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We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.
We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that
The period of the tangent function is because the graph repeats itself on intervals of where is a constant. If we graph the tangent function on to we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.
We can determine whether tangent is an odd or even function by using the definition of tangent.
Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in [link] .
0 | |||||||||
undefined | –1 | 0 | 1 | undefined |
These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when we can use a table to look for a trend. Because and we will evaluate at radian measures as shown in [link] .
1.3 | 1.5 | 1.55 | 1.56 | |
3.6 | 14.1 | 48.1 | 92.6 |
As approaches the outputs of the function get larger and larger. Because is an odd function, we see the corresponding table of negative values in [link] .
−1.3 | −1.5 | −1.55 | −1.56 | |
−3.6 | −14.1 | −48.1 | −92.6 |
We can see that, as approaches the outputs get smaller and smaller. Remember that there are some values of for which For example, and At these values, the tangent function is undefined, so the graph of has discontinuities at At these values, the graph of the tangent has vertical asymptotes. [link] represents the graph of The tangent is positive from 0 to and from to corresponding to quadrants I and III of the unit circle.
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