Secant, Cosecant and Cotangent (9709 Pure Mathematics 3) — Mathematics

Introduction

Trigonometry at A-Level goes beyond $\sin$ , $\cos$ and $\tan$ . The three reciprocal trigonometric functions — secant ( $\sec$ ), cosecant ( $\cosec$ ) and cotangent ( $\cot$ ) — appear regularly in 9709 Paper 3 questions on solving equations, proving identities and integration (in later topics). Understanding their definitions, graphs, domains and key properties is therefore essential exam preparation.

Core Concept

Definitions

The three reciprocal functions are defined directly from the three primary functions:

\sec\theta = \frac{1}{\cos\theta}, \qquad \cosec\,\theta = \frac{1}{\sin\theta}, \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}

Each function is undefined whenever its denominator equals zero:

Function	Undefined when	Excluded values
$\sec\theta$	$\cos\theta = 0$	$\theta = 90° + 180°n,\ n \in \mathbb{Z}$
$\cosec\,\theta$	$\sin\theta = 0$	$\theta = 180°n,\ n \in \mathbb{Z}$
$\cot\theta$	$\sin\theta = 0$	$\theta = 180°n,\ n \in \mathbb{Z}$

Ranges

Function	Range
$\sec\theta$	$\sec\theta \leq -1$ or $\sec\theta \geq 1$
$\cosec\,\theta$	$\cosec\,\theta \leq -1$ or $\cosec\,\theta \geq 1$
$\cot\theta$	$\cot\theta \in \mathbb{R}$ (all real values)

This follows directly from the fact that $-1 \le \cos\theta \le 1$ and $-1 \le \sin\theta \le 1$ , so their reciprocals can never lie strictly between $-1$ and $1$ .

Graph Behaviour

$y = \sec\theta$ is the reciprocal of $y = \cos\theta$ . It has:

Vertical asymptotes at every zero of $\cos\theta$ (i.e. $\theta = \pm90°, \pm270°, \ldots$ ).
Local minima of $+1$ and local maxima of $-1$ , coinciding with the peaks/troughs of $\cos\theta$ .
Period $360°$ .

$y = \cosec\,\theta$ is the reciprocal of $y = \sin\theta$ . It has:

Vertical asymptotes at every zero of $\sin\theta$ (i.e. $\theta = 0°, \pm180°, \pm360°, \ldots$ ).
Local minima of $+1$ and local maxima of $-1$ .
Period $360°$ .

$y = \cot\theta$ is the reciprocal of $y = \tan\theta$ . It has:

Vertical asymptotes where $\sin\theta = 0$ (same as $\cosec$ ).
Decreasing on each interval between asymptotes (unlike $\tan$ , which is increasing).
Period $180°$ .
Note: $\cot\theta = 0$ wherever $\cos\theta = 0$ , i.e. at $\theta = 90° + 180°n$ .

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