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Cosec A, sec A and cot A

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to keep the same fraction and only change the name. That is wrong because reciprocal ratios must be inverted.

Definition

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

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.

Rule to remember

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

Memory hook

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

Examples and method

Worked example

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

Method to apply

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

Diagram support

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.

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

A common wrong answer is to keep the same fraction and only change the name. That is wrong because reciprocal ratios must be inverted.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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

2-mark answer

Cosec A, sec A, and cot A are the reciprocal trigonometric ratios of sine, cosine, and tangent. cosec A = 1/sin A = hypotenuse/opposite, sec A = 1/cos A = hypotenuse/adjacent, cot A = 1/tan A = adjacent/opposite. 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-mark answer

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. cosec A = 1/sin A = hypotenuse/opposite, sec A = 1/cos A = hypotenuse/adjacent, cot A = 1/tan A = adjacent/opposite. If a right triangle gives tan A = 2/3, then cot A = 3/2. This is because cot A is the reciprocal of tan A. CBSE may ask to convert a basic ratio into its reciprocal, or choose the correct reciprocal ratio from side lengths. A common wrong answer is to keep the same fraction and only change the name. That is wrong because reciprocal ratios must be inverted.
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