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Secant

Secant is the reciprocal of cosine, or in mathematical terms, $\sec{\theta}=\frac{1}{\cos{\theta}}$.

Strategies involving Secant

Memorizing the reciprocal of the secant and cosecant can be quite annoying. A good rule is this: with reciprocals, sine and cosine have their "co" switched, but tangent keeps its "co".

It is not very effective to memorize another entire set of values for a function; instead use the definition to derive them from cosine.

If you see secants ($\sec{\text{expression}}$) in a problem, replace them with $\frac{1}{\cos{\text{expression}}}$, as with the other inverse trig functions.

The Secant Function

The next lesson in the series of trigonometric functions is the secant function. Secant is the reciprocal of the cosine function, which you just learned about. We know from SOHCAHTOA that cosine is the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

That means that the reciprocal of that fraction is the ratio of the hypotenuse to the adjacent side.

This function looks similar to the cosecant function! To help you understand the graphs of the cosecant and secant functions, try looking closely at the graphs to see how they're similar or different.

You will also learn how to calculate their exact values. You can also use the unit circle to find secant.

(Video) The Domain, Range, and Vertical Asymptote of Secant

by Brian McLogan

In this video by Brian McLogan, you will learn what the domain, range, and vertical asymptotes of the secant function are.

Summary

Let's start with the domain and range of the secant function. To do that, we will first graph the cosine function. The cosine function looks like:

From here, we know that every $x$-intercept becomes a vertical asymptote on the secant function. From here, we know that our secant graph approaches our vertical asymptotes. So, our secant graph looks like:

This function is going to repeat in both the positive and negative directions. To get the domain, we need to find all of the $x$ values that are part of this function. Because the function goes on forever in both directions, it is easier to first look at which values of $x$ the secant function cannot be, which is where the vertical asymptotes come into play!

We can see that there are vertical asymptotes at $\frac{\pi}{2}$, $-\frac{\pi}{2}$, $\frac{3\pi}{2}$, and $-\frac{3\pi}{2}$. This pattern goes on forever in both directions.

So, we know that our domain is all real numbers, except for those asymptotes. We know that the distance between each asymptote is $\pi$. So we can represent the domain as: $x= \frac{\pi}{2} + \pi n$.

Next, we will look at the range of the function, which is all possible $y$ values. There are no horizontal asymptotes, so we have to think through this a little differently. If we look at our graph, we notice that our range is: $(-\infty, -1]$ and $[1, \infty)$.