Easy Ways to Graph Secant and Cosecant Functions

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Secant and Cosecant Transcript

Let’s start with the first example right away.  We have to draw a graph of the secant function.  In particular, we have to label all zeros, max’s, mins, and asymptotes and figure what the period of the secant function is.  Remember now that the secant function, sec(?), by definition is 1/cos(?).  A really good place to start here when you’re trying to understand sec(?) is with the graph of cos(?).  Let me start with the graph of cos(?).  There’s ?, there’s 2?, and I’m going to extend this out a bit, 3?, and -?.  Now, remember that the cosine function starts at 1, goes down to 0 at ?/2, so there’s ?/2, goes down to -1 at ?, comes back to 0 at 3?/2, back up to 1 at 2?.  The period of cosine is 2?, so it’s repeating itself after 2?.  I’m not drawing secant yet, I’m drawing cos(x), y=cos(x). In black here, I’ve got, we’ll call it cos(?).

Now, I’m going to draw the secant function in red. That means we’re doing 1/cos(x), or 1/cos(?). In particular, 1/1 is 1. Then as the cosine goes down to 0, secant is 1/cos, so it goes up to infinity there, and does the same thing on the other side, so secant looks like that. When the cosine is negative, when the cosine is -1, secant is -1. When the cosine goes to 0, secant blows up but since cosine is negative here, secant goes to negative infinity when the cosine is negative. Then when cosine is positive again, secant is positive again, going up to positive infinity on both ends there. That’s what the secant function looks like. While the cosine goes between -1 and 1, secant is just the reciprocal of that, so it goes 1 up to negative infinity, and from -1 down to negative infinity. Now, I’ve got the secant graph in red. Let me label all the things that we’ve been asked to label. First of all, zeros, while the sec(?) has no zeros because it never crosses the x-axis, so there are no zeros to label. Max’s, all local max’s of the secant function, well here’s one down here at (?,-1) is a local max. Minimum at (0,1) is a local min, (2?,1) is a local min, and so on, as a local min and a local max, every ? units.

We’ve got the max’s and the mins. The asymptotes are the places where the secant blows up to infinity or drops down to negative infinity and so that’s an asymptote at -?/2, and at ?/2. We have an asymptote at ?/2, and again at 3?/2, and again at 5?/2. Basically, every ? units, we have an asymptote where the secant function blows up to infinity or drops down to negative infinity. Finally, what is the period of secant function. The secant function is really dependent on the cosine function, and the cosine function repeats itself once every 2?. The period of cosine is 2?. The period of secant is also 2?. You can see that from the graph, it starts repeating itself after a multiple of 2?. That’s what the graph of the secant function looks like. It kind of has this hues and this upside down hues really dependent on the cosine function because the secant is just 1/cos(?).

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