The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.

1) What does this statement mean?

• When we calculate sine, cosine, or tangent of an angle in a right-angled triangle, we use the sides of the triangle.
• For example, in a right triangle:
  • sin A = Perpendicular / Hypotenuse
  • cos A = Base / Hypotenuse
  • tan A = Perpendicular / Base
• Even if we make a bigger or smaller triangle with the same angle, these ratios stay the same.
• This is because the sides increase or decrease in the same proportion. So the fraction stays constant.

2) Let’s see an example

Example 1:

• Take a right-angled triangle ABC with angle A = 30°.
• Suppose the side opposite to A is 1 unit and the hypotenuse is 2 units.

Then:

• sin A = 1/2 = 0.5
• cos A = √3/2 ≈ 0.866 (using Pythagoras theorem to find other side)
• tan A = 1/√3 ≈ 0.577

Now, let’s make a bigger triangle:

• Keep the angle A = 30°.
• Make the opposite side 5 units.
• Hypotenuse will now be 10 units (same proportion).

Then:

• sin A = 5/10 = 1/2 = 0.5 again!
• The value is the same.
• Same for cos A and tan A.

✅ So the actual length does not matter — only the angle matters for trigonometric ratios.

Understanding  Notations :

3) What about squares and powers?

In the note, you saw:

“For the sake of convenience, we may write sin²A instead of (sin A)².”

This means:

• If you calculate sin A = 1/2,
• Then sin²A means (1/2)² = 1/4.

It’s just a shorter way to write.

So:

• sin²A = (sin A)²
• cos²A = (cos A)²
• tan²A = (tan A)²

It saves space and is easy to write in books and exams.

4) Important difference: cosec A and sin⁻¹ A

The note says:

“cosec A = (sin A)⁻¹ ≠ sin⁻¹ A”

This confuses many students. Let’s clear it:

• cosec A means the reciprocal of sin A.
  • For example, if sin A = 1/2,
  • Then cosec A = 1 / (1/2) = 2.
• sin⁻¹ A means inverse sine (arc sine).
  • It’s NOT the same as reciprocal.
  • sin⁻¹ 1/2 means: “Which angle has sine equal to 1/2?”
  • So sin⁻¹ (1/2) = 30°.

So don’t mix them up:

• (sin A)⁻¹ = cosec A → reciprocal
• sin⁻¹ A → inverse sine (find the angle)

5) What about other trigonometric ratios?

The same logic applies to:

• sec A = 1 / cos A
• cot A = 1 / tan A

These are just reciprocals of the basic ratios.

6) What is theta (θ)?

Sometimes, instead of A, we use the Greek letter θ (theta) to show an unknown angle.

• For example, sin θ = 1/2 means we don’t know the angle yet.
• When you solve, you get θ = 30°.

7) When do we learn inverse trigonometry?

Inverse trigonometry (like sin⁻¹) is taught in Class 11 & Class 12.

• It helps when you know the ratio but need the angle.
• Example: If sin θ = 0.5, then θ = sin⁻¹ (0.5) = 30°

8) Real-life example

• Builders use trigonometry to find heights of towers.
• If the angle of elevation is known and the distance from the tower is known, they can calculate the height.

9) Some important tips

• Always remember: sin²A means (sin A)², not sin A × 2.
• Cosec, sec, and cot are just reciprocals.
• Do not confuse reciprocal with inverse.
• Use standard trigonometric tables for calculations

 Summary

• Trigonometric ratios depend only on the angle, not on the actual size of the triangle.
• Notation like sin²A means the square of sin A.
• Reciprocal is different from inverse.
• Theta (θ) is commonly used for angles.