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medium difficulty5 min

Cosec A, sec A and cot A

Cosec A, sec A, and cot A are the reciprocal trigonometric ratios of sine, cosine, and tangent.

In one line: co-sec, se-c, and co-t all feel like the flip side of sine, cosine, and tangent.

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Understand it

These ratios are useful because they flip the basic ratios. Cosec A is hypotenuse over opposite, sec A is hypotenuse over adjacent, and cot A is adjacent over opposite. In exams, students often lose marks by confusing these reciprocals with the basic ratios.

Example

If sin A = 3/5, then cosec A = 5/3. If cos A = 4/5, then sec A = 5/4. If tan A = 3/4, then cot A = 4/3.

Formula and rule

cosec A = 1/sin A = hypotenuse/opposite, sec A = 1/cos A = hypotenuse/adjacent, cot A = 1/tan A = adjacent/opposite.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1If a right triangle gives tan A = 2/3, then cot A = 3/2.
  2. 2This is because cot A is the reciprocal of tan A.

Method you can reuse

  1. 1Write the basic ratio first if needed.
  2. 2Decide whether the question asks for reciprocal form.
  3. 3Flip the fraction carefully.
  4. 4Check side names against the chosen angle.

Sketch it in your answer

Draw one right triangle and keep the same angle A. Use the same labels as sine, cosine, and tangent, then flip the fractions for the reciprocal ratios.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If cos A = 7/25, what is sec A?

Think first, then show answer
sec A is the reciprocal of cos A, so sec A = 25/7. The numerator and denominator are exchanged.

Common wrong answer

A common wrong answer is to keep the same fraction and only change the name.

Why it loses marks: Reciprocal ratios must be inverted.

Don't mix this up

Students sometimes write sec A as adjacent over hypotenuse, but that is cosine, not secant.

Exam tip

When you see cosec, sec, or cot, ask yourself which basic ratio must be flipped.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

CBSE may ask to convert a basic ratio into its reciprocal, or choose the correct reciprocal ratio from side lengths.
  1. 1

    Enough for 1 mark

    Cosec A, sec A, and cot A are the reciprocal trigonometric ratios of sine, cosine, and tangent.

  2. 2

    Add for 2 marks

    Example: If sin A = 3/5, then cosec A = 5/3. If cos A = 4/5, then sec A = 5/4. If tan A = 3/4, then cot A = 4/3.

  3. 3

    Add for 3 marks

    These ratios are useful because they flip the basic ratios.

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