Key features of cot, sec and cosec curves

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

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

As mentioned, both tangent and cotangent have the range $-\infty−∞<y<∞ so that neither function has an absolute maximum or minimum.

It is also true that secant and cosecant can attain any large value by taking points close to the asymptotes. Therefore, for both of these functions there is also no maximum or minimum.

We can, however, talk about local maxima and minima as distinct from a global or absolute maximum or minimum.

Both sine and cosine have the range $-1\le y\le1$−1≤y≤1. The reciprocals of numbers in this range must be greater than or equal to $1$1 or less than or equal to $-1$−1. You could confirm by looking at the graphs that as $\sin\left(x\right)$sin(x) approaches its maximum value of $1$1, its reciprocal $\csc\left(x\right)$csc(x) must approach a local minimum of $1$1. The same fact is also true for the cosine and secant functions.

Similarly, as sine and cosine approach their minimum values, the reciprocal functions, cosecant and secant, must approach local maxima.

Shown below are the graphs of $\cos\left(x\right)$cos(x) and $\sec\left(x\right)$sec(x) illustrating this pattern.

The graph of $y=\cos\left(x\right)$y=cos(x) (in green) and $y=\sec\left(x\right)$y=sec(x) (in blue).

Increasing or decreasing?

As we have already noticed in terms of local maxima and minima, taking the reciprocal of a function inverts the relative size of the function values. That is, when we take the reciprocal of two numbers, the bigger number becomes the smaller and the smaller number becomes the bigger one. We could write the fact like this:

If $a>b$a>b, then $\frac{1}{a}<\frac{1}{b}$1a​<1b​.

This means that if, going left to right on the graph of a function, we move from a higher $y$y-value to a lower $y$y-value, the same movement on the graph of the reciprocal will be from a lower $y$y-value to a higher $y$y-value. The increasing trend on the first function's graph turns into a decreasing trend on the graph of the reciprocal. Likewise, a decreasing trend turns into an increasing one when you take the reciprocal of a function.

We can see this fact in the graphs of $y=\tan\left(x\right)$y=tan(x) and $y=\cot\left(x\right)$y=cot(x) below.

Where $y=\tan\left(x\right)$y=tan(x) (in green) is increasing, its reciprocal $y=\cot\left(x\right)$y=cot(x) (in blue) is decreasing.

Period

The period of a function is the distance on the $x$x-axis between repeated parts of its graph. Since the cosecant, secant and cotangent functions are the reciprocals of functions that do repeat, then these reciprocal functions must also repeat. In fact, they will repeat at the same rate as the function to which they are the reciprocal.

Exploration

Consider a function, $f(x)$f(x) that has a period of $\alpha$α. This means that $f(x)=f(x+\alpha)$f(x)=f(x+α) for all values of $x$x. and that $\alpha$α is the smallest positive value for which this fact is true.

So, it must be the case that $\frac{1}{f(x)}=\frac{1}{f(x+\alpha)}$1f(x)​=1f(x+α)​. If we say that $g(x)$g(x) is the reciprocal function $\frac{1}{f(x)}$1f(x)​, then we have that $g(x)=g(x+\alpha)$g(x)=g(x+α). So, $g(x)$g(x) has the same period as $f(x)$f(x).

Practice questions

Question 1

Question 2

Question 3