Sec, cosec and cot: the reciprocal three

8 minA Level3.3.1

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can define $\sec\theta$ , $\operatorname{cosec}\theta$ and $\cot\theta$ , say where each one is undefined, sketch their graphs, and use them to find exact values and simplify expressions.

Three new names for three old ideas

Each new function is 1 divided by a function you already know. That's called a reciprocal: the reciprocal of 4 is $\tfrac14$ .

New function	Read as	Means
$\sec\theta$	"secant"	$\dfrac{1}{\cos\theta}$
$\operatorname{cosec}\theta$	"cosecant"	$\dfrac{1}{\sin\theta}$
$\cot\theta$	"cotangent"	$\dfrac{1}{\tan\theta} = \dfrac{\cos\theta}{\sin\theta}$

The trap: the pairings cross over. sec goes with cos, and cosec goes with sin.

Memory trick: the third letter of each name. sec → c → cos. cosec → s → sin. cot → t → tan.

Definition

The reciprocal trigonometric functions

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

Each one is undefined wherever its bottom line is zero, because you can't divide by 0:

$\sec\theta$ is undefined where $\cos\theta = 0$ (at $90°$ , $270°$ , …);
$\operatorname{cosec}\theta$ is undefined where $\sin\theta = 0$ (at $0°$ , $180°$ , $360°$ , …);
$\cot\theta = \dfrac{\cos\theta}{\sin\theta}$ is undefined where $\sin\theta = 0$ too.

On a graph, each undefined value becomes a dashed vertical line called an asymptote: the curve gets closer and closer to it but never touches it.

Reading the graphs: four facts

1. sec and cosec never take values between $-1$ and $1$ .

$\cos\theta$ is always between $-1$ and $1$ . Dividing 1 by a number of size 1 or less gives a number of size 1 or more. For example, $\dfrac{1}{0.5} = 2$ .

So $\sec\theta \geqslant 1$ or $\sec\theta \leqslant -1$ . The same is true for $\operatorname{cosec}\theta$ .

2. Undefined points are vertical asymptotes.

sec: at $90°$ and $270°$ .
cosec and cot: at $0°$ , $180°$ and $360°$ .

3. Where the old function is 0, the new one blows up, and the other way round.

For example, $\tan\theta$ is undefined at $90°$ (because $\cos 90° = 0$ ). There, $\cot\theta = \dfrac{\cos\theta}{\sin\theta} = \dfrac{0}{1} = 0$ .

4. Periods. $\sec\theta$ and $\operatorname{cosec}\theta$ repeat every $360°$ , like $\cos$ and $\sin$ . $\cot\theta$ repeats every $180°$ , like $\tan$ .

Worked example: Example 1: exact values

Find the exact values of $\sec 60°$ , $\operatorname{cosec} 30°$ and $\cot 45°$ .

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Quick check

Multiple choiceQuestion 1 · Recall

Answer the 3 questions above to continue.