1.05h–iPureCore

Secant, Cosecant and Cotangent

The three reciprocal trig functions — secant, cosecant and cotangent — are the reciprocals of cosine, sine and tangent. They appear throughout Year 2 trigonometry, so it pays to know their definitions cold.

30 min Trigonometry

The big picture

$\sec x$ has vertical asymptotes where $\cos x=0$ and range $(-\infty,-1]\cup[1,\infty)$ . $\operatorname{cosec}x$ has the same range, with asymptotes where $\sin x=0$ . Neither graph can cross the x-axis. For $\cot x=\cos x/\sin x$ , exclude $x=k\pi$ . Its period is $\pi$ and it can equal zero where cosine is zero. Start from definitions instead of treating a reciprocal as an inverse function. To sketch secant, mark the cosine zeros as asymptotes, then mark points where cosine is 1 or -1. The reciprocal branches stay beyond y = 1 or y = -1. Reflecting a sine graph in y = x gives an inverse relation, which is a different operation altogether.

Edexcel A level Maths: 6.1 Secant, Cosecant, CotangentCurated by Kepler, not produced by us. Watch it, then work through the notes below.

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What you'll be able to do

Define sec, cosec and cot
Match each to its reciprocal cos, sin, tan
Evaluate them at standard angles
Identify where they are undefined

The definitions

$\sec\theta = \dfrac{1}{\cos\theta}$ , $\operatorname{cosec}\theta = \dfrac{1}{\sin\theta}$ , and $\cot\theta = \dfrac{1}{\tan\theta} = \dfrac{\cos\theta}{\sin\theta}$ .

\sec\theta = \tfrac{1}{\cos\theta}, \quad \operatorname{cosec}\theta = \tfrac{1}{\sin\theta}, \quad \cot\theta = \tfrac{\cos\theta}{\sin\theta}

Reciprocal trig functions.

Tip — Memory aid: the third letter tells you the reciprocal — seCant↔Cos, coSecant↔Sin, coTangent↔Tan.

Where they are undefined

Each is undefined where its denominator is zero: $\sec\theta$ where $\cos\theta = 0$ (odd multiples of $\tfrac{\pi}{2}$ ), $\operatorname{cosec}\theta$ where $\sin\theta = 0$ (multiples of $\pi$ ), $\cot\theta$ where $\sin\theta = 0$ .

Example 1Find

\sec 60°

\cos 60° = \tfrac12

\sec 60° = \tfrac{1}{1/2} = 2

Answer

2

Graphs, asymptotes and equations

For $\cot x=\cos x/\sin x$ , exclude $x=k\pi$ . Its period is $\pi$ and it can equal zero where cosine is zero. Start from definitions instead of treating a reciprocal as an inverse function.

To sketch secant, mark the cosine zeros as asymptotes, then mark points where cosine is 1 or -1. The reciprocal branches stay beyond y = 1 or y = -1. Reflecting a sine graph in y = x gives an inverse relation, which is a different operation altogether.

Example 1Solve

\sec x=-2

for

0\leq x<2\pi

1/\cos x=-2

gives

\cos x=-1/2

2Cosine is negative in quadrants II and III.

3The reference angle is

\pi/3

, so

x=2\pi/3,4\pi/3

. Neither is an asymptote.

Answer

x=2\pi/3

4\pi/3

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

Thinking sec is the reciprocal of sin.

sec ↔ cos, cosec ↔ sin (the “co” swaps).

Writing cot θ = 1/tan θ only.

cot θ = cos θ / sin θ too — useful when tan θ = 0 handling.

Key results to remember

\sec\theta = \tfrac{1}{\cos\theta}

Secant.

\operatorname{cosec}\theta = \tfrac{1}{\sin\theta}

Cosecant.

\cot\theta = \tfrac{\cos\theta}{\sin\theta}

Cotangent.

Key takeaways

sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = cos θ/sin θ.
Undefined where the denominator is zero.
The “co” swaps: sec↔cos, cosec↔sin.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

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