# 8.2 Graphs of the other trigonometric functions

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In this section, you will:
• Analyze the graph of  y=tan x.
• Graph variations of  y=tan x.
• Analyze the graphs of  y=sec x  and  y=csc x.
• Graph variations of  y=sec x  and  y=csc x.
• Analyze the graph of  y=cot x.
• Graph variations of  y=cot x.

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.

## Analyzing the graph of y = tan x

We will begin with the graph of the tangent    function, plotting points as we did for the sine and cosine functions. Recall that

$\mathrm{tan}\text{\hspace{0.17em}}x=\frac{\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}$

The period    of the tangent function is $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ because the graph repeats itself on intervals of $\text{\hspace{0.17em}}k\pi \text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is a constant. If we graph the tangent function on $\text{\hspace{0.17em}}-\frac{\pi }{2}\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\frac{\pi }{2},\text{\hspace{0.17em}}$ 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] .

 $x$ $-\frac{\pi }{2}$ $-\frac{\pi }{3}$ $-\frac{\pi }{4}$ $-\frac{\pi }{6}$ 0 $\frac{\pi }{6}$ $\frac{\pi }{4}$ $\frac{\pi }{3}$ $\frac{\pi }{2}$ $\mathrm{tan}\left(x\right)$ undefined $-\sqrt{3}$ –1 $-\frac{\sqrt{3}}{3}$ 0 $\frac{\sqrt{3}}{3}$ 1 $\sqrt{3}$ 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 $\text{\hspace{0.17em}}\frac{\pi }{3} we can use a table to look for a trend. Because $\text{\hspace{0.17em}}\frac{\pi }{3}\approx 1.05\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{\pi }{2}\approx 1.57,\text{\hspace{0.17em}}$ we will evaluate $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ at radian measures $\text{\hspace{0.17em}}1.05 as shown in [link] .

 $x$ 1.3 1.5 1.55 1.56 3.6 14.1 48.1 92.6

As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}\frac{\pi }{2},\text{\hspace{0.17em}}$ the outputs of the function get larger and larger. Because $\text{\hspace{0.17em}}y=\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is an odd function, we see the corresponding table of negative values in [link] .

 $x$ −1.3 −1.5 −1.55 −1.56 $\mathrm{tan}\text{\hspace{0.17em}}x$ −3.6 −14.1 −48.1 −92.6

We can see that, as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}-\frac{\pi }{2},\text{\hspace{0.17em}}$ the outputs get smaller and smaller. Remember that there are some values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ For example, $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{2}\right)=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{3\pi }{2}\right)=0.\text{\hspace{0.17em}}$ At these values, the tangent function is undefined, so the graph of $\text{\hspace{0.17em}}y=\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ has discontinuities at At these values, the graph of the tangent has vertical asymptotes. [link] represents the graph of $\text{\hspace{0.17em}}y=\mathrm{tan}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ The tangent is positive from 0 to $\text{\hspace{0.17em}}\frac{\pi }{2}\text{\hspace{0.17em}}$ and from $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\frac{3\pi }{2},\text{\hspace{0.17em}}$ corresponding to quadrants I and III of the unit circle.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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