Domain and range of cot, sec and cosec curves

When finding the domain and range of the reciprocal trigonometric ratios cosecant ($\csc$csc), secant ($\sec$sec) and cotangent ($\cot$cot), it is important to notice the pattern of where these ratios are undefined.

These functions are defined on most of the real numbers, but there are infinitely many missing numbers where the functions are undefined. To see why this is so, we turn to the definitions of these three functions as reciprocals of the functions sine, cosine and tangent respectively.

It will be helpful to refer to the graphs of cosecant, secant and cotangent that are sketched in a previous chapter. Here they are, again.

 

Cosecant

The cosecant function is defined by $\csc x=\frac{1}{\sin x}$cscx=1sinx​. Hence, $\csc x$cscx is undefined when $\sin x=0$sinx=0. That is, when $x=n\pi$x=nπ for all integers $n$n.

The domain of the cosecant function will therefore be the set of real numbers excluding $n\pi$nπ. In the graph of $\csc x$cscx, there are asymptotes at these locations. In between each of the asymptotes are the intervals on which $\csc x$cscx is defined.

So, to represent the entire domain in interval notation, we can use an infinite union of intervals like this:

$\dots\cup\left(-\pi,0\right)\cup\left(0,\pi\right)\cup\left(\pi,2\pi\right)\cup\dots$…∪(−π,0)∪(0,π)∪(π,2π)∪…

For the range of the cosecant function, we observe that the sine function only takes values that are between $-1$−1 and $+1$+1. It follows from the definition that $\csc x$cscx can only take values that are greater than $1$1 or less than $-1$−1.

 

Secant

The secant function is defined by $\sec x=\frac{1}{\cos x}$secx=1cosx​. Hence, $\sec x$secx is undefined when $\cos x=0$cosx=0. That is, when $x=n\pi+\frac{\pi}{2}$x=nπ+π2​ for all integers $n$n.

The domain of the secant function will therefore be the set of real numbers excluding $n\pi+\frac{\pi}{2}$nπ+π2​. In the graph of $\sec x$secx, there are asymptotes at these locations. In between each of the asymptotes are the intervals on which $\sec x$secx is defined.

So, to represent the entire domain in interval notation, we can use an infinite union of intervals like this:

$\dots\cup\left(-\frac{3\pi}{2},-\frac{\pi}{2}\right)\cup\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\cup\dots$…∪(−3π2​,−π2​)∪(−π2​,π2​)∪(π2​,3π2​)∪…

The secant function has the same range as the cosecant function. That is, the function can take all real values except those that are strictly between $-1$−1 and $+1$+1. This is a consequence of the fact that the cosine function only takes values that are between $-1$−1 and $+1$+1.

 

Cotangent

By definition, $\cot x=\frac{1}{\tan x}$cotx=1tanx​. The function must be left undefined wherever $\tan x=0$tanx=0. That is, when $x=n\pi$x=nπ for each integer $n$n.

The domain of the cotangent function, therefore, is the set of real numbers excluding $n\pi$nπ for all integers $n$n. The excluded numbers are the locations of the vertical asymptotes that can be seen in the graph.

Like the cosecant function, we can write the entire domain as the following union of intervals.

$\dots\cup\left(-\pi,0\right)\cup\left(0,\pi\right)\cup\left(\pi,2\pi\right)\cup\dots$…∪(−π,0)∪(0,π)∪(π,2π)∪…

The range, on the other hand, is the full set of real numbers. In other words, every real number is the image of some number under the operation of the cotangent function.

In the graph of the cotangent function, it is clear that the values extend from $-\infty$−∞ to $+\infty$+∞ with no gaps.

 

Transformations

The domain and range of a function are affected in different ways by transformations. In particular, when we transform the graphs of $\csc x$cscx, $\sec x$secx or $\cot x$cotx, their asymptotes, intercepts and local maxima and minima are also transformed accordingly.

In order to determine the domain of a transformed function, we look at how the $x$x-values of the asymptotes are altered by either horizontal dilation or horizontal translation.

To find the range of a transformed function, we look at how the $y$y-values of any local maxima or minima are altered by either vertical dilation or vertical translation.

 

Restrictions

Remember that the natural domain of a function is the largest possible domain on which a function can be defined. In defining a function for some particular purpose, we can always restrict the domain to some smaller set that meets the immediate need.

For example, we may only be interested in a function defined on an interval like $\left[-2\pi,2\pi\right]$[−2π,2π], although we would still have to exclude the points in the interval for which the function was undefined.

 

Practice questions

Question 1

Question 2

Question 3