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6.1PureCore

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
1

The definitions

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

sec⁡θ=1cos⁡θ,cosec⁡θ=1sin⁡θ,cot⁡θ=cos⁡θsin⁡θ\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.

2

Where they are undefined

Each is undefined where its denominator is zero: sec⁡θ\sec\theta where cos⁡θ=0\cos\theta = 0 (odd multiples of π2\tfrac{\pi}{2}), cosec⁡θ\operatorname{cosec}\theta where sin⁡θ=0\sin\theta = 0 (multiples of π\pi), cot⁡θ\cot\theta where sin⁡θ=0\sin\theta = 0.

Example 1Find sec⁡60°\sec 60°.
1cos⁡60°=12\cos 60° = \tfrac12.
2sec⁡60°=11/2=2\sec 60° = \tfrac{1}{1/2} = 2.
Answer
22

Formula recap

sec⁡θ=1cos⁡θ\sec\theta = \tfrac{1}{\cos\theta}
Secant.
cosec⁡θ=1sin⁡θ\operatorname{cosec}\theta = \tfrac{1}{\sin\theta}
Cosecant.
cot⁡θ=cos⁡θsin⁡θ\cot\theta = \tfrac{\cos\theta}{\sin\theta}
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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. sec⁡θ\sec\theta is defined as:

    • 1cos⁡θ\tfrac{1}{\cos\theta}
    • 1sin⁡θ\tfrac{1}{\sin\theta}
    • 1tan⁡θ\tfrac{1}{\tan\theta}
    • cos⁡θ\cos\theta
    Show answer and explanation

    Answer: 1cos⁡θ\tfrac{1}{\cos\theta}

    sec ↔ cos.

  2. cosec⁡θ\operatorname{cosec}\theta is:

    • 1sin⁡θ\tfrac{1}{\sin\theta}
    • 1cos⁡θ\tfrac{1}{\cos\theta}
    • 1tan⁡θ\tfrac{1}{\tan\theta}
    • sin⁡θ\sin\theta
    Show answer and explanation

    Answer: 1sin⁡θ\tfrac{1}{\sin\theta}

    cosec ↔ sin.

  3. cot⁡θ\cot\theta equals:

    • cos⁡θsin⁡θ\tfrac{\cos\theta}{\sin\theta}
    • sin⁡θcos⁡θ\tfrac{\sin\theta}{\cos\theta}
    • 1cos⁡θ\tfrac{1}{\cos\theta}
    • tan⁡θ\tan\theta
    Show answer and explanation

    Answer: cos⁡θsin⁡θ\tfrac{\cos\theta}{\sin\theta}

    Reciprocal of tan.

  4. sec⁡60°\sec 60° equals:

    • 22
    • 12\tfrac12
    • 23\tfrac{2}{\sqrt3}
    • 11
    Show answer and explanation

    Answer: 22

    1/cos60° = 1/(1/2).

  5. cosec⁡θ\operatorname{cosec}\theta is undefined when:

    • sin⁡θ=0\sin\theta = 0
    • cos⁡θ=0\cos\theta = 0
    • tan⁡θ=0\tan\theta = 0
    • never
    Show answer and explanation

    Answer: sin⁡θ=0\sin\theta = 0

    Denominator zero.

Next step

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