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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.

25 min Trigonometric Functions

Edexcel A level Maths: 6.1 Secant, Cosecant, CotangentCurated by Kepler, not produced by us. Watch it before 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

$sec θ = \frac{1}{cos θ}$ , $cosec θ = \frac{1}{sin θ}$ , and $cot θ = \frac{1}{tan θ} = \frac{cos θ}{sin θ}$ .

sec θ = \frac{1}{c o s θ}, cosec θ = \frac{1}{s i n θ}, cot θ = \frac{c o s θ}{s i n θ}

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: $sec θ$ where $cos θ = 0$ (odd multiples of $\frac{π}{2}$ ), $cosec θ$ where $sin θ = 0$ (multiples of $π$ ), $cot θ$ where $sin θ = 0$ .

cos 60° = \frac{1}{2}

sec 60° = \frac{1}{1/2} = 2

Answer

2

Formula recap

sec θ = \frac{1}{c o s θ}

Secant.

cosec θ = \frac{1}{s i n θ}

Cosecant.

cot θ = \frac{c o s θ}{s i n θ}

Cotangent.

Common mistakes to avoid

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 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.

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Secant, Cosecant and Cotangent

The definitions

Where they are undefined

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Secant, Cosecant and Cotangent

The definitions

Where they are undefined

Practise this topic

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