\( \definecolor{flatred}{RGB}{192, 57, 43} \definecolor{flatblue}{RGB}{41, 128, 185} \definecolor{flatgreen}{RGB}{39, 174, 96} \definecolor{flatpurple}{RGB}{142, 68, 173} \definecolor{flatlightblue}{RGB}{52, 152, 219} \definecolor{flatlightred}{RGB}{231, 76, 60} \definecolor{flatlightgreen}{RGB}{46, 204, 113} \definecolor{flatteal}{RGB}{22, 160, 133} \definecolor{flatpink}{RGB}{243, 104, 224}\)

February 24, 2021

Graphs of Other Circular Functions

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Textbook Reference: Section 6.2 Graphs of the Other Trigonometric Functions

Now that you've got a feel for the graphs of the sine and cosine functions, let's take a look at the graphs of the other four circular functions. While these graphs aren't usually used to represent physical things, they do give us a good visual idea of how the functions behave at a glance — especially what happens as the values get very large.

The tangent and cotangent functions

Let's start with the tangent function.


That's a pretty wild-looking graph! What the heck is going on?

Let's break it down.

Remember our definition of the tangent function: \[ {\color{flatgreen}\tan(\theta)}=\dfrac{\color{flatblue}\sin(\theta)}{\color{flatred}\cos(\theta)} \] This means we can understand the tangent by thinking of what happens to the sine and cosine.

  • When \(\color{flatblue}\sin(\theta)=0\), the numerator is zero, so we'll have \(\color{flatgreen}\tan(\theta)=0\). This happens at \(0\), \(\pm\pi\), \(\pm2\pi\), and so on.
  • When \({\color{flatblue}\sin(\theta)}={\color{flatred}\cos(\theta)}\), the numerator and denominator are the same, so we'll have \(\color{flatgreen}\tan(\theta)=1\). Similarly, when \({\color{flatblue}\sin(\theta)}={-\color{flatred}\cos(\theta)}\), the numerator and denominator are opposites, so we'll have \(\color{flatgreen}\tan(\theta)=-1\). These happen at \(\pm\tfrac\pi4\), \(\pm\tfrac{3\pi}4\), \(\pm\tfrac{5\pi}4\), and so on.
  • When \(\color{flatred}\cos(\theta)=0\), the denominator is zero, so \(\color{flatgreen}\tan(\theta)\) will be undefined. Remember, however, that as a number shrinks to zero, its reciprocal blows up to infinity (in either direction), so we can also think of it as \(\color{flatgreen}\tan(\theta)=\overset\sim\infty\). This shows up on the graph as a bunch of vertical asymptotes, at \(\pm\tfrac\pi2\), \(\pm\tfrac{3\pi}2\), \(\pm\tfrac{5\pi}2\), and so on.
    • I prefer to call these vertical asymptotes "poles" instead. First of all, it's more consistent with what they're called in the rest of math when functions blow up to infinity. Second of all, it's shorter and easier to spell.

It also helps to remember that the tangent function tells us the slope of an angle! As we go around the unit circle, the slope increases rapidly, until we have an undefined (or infinite) slope at \(\tfrac\pi2\). After this, the slope is steep but negative, which makes the graph appear to "wrap around" infinity and come out the other side on the bottom of the graph! Eventually we hit zero again and the cycle repeats all over again.

By the way, notice that the period of the tangent function is actually \(\pi\). That's because it only takes half of a circle to get to another point with the same slope — for example, the slope is \(1\) at both \(\tfrac\pi4\) and \(\tfrac{5\pi}4\).

The cotangent function looks similar:

You'll compare the cotangent function to the tangent function in the Preview Activity below.

The secant and cosecant functions

Take a look at the graph of the cosecant function.


This graph is even weirder than the last ones. How can we break this one down?

