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Identity sin^2 A + cos^2 A = 1

The identity sin squared A plus cos squared A equals 1 is a fundamental trigonometric relation for every angle A.

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Student-friendly explanation

This identity comes from a right triangle. If the hypotenuse is taken as the common denominator, the squared opposite and adjacent parts add up to the square of the hypotenuse. After dividing by hypotenuse squared, the result becomes sin squared A plus cos squared A equals 1.

How to write this in exams

  1. 1

    Start with the exact idea

    The identity sin squared A plus cos squared A equals 1 is a fundamental trigonometric relation for every angle A.

  2. 2

    Then show how to use it

    1. Square the known ratio. 2. Subtract from 1 if the missing ratio is required. 3. Take the square root. 4. Choose the positive value for an acute angle.

  3. 3

    Add one concrete example

    If sin A = 3/5, then cos A = 4/5, and 9/25 + 16/25 = 25/25 = 1.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to forget the squares and use sin A plus cos A equals 1. That is not the correct identity.

Definition

The identity sin squared A plus cos squared A equals 1 is a fundamental trigonometric relation for every angle A.

Example

If sin A = 3/5, then cos A = 4/5, and 9/25 + 16/25 = 25/25 = 1.

Rule to remember

sin^2 A + cos^2 A = 1 for every angle A. This is a basic identity and is often used as a starting point for other identities.

Memory hook

Squares matter here: sine squared plus cosine squared gives one full unit.

Examples and method

Worked example

If sin A = 5/13, then cos A = 12/13 by the identity, because 25/169 + 144/169 = 169/169 = 1.

Method to apply

1. Square the known ratio. 2. Subtract from 1 if the missing ratio is required. 3. Take the square root. 4. Choose the positive value for an acute angle.

Diagram support

A right triangle with one angle A and the side labels opposite, adjacent, and hypotenuse helps students see why the identity works.

How CBSE asks it

CBSE may ask you to find a missing ratio, verify a value pair, or simplify an expression using this identity.

Avoid common mistakes

Common confusion

Students sometimes write sin A plus cos A equals 1, but the squares are essential and cannot be removed.

Common wrong answer

A common wrong answer is to forget the squares and use sin A plus cos A equals 1. That is not the correct identity.

Exam tip

Use this identity to check standard values or simplify expressions that contain both sin squared and cos squared terms.

Quick check

If sin A = 4/5, what should cos A be for this identity to hold?

cos A should be 3/5 because (4/5)^2 + (3/5)^2 = 16/25 + 9/25 = 1. The identity must remain true.

Answer writing and exam use

1-mark answer

The identity sin squared A plus cos squared A equals 1 is a fundamental trigonometric relation for every angle A.

2-mark answer

The identity sin squared A plus cos squared A equals 1 is a fundamental trigonometric relation for every angle A. sin^2 A + cos^2 A = 1 for every angle A. This is a basic identity and is often used as a starting point for other identities. If sin A = 3/5, then cos A = 4/5, and 9/25 + 16/25 = 25/25 = 1.

3-mark answer

This identity comes from a right triangle. If the hypotenuse is taken as the common denominator, the squared opposite and adjacent parts add up to the square of the hypotenuse. After dividing by hypotenuse squared, the result becomes sin squared A plus cos squared A equals 1. sin^2 A + cos^2 A = 1 for every angle A. This is a basic identity and is often used as a starting point for other identities. If sin A = 5/13, then cos A = 12/13 by the identity, because 25/169 + 144/169 = 169/169 = 1. CBSE may ask you to find a missing ratio, verify a value pair, or simplify an expression using this identity. A common wrong answer is to forget the squares and use sin A plus cos A equals 1. That is not the correct identity.
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