Trigonometric Graphs

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Domain and range of cot, sec and cosec curves

Lesson

When finding the domain and range of the reciprocal trigonometric ratios cosecant ($\csc$`c``s``c`), secant ($\sec$`s``e``c`)* *and cotangent ($\cot$`c``o``t`)*, *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.

The cosecant function is defined by $\csc x=\frac{1}{\sin x}$`c``s``c``x`=1`s``i``n``x`. Hence, $\csc x$`c``s``c``x` is undefined when $\sin x=0$`s``i``n``x`=0. That is, when $x=n\pi$`x`=`n`π for all integers $n$`n`.

The domain of the cosecant function will therefore be the set of real numbers excluding $n\pi$`n`π. In the graph of $\csc x$`c``s``c``x`, there are asymptotes at these locations. In between each of the asymptotes are the intervals on which $\csc x$`c``s``c``x` 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 $\csc x$`c``s``c``x` can only take values that are greater than $1$1 or less than $-1$−1.

The secant function is defined by $\sec x=\frac{1}{\cos x}$`s``e``c``x`=1`c``o``s``x`. Hence, $\sec x$`s``e``c``x` is undefined when $\cos x=0$`c``o``s``x`=0. That is, when $x=n\pi+\frac{\pi}{2}$`x`=`n`π+π2 for all integers $n$`n`.

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