Many students think expressions like sin A, cos A, etc., mean “sin × A” or “cos × A”. But this is completely incorrect. These are functions, not multiplications.

✅ These Ratios Do NOT Mean Multiplication

Let’s break this down clearly for all six trigonometric ratios.

✅ 1. Sin A ≠ sin × A

❌ Wrong:

• “sin A” is not the result of multiplying “sin” and “A”.
• You can’t multiply a function name with an angle.

✅ Correct:

• sin A means: apply the sine function to the angle A.
• In a right-angled triangle,
  sin A = (Perpendicular) / (Hypotenuse)

🧮 Example:

If Perpendicular = 3, Hypotenuse = 5:
Then, sin A = 3/5
✔ No multiplication involved — it’s a ratio of sides.

✅ 2. Cos A ≠ cos × A

❌ Wrong:

• “cos A” is not the product of “cos” and “A”.

✅ Correct:

• cos A means: take the cosine of angle A.
• In a right-angled triangle,
  cos A = (Base) / (Hypotenuse)

🧮 Example:

If Base = 4, Hypotenuse = 5:
Then, cos A = 4/5

✅ 3. Tan A ≠ tan × A

❌ Wrong:

• It’s not “tan multiplied by A”.

✅ Correct:

• tan A = (Perpendicular) / (Base)

🧮 Example:

If Perpendicular = 3, Base = 4:
Then, tan A = 3/4

✅ 4. Cosec A ≠ cosec × A

• Cosec (or “csc”) A is the reciprocal of sin A

✅ Correct:

• cosec A = Hypotenuse / Perpendicular

🧮 Example:

If sin A = 3/5, then
cosec A = 5/3

✅ 5. Sec A ≠ sec × A

• Sec A is the reciprocal of cos A

✅ Correct:

• sec A = Hypotenuse / Base

🧮 Example:

If cos A = 4/5, then
sec A = 5/4

✅ 6. Cot A ≠ cot × A

• Cot A is the reciprocal of tan A

✅ Correct:

• cot A = Base / Perpendicular

🧮 Example:

If tan A = 3/4, then
cot A = 4/3

Trigonometric Ratios Belong to Acute Angles in Right-Angled Triangles

✅ 1. What Is a Right-Angled Triangle?

A right-angled triangle is a triangle where one angle is exactly 90°, and the other two angles are acute (less than 90°).

Let’s say:

• Triangle ABC is a right-angled triangle,
• ∠C = 90°,
• Then ∠A and ∠B are acute angles.

We define trigonometric ratios with respect to one of the acute angles, like angle A.

✅ 2. Trigonometric Ratios Are Defined for Acute Angles (less than 90°)

In a right triangle, trigonometric ratios are used to relate the sides of the triangle to one of the acute angles.

📌 Final Summary:

• Trigonometric ratios like sin A, cos A, tan A, etc. are not multiplications.
• They are functions applied to an acute angle A in a right-angled triangle.
• These ratios relate the sides of the triangle to the angle A.
• You cannot treat “sin A” as “sin × A” — that’s mathematically incorrect.
• These definitions do not apply if A is not acute or the triangle is not right-angled.

🧑‍🏫 Author Bio:

Aman Sir, founder of Maths Vidya Institute, is a passionate mathematics educator with over 14 years of teaching experience, helping students master concepts from Class 9 to 12 with clarity and confidence.