Now that we have covered how to graph sine and cosine, and how to graph tangent and cotangent, it’s time to learn how to graph the secant and cosecant functions. In order to do this, you HAVE to know how to graph sine and cosine. If you do not comfortably know how to do this, please go review my posts on how to do this.
We have to understand how to graph sine and cosine because they are the base graphs for our secant and cosecant graphs.
The secant and cosecant graphs are, in my opinion, the most fun to graph, so I hope that you feel the same way.
Here’s what our graph for the secant function looks like.
y = \text{sec}x \\ \text{Base graph: } y = \text{cos}x
In order to graph the secant function, we have to first draw the graph for y=cosx. You can see this represented with the dashed line. Our vertical asymptotes are going to be where the cosecant function crosses the x-axis. These are going to be the barriers for our secant graph.
The peaks and troughs of our cosine function are going to be where we base our secant function from. At each peak and trough, we are going to have this U-shaped line that approaches our asymptotes.
Let’s take a look at what it looks like to graph the cosecant function.
y = \text{csc}x \\ \text{Base graph: } y = \text{sin}x
Here we see that our base graph is the graph of y = sinx. Our graph of cosecant is based off the peaks and troughs of y = sinx, similar to our graph of secant. We also have vertical asymptotes where the graph of y = sinx touches the x-axis and separated by half a period.
Now it should be plain how we are forming these graphs, but just so we can further understand these functions, let’s have one more example.
y = \text{csc}\frac{1}{2}x
We graph sine according to the function, which gives us a horizontal stretch, and then we graph our cosecant function.
And there we have it! That’s how we graph our secant and cosecant functions! Isn’t that so much fun?
Please leave a comment if you found this post helpful (or if you didn’t). See you next time.