents the original function, , and the dashed graph represents the reflected function, .
In particular, we note that both the roots and the vertical asymptotes remain the same after reflection. Hence, the graph of should have roots at and vertical asymptotes at for any .
We can rule out options B, D, and E since these graphs do not demonstrate roots at , particularly at .
We can see that the difference between the graphs in the two remaining options, A and C, is that the function is negative for in A while it is positive over this interval in C. In the reflective graph of we constructed, we can see that our function takes negative values over .
This leads to option A.
In our final example, we will apply function transformations to the sine function to obtain a new graph.
Example 6: 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, .
- The graph of the tangent function demonstrates the following characteristics:
- The -intercept of the function is 0, and it increases without bound until .
- The roots of are the same as those of . These are , or , for any .
- The graph of has vertical asymptotes at the roots of , which are , or , for any .
- The graph of is unbounded.
- The function is periodic, with a period of , or radians.
- is an odd function; that is, .
