NZ Level 8 (NZC) Level 3 (NCEA) [In development]
Key features of cot, sec and cosec curves
Lesson

The functions cotangent, secant and cosecant are defined as the reciprocal functions of tangent, cosine and sine respectively.

 $\cot\left(x\right)$cot(x) $=$= $\frac{1}{\tan\left(x\right)}$1tan(x)​ $\sec\left(x\right)$sec(x) $=$= $\frac{1}{\cos\left(x\right)}$1cos(x)​ $\csc\left(x\right)$csc(x) $=$= $\frac{1}{\sin\left(x\right)}$1sin(x)​

We can use the properties of $\tan\left(x\right)$tan(x), $\cos\left(x\right)$cos(x) and $\sin\left(x\right)$sin(x) to deduce the properties of their reciprocals.

## 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 which is undefined. Similarly, when $\cos\left(x\right)=0$cos(x)=0, the definition requires the impossible expression $\frac{1}{0}$10 for $\sec\left(x\right)$sec(x).

So, there must be discontinuities in the $\sec\left(x\right)$sec(x) and $\csc\left(x\right)$csc(x) functions corresponding to the points at which $\cos\left(x\right)$cos(x) and $\sin\left(x\right)$sin(x) are zero.

We see also that when sine or cosine are close to but not quite equal to zero, the corresponding values of cosecant and secant can be made very large in the positive or negative direction, depending on which side of zero the variable $x$x is.

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) is asymptotic at $x=\frac{\pi}{2}\pm n\pi$x=π2±nπ, where $n$n is an integer.

Similarly, $\csc\left(x\right)$csc(x) has vertical asymptotes wherever $\sin\left(x\right)=0$sin(x)=0. That is, at $x=0,\pm\pi,\pm2\pi,...$x=0,±π,±2π,... and so on.

The range of the tangent function is the whole of the real numbers: $-\infty<y<. It has vertical asymptotes wherever the cosine function is zero due to the fact that$\tan\left(x\right)=\frac{\sin\left(x\right)}{\cos\left(x\right)}$tan(x)=sin(x)cos(x). The reciprocal of$\tan\left(x\right)$tan(x) must also have the range$-\infty<y< but its vertical asymptotes occur where $\sin\left(x\right)=0$sin(x)=0, because $\cot\left(x\right)=\frac{\cos\left(x\right)}{\sin\left(x\right)}$cot(x)=cos(x)sin(x).

The following graph illustrates the asymptotes. Notice that both $\cot\left(x\right)$cot(x) and $\csc\left(x\right)$csc(x) share the same asymptotes because they both have a denominator of $\sin\left(x\right)$sin(x).

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

## Maxima and minima

### Outcomes

#### M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions