Kosekan

So far, we have learned about sine, cosine, and tangent, but now it's time to learn about the inverses of these functions. Let's start with the cosecant function!

##Cosecant

Cosecant is the reciprocal of sine: $\csc(\theta) = \frac{1}{sin(\theta)}$.

Just like the other trigonometric functions, it is a periodic function, with properties such as an amplitude, midline, and period. Keep in mind that $\csc(x)$ looks very different from $\sin(x)$, so these parts on the graph will also look very different.

Thinking of a triangle, $\sin = \frac{opposite}{hypotenuse}$.

Therefore, $\csc = \frac{hypotenuse}{opposite}$.

Cosecant can be used to solve equations such as $\csc(\theta) + \pi = \frac{4\pi}{2}$.

It can also be used in trig identities, which you will learn about in later math courses!

The above is a graph of the $\csc(x)$ function. While this looks very complicated, the process to graphing it is very similar to how we graphed $\sin(x)$.

Keep going to learn more inverse trig functions as well as how to graph these! You will also learn how to calculate their exact values.

The Cosecant Function

The next item in our set of trigonometric functions is the cosecant function! The cosecant of an angle is the reciprocal of the sine. From SOHCAHTOA, we know that sine is the ratio of the opposite angle to the hypotenuse of a right triangle.

$$\text{sin}{x} = \frac{\text{opposite}}{\text{hypotenuse}}$$

The reciprocal of the sine function would mean that the cosecant is the ratio of the hypotenuse to the opposite angle.

$$\text{csc}{x} = \frac{\text{hypotenuse}}{\text{opposite}}$$

The graph kind of looks likes a grid!

If you want to take this concept a step further, you could explore the table of cosecants. This table will be entirely the reciprocals of the sine function!

You can also use the unit circle to find cosecant.

(Video) Graphing the Cosecant Function

by Brian McLogan

In this video by Brian McLogan, you will learn how to graph the cosecant function.

Summary

Graph the following function: $y=3 \text{ csc }(4x)$

Cosecant is the reciprocal of sine! To graph the cosecant function, let's first try to graph the sine function, $y=3 \text{ sin }(4x)$. The amplitude is: $3$, the period is $\frac{\pi}{2}$. The graph looks like:

The above sine graph is NOT the solution to our problem! But we will use this sine graph to graph the cosecant function. Let's look at this graph over two periods, and let's make it dashed.

We know that $\text{ csc }(x) = \frac{1}{\text{ sin }(x)}$, so we can say that:

$$\text{ csc }(\frac{\pi}{4}) = \frac{1}{sin(\frac{\pi}{4})} = \text{ csc }(\frac{\pi}{4}) = \frac{1}{0}$$

This is undefined! So we know that there will be a vertical asymptote at $x=\frac{\pi}{4}$. We can extend this to: at all of the $x$-intercepts on the sine graph, there will be a vertical asymptote on the cosecant graph. On the next graph, the asymptotes are labeled in green.

We now have everything we need to graph the cosecant function. It looks like:

Without all of the extraneous information, our cosecant graph looks like: