# 9.04 Key features of secant, cosecant, and cotangent graphs

Lesson

### The shape of secant, cosecant and cotangent

Recall that the cosecant function at a point $x$x is written as $\csc x$cscx and it is defined by $\csc x=\frac{1}{\sin x}$cscx=1sinx. Similarly, the secant function is defined by $\sec x=\frac{1}{\cos x}$secx=1cosx. And, the cotangent function is defined by $\cot x=\frac{\cos x}{\sin x}$cotx=cosxsinx. The graph of each function is drawn below.

 Graph of $y=\csc x$y=cscx

 Graph of $y=\sec x$y=secx

 Graph of $y=\cot x$y=cotx

### Asymptotes

The sine and cosine functions vary continuously between $-1$1 and $1$1, passing through zero twice in every period. When $\sin\left(x\right)=0$sin(x)=0 we should have $\csc\left(x\right)=\frac{1}{0}$csc(x)=10 and $\cot x=\frac{1}{0}$cotx=10 which are undefined. Similarly, when $\cos\left(x\right)=0$cos(x)=0, the we would get that $\sec\left(x\right)=\frac{1}{0}$sec(x)=10.

We say that the secant function has vertical asymptotes at the points where the cosine function is zero. That is, $\sec\left(x\right)$sec(x) has asymptotes at $x=\frac{\pi}{2}+n\pi$x=π2+nπ, or $x=90^\circ+180^\circ n$x=90°+180°n, where $n$n is an integer.

Similarly, $\csc\left(x\right)$csc(x) and $\cot x$cotx have vertical asymptotes wherever $\sin\left(x\right)=0$sin(x)=0. That is, at $x=n\pi$x=nπ, or $x=180^\circ n$x=180°n, where $n$n is an integer.

Comparing the location of the asymptotes of each reciprocal trigonometric function.

#### Worked example

##### Question 1

At what values of $x$x is the function $y=\cot x$y=cotx undefined?

Think: The function is defined by $\cot x=\frac{\cos x}{\sin x}$cotx=cosxsinx. It is undefined whenever the denominator is zero.

Do: The denominator is zero when $\sin x=0$sinx=0.

This occurs at $x=0^\circ,180^\circ,360^\circ,...$x=0°,180°,360°,... and, to be complete, when $x=180^\circ\times n$x=180°×n, for all integer values of $n$n.

### Outcomes

#### F.TF.B.5

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.