Reciprocal Trigonometric Functions: cosec(x), sec(x), cotan(x)

When we take 1 divided by each of the original trigonometric functions we obtain the reciprocal trigonometric functions:  

$\text{cosec}(x)=\frac{1}{\sin(x)},\hspace{30pt}\text{sec}(x)=\frac{1}{\cos(x)}, \hspace{30pt}\text{cot}(x)=\frac{1}{\tan(x)}$.

An easy way to remember which one is which is to look at the 3rd letter in each reciprocal function. For example, the third letter of cosec is s and so this is one over sin. We can solve equations involving reciprocal trigonometric functions in the same way we solve trigonometric equations. See Example 1. You might also notice that $\text{cot}(x)=\frac{1}{\tan(x)}=\frac{\cos(x)}{\sin(x)}$.

These reciprocal functions can be evaluated for angles given in degrees and radians. For example, $\text{cosec}(270^\circ)=\frac{1}{\sin(270^\circ)}=\frac{1}{-1}=-1$ or $\text{sec}(2\pi)=\frac{1}{\cos(2\pi)}=\frac{1}{1}=1$. However, like the graph of $\tan(x)$, all of the reciprocal trigonometric functions have periodic asymptotes. We cannot evaluate them when the denominator is 0. See graphs below.

Graphs of the Reciprocal Trigonometric Functions

Here are the graphs of the $\text{cosec}(x)$, $\text{sec}(x)$ and $\text{cotan}(x)$ respectively. As you can see the angles are given in radians:

reciprocal trigonometric functions

reciprocal trigonometric functions
reciprocal trigonometric functions

Consider the graph of $\text{cosec}(x)$, for example. It is defined by $\frac{1}{\sin(x)}$ and so has asymptotes whenever $\sin(x)=0$. As you can see from the graph, this occurs for multiples of $\pi$ i.e, when $x=n\pi, n\in{\mathbb Z}$. As $x$ tends towards these values, the curve grows infinitely positive if $\sin(x)$ is positive and infinitely negative if $\sin(x)$ is negative. This asymptotic behaviour can be seen in the other two curves as well. 

Now consider the graph of $\text{sec}(x)$. This graph has periodic maxima and minima at multiples of $\pi$. For example, there is a maximum at $(\pi,-1)$ since $\text{sec}(\pi)=\frac{1}{\cos(\pi)}=\frac{1}{-1}=-1$. Notice that $\cos(x)$ takes values between $-1$ and $1$ and so $\frac{1}{\cos(x)}$ will take values less than $-1$ or greater than $1$. This follows from the fact that dividing by a number less than 1 gives a result greater than 1, etc. Hence, the domain and range for $\text{sec}(x)$ are:

DOMAIN of $\text{sec}(x):\hspace{30pt} \left\lbrace x\in {\mathbb R}: x\neq \frac{(2n+1)\pi}{2},n\in{\mathbb Z}\right\rbrace$

RANGE of $\text{sec}(x): \hspace{30pt}\left\lbrace y\in {\mathbb R}:y\leq-1,y\geq 1\right\rbrace$

where we note that we cannot evaluate $\cos(x)$ nor $\text{sec}(x)$ at odd multiples of $\frac{\pi}{2}$. See more on domain and range. The domain of both $\text{cosec(x)}$ and $\text{cotan}(x)$ is $\left\lbrace x\in {\mathbb R}: x\neq n\pi,n\in{\mathbb Z}\right\rbrace$. The range of $\text{cosec(x)}$ is the same as that of $\text{sec(x)}$. Since the range of $\tan(x)$ is $y\in{\mathbb R}$ with no exceptions, the range of $\text{cotan}(x)$ is also $y\in{\mathbb R}$. See Example 2 for a transformation example.

Reciprocal Trigonometric Function Identities

There are two more trigonometric identities you should learn. See other trigonometric identities.  These two new identities are useful when solving certain trigonometric equations:

$1+\tan^2(x)=\text{sec}^2(x)$

$1+\text{cot}^2(x)=\text{cosec}^2(x)$

See Example 3. There is a quirky way to remember these identities. ‘The one with the tan is sexy’ and ‘the one with the coat is cosy’. They both come from the trigonometric identity $\sin^2(x)+\cos^2(x)=1$. Dividing both sides by $\cos^2(x)$ gives the first identity and by $\sin^2(x)$ gives the second. 

Examples

Solve $5\text{cosec}(2x)=10$ for $-2\pi<x<2\pi$.

Sketch and state the domain and range of the graph of $y=\text{cosec}(2x)+1$.

  1. Given that $\tan(\theta)=-\frac{3}{4}$ and $\theta$ is obtuse, find the value of $\sec(\theta)$.
  2. Show that $\text{sec}^2(\theta)+\text{cosec}^2(\theta)\equiv\text{cosec}^2(\theta)\text{sec}^2(\theta)$

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