Secant, cosecant and cotangent

In a nutshell

These three trigonometric functions are called the reciprocal trigonometric functions because of their reciprocal relationships with the functions sine, cosine and tangent, respectively.

Definitions

$\boxed{\cosec(x)=\frac1{\sin(x)}}$

$\boxed{\sec(x)=\frac1{\cos(x)}}$

$\boxed{\cot(x)=\frac1{\tan(x)}}$

Note: These trigonometric functions do not have a button on the calculator, so you have to use the $\sin$ , $\cos$ and $\tan$ functions.

Example 1

Use your calculator to find $\cot(48^\circ)$ to $3\text{ s.f.}$

With your calculator in degrees mode, type in $\frac1{\tan(48)}$ :

$\cot(48)=\frac1{\tan(48)}=\frac1{1.111}=\underline{0.900}\text{ (to }3\text{ s.f.})$ 

Example 2

Without using a calculator, find the exact value of $\cosec(210^\circ)$ .

First identify the trigonometric function that you will be able to use. Since $\cosec(210)=\frac1{\sin(210)}$ , you will find $\sin(210^\circ)$ next. To do so, use that $210=180+30$ . By using the sine graph you have

Maths; Trigonometric functions; KS5 Year 13; Secant, cosecant and cotangent

$\sin(210^\circ)=-\sin(30^\circ)=-\frac{1}2$ 

Alternatively, you could have used the double angle formula:

$\begin{aligned}\sin(210)&=\sin(180+30)\\&=\sin(180)\cos(30)+\sin(30)\cos(180)\\&=0+\left(\frac12\right)(-1)\\&={-\frac12}\end{aligned}$ 

Hence you have that $\cosec(210^\circ)=\dfrac1{-\frac12}=\underline{-2}$ .

Tool to remember

One way to remember which reciprocal trigonometric function relates to which trigonometric function is to look at the third letter of the reciprocal function. It corresponds to the first letter of the related trigonometric function:

$\begin{aligned}\text{co\underline{s}ec}(x)&=\frac1{\text{\underline{s}in}(x)}\\\text{se\underline{c}}(x)&=\frac1{\text{\underline{c}os}(x)}\\\text{co\underline{t}}(x)&=\frac1{\text{\underline{t}an}(x)}\\\end{aligned}$