\sec x=\frac{1}{\cos x}, undefined where \cos x=0.

Define \operatorname{cosec}\,x.

\operatorname{cosec}\,x=\frac{1}{\sin x}, undefined where \sin x=0.

Define \cot x in two ways.

\cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x}.

Where are the asymptotes of y=\sec x?

At x=\frac{\pi}{2}+n\pi (where \cos x=0).

Range of \sec x and \operatorname{cosec}\,x?

y\le-1 or y\ge1; no values between -1 and 1.

Period of \sec x, \operatorname{cosec}\,x, \cot x?

2\pi, 2\pi and \pi respectively.

Restricted domain used for \sec x?

0\le x\le\pi, x\ne\frac{\pi}{2}.

Restricted domain used for \operatorname{cosec}\,x?

-\frac{\pi}{2}\le x\le\frac{\pi}{2}, x\ne0.

Secant, cosecant and cotangentEdexcel International A Level Maths: Flashcards

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Define $\sec x$.

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Define $\sec x$ .: $\sec x=\frac{1}{\cos x}$ , undefined where $\cos x=0$ .
Define $\operatorname{cosec}\,x$ .: $\operatorname{cosec}\,x=\frac{1}{\sin x}$ , undefined where $\sin x=0$ .
Define $\cot x$ in two ways.: $\cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x}$ .
Where are the asymptotes of $y=\sec x$ ?: At $x=\frac{\pi}{2}+n\pi$ (where $\cos x=0$ ).
Where are the asymptotes of $y=\operatorname{cosec}\,x$ and $y=\cot x$ ?: At $x=n\pi$ (where $\sin x=0$ ).
Range of $\sec x$ and $\operatorname{cosec}\,x$ ?: $y\le-1$ or $y\ge1$ ; no values between $-1$ and $1$ .
Range of $\cot x$ ?: All real numbers.
Period of $\sec x$ , $\operatorname{cosec}\,x$ , $\cot x$ ?: $2\pi$ , $2\pi$ and $\pi$ respectively.
Restricted domain used for $\sec x$ ?: $0\le x\le\pi$ , $x\ne\frac{\pi}{2}$ .
Restricted domain used for $\operatorname{cosec}\,x$ ?: $-\frac{\pi}{2}\le x\le\frac{\pi}{2}$ , $x\ne0$ .
Restricted domain used for $\cot x$ ?: $0<x<\pi$ .
Exact values of $\sec60^{\circ}$ and $\cot45^{\circ}$ ?: $2$ and $1$ .
Does $\sec\theta=\frac12$ have any solutions?: No, since $|\sec\theta|\ge1$ .

Exam questions on Secant, cosecant and cotangent

The angle $\theta$ is acute and $\sin\theta=\frac{5}{13}$ .
Find the exact value of $\sec\theta+\operatorname{cosec}\,\theta$ .2 marks
Throughout, $x$ is measured in radians, and each function is considered only where it is defined.
Write $\dfrac{\operatorname{cosec}\,x}{\cot x}$ as a single trigonometric function.2 marks
Angles are measured in degrees and $0\le\theta<360$ .
Solve $\sec\theta=-2$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).