Well, remember that the cosecant function is the reciprocal of the sine function: \[ {\color{flatlightblue}\csc(\theta)}=\dfrac{1}{\color{flatblue}\sin(\theta)} \] So, maybe we can understand it by comparing those two graphs. Let's put them both on the same graph:

Some things to notice here:

  • The cosecant and sine graphs touch when they have values of \(1\) or \(-1\).
    This makes sense, because \(\dfrac{1}{\color{flatblue}1}=\color{flatlightblue}1\) and \(\dfrac{1}{\color{flatblue}-1}=\color{flatlightblue}-1.\)
  • As the sine gets smaller (that is, closer to zero), the cosecant gets larger (that is, closer to infinity). As a result, when \(\color{flatblue}\sin(\theta)=0\), the value of \(\color{flatlightblue}\csc(\theta)\) is undefined — or, once again, you can often think of it as \(\color{flatlightblue}\overset\sim\infty\). Graphically, this means that wherever \(\color{flatblue}y=\sin(x)\) has an \(x\)-intercept, \(\color{flatlightblue}y=\csc(x)\) has a pole.

And, as you might imagine, the graph of secant has a similar shape:

And we have a similar relationship to the cosine:

Take a careful look at these graphs to see how they relate to each other.

By the way, here's a useful intuition to have for these reciprocal graphs:

Reciprocals swap large values with small values.

If \(\color{flatred}\cos(\theta)\) is small, you can expect \(\color{flatlightred}\sec(\theta)\) to be large. Likewise, if \(\color{flatgreen}\tan(\theta)\) is large, you can expect \(\color{flatlightgreen}\cot(\theta)\) to be small.

Putting it all together

If you want to see all the graphs together, you can take a look at this Desmos link.

One side note about using Desmos by the way — it won't actually draw asymptotes by itself. In order to get them to appear on the graphs above, I actually had to tell Desmos where to put them (which you will see if you open them up yourself and look at how they're set up.

By the way, you might understandbly be feeling a bit overwhelmed because of how many different graphs we've looked at.

If so, let me put this in a broader perspective.

It's essential to be able to sketch out the basic graphs of \(\color{flatblue}y=\sin(x)\) and \(\color{flatred}y=\cos(x)\), since they're arguably the two most important circular functions and you want to have a good intuition for how they work.

However, producing the graphs of the other four functions from memory isn't nearly as important — it's enough to have an overall idea of what they look like, and let technology do the rest if necessary. What's far more important is that you understand where the graphs come from, why they look like they do, and how they're related.

Preview Activity 7

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

  1. Compare and contrast the graphs of \(\color{flatgreen}y=\tan(x)\) and \(\color{flatlightgreen}y=\cot(x)\). Pay special attention to where the zeros and poles are.
  2. The value of \(\tfrac\pi2\) is about \(1.5707963268\). Complete the following table.
    \(x\) 1.5 1.57 1.5707 1.57079 1.570796

    How does this support the behavior of \(\color{flatlightred}\sec(x)\) as \(x\) gets close to \(\tfrac\pi2\)?
  3. Look at the graphs of all six circular functions. What kinds of symmetry do you notice in the various graphs?
  4. Answer AT LEAST one of the following questions:
    1. What was something you found interesting about this reading?
    2. What was an "a-ha" moment you had while doing this reading?
    3. What was the muddiest point of this reading for you?
    4. What question(s) do you have about anything you've read?


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1 comment :

  1. I've never learned the term "pole", despite getting a degree in math, so I had to look it up. Apparently at some point I forgot that "asymptote" means something more general.

    Pole - The point on the x-axis of a vertical asymptote; a singularity of the function
    Asymptote - Any line the function gets arbitrarily close to without touching

    It seems odd that neither mathworld nor math wiki mention either term in the other's page
    https://mathworld.wolfram.com/Pole.html - https://mathworld.wolfram.com/Asymptote.html
    https://math.wikia.org/wiki/Pole - https://math.wikia.org/wiki/Asymptote

    The one search result that also uses the relationship is from another course


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