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Trigonometric ratios

Trigonometric ratios are the fixed ratios between the sides of a right triangle for a chosen acute angle.

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Student-friendly explanation

For one acute angle in a right triangle, the sides keep a fixed proportion. That is why ratios like sine, cosine, and tangent depend on the angle, not on the size of the triangle. If two right triangles have the same acute angle, the ratios for that angle stay the same.

How to write this in exams

  1. 1

    Start with the exact idea

    Trigonometric ratios are the fixed ratios between the sides of a right triangle for a chosen acute angle.

  2. 2

    Then show how to use it

    1. Mark the given angle. 2. Find the right angle and identify the hypotenuse. 3. Name the opposite and adjacent sides with respect to the given angle. 4. Substitute the correct sides into the required ratio.

  3. 3

    Add one concrete example

    If two right triangles both have angle A equal to 30 degrees, then the ratio opposite side divided by hypotenuse for angle A will be the same in both triangles.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to use the same side names for every angle. That is wrong because opposite and adjacent change when the chosen angle changes.

Definition

Trigonometric ratios are the fixed ratios between the sides of a right triangle for a chosen acute angle.

Example

If two right triangles both have angle A equal to 30 degrees, then the ratio opposite side divided by hypotenuse for angle A will be the same in both triangles.

Rule to remember

For a right triangle with acute angle A: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent. These are ratios of sides.

Memory hook

Think: opposite is across, adjacent is beside, hypotenuse is the longest side.

Examples and method

Worked example

In a right triangle, if the side opposite angle A is 3 cm and the hypotenuse is 5 cm, then sin A = 3/5. The ratio is a number between 0 and 1 for an acute angle.

Method to apply

1. Mark the given angle. 2. Find the right angle and identify the hypotenuse. 3. Name the opposite and adjacent sides with respect to the given angle. 4. Substitute the correct sides into the required ratio.

Diagram support

Draw a right triangle, mark one acute angle as A, label the side opposite A, the side adjacent to A, and the hypotenuse opposite the right angle.

How CBSE asks it

CBSE may ask you to identify the correct ratio from a triangle, calculate a ratio from given sides, or choose the side pair used in a trigonometric ratio.

Avoid common mistakes

Common confusion

Students often mix up opposite side and adjacent side, especially when the triangle is drawn in a different orientation.

Common wrong answer

A common wrong answer is to use the same side names for every angle. That is wrong because opposite and adjacent change when the chosen angle changes.

Exam tip

Always identify the chosen angle first, then label opposite, adjacent, and hypotenuse before writing any ratio.

Quick check

Why are trigonometric ratios called ratios and not side lengths?

They are called ratios because each one compares two side lengths, such as opposite divided by hypotenuse or opposite divided by adjacent. The value depends on the angle, not on the actual size of the triangle.

Study the trigonometric ratios diagram carefully

Use the labelled diagram to keep trigonometric ratios clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of trigonometric ratios in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on trigonometric ratios.

Revision cue

Revise trigonometric ratios through the labels before writing the answer.

Answer writing and exam use

1-mark answer

Trigonometric ratios are the fixed ratios between the sides of a right triangle for a chosen acute angle.

2-mark answer

Trigonometric ratios are the fixed ratios between the sides of a right triangle for a chosen acute angle. For a right triangle with acute angle A: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent. These are ratios of sides. If two right triangles both have angle A equal to 30 degrees, then the ratio opposite side divided by hypotenuse for angle A will be the same in both triangles.

3-mark answer

For one acute angle in a right triangle, the sides keep a fixed proportion. That is why ratios like sine, cosine, and tangent depend on the angle, not on the size of the triangle. If two right triangles have the same acute angle, the ratios for that angle stay the same. For a right triangle with acute angle A: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent. These are ratios of sides. In a right triangle, if the side opposite angle A is 3 cm and the hypotenuse is 5 cm, then sin A = 3/5. The ratio is a number between 0 and 1 for an acute angle. CBSE may ask you to identify the correct ratio from a triangle, calculate a ratio from given sides, or choose the side pair used in a trigonometric ratio. A common wrong answer is to use the same side names for every angle. That is wrong because opposite and adjacent change when the chosen angle changes.
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