Let’s break this down step by step so you fully understand why this is always true.

📌 What Is a Right Triangle?

• A right triangle is a triangle with one 90-degree angle.
• The side opposite this right angle is called the hypotenuse.
• The other two sides are called the perpendicular and the base — they are always shorter than the hypotenuse.

📌 Basic Trigonometric Ratios

In a right triangle, we mainly use three ratios:

1. Sine (sin)
   • sin A = Perpendicular / Hypotenuse
2. Cosine (cos)
   • cos A = Base / Hypotenuse
3. Tangent (tan)
   • tan A = Perpendicular / Base

The line in the screenshot talks about sin A and cos A, so let’s stick to those.

📌 Why Is the Hypotenuse Important?

The hypotenuse is always the longest side in a right triangle.
This means:

• The perpendicular and base will always be shorter than the hypotenuse.
• So, when you divide a shorter side by the hypotenuse, the result is always a fraction less than 1.

📌 How sin A and cos A Work as Fractions

Let’s look at an example:

Example Triangle:

• Hypotenuse = 10 units
• Perpendicular = 6 units
• Base = 8 units

So,

• sin A = Perpendicular / Hypotenuse = 6/10 = 0.6
• cos A = Base / Hypotenuse = 8/10 = 0.8

Both values are less than 1.
This is always true because:

• The numerator (Perpendicular or Base) is smaller than the denominator (Hypotenuse).
• The only exception is if the side equals the hypotenuse — which only happens at a special limit like 90 degrees or 0 degrees.

📌 Special Cases When sin A or cos A = 1

There are certain angles where sin A or cos A equals 1:

• sin 90° = 1
• cos 0° = 1

Why?

• If the angle A is 90°, then the perpendicular equals the hypotenuse.
• If the angle A is 0°, then the base equals the hypotenuse.

But in a real right triangle, you never have both sides equal to the hypotenuse — so this only happens in special angle limits.

📌 Visual Example: Pythagoras Theorem

In a right triangle, the sides follow Pythagoras Theorem:

Hypotenuse² = Perpendicular² + Base²

Using the same example:

• Perpendicular = 6 units
• Base = 8 units

So,

Hypotenuse² = 6² + 8²
Hypotenuse² = 36 + 64
Hypotenuse² = 100
Hypotenuse = √100 = 10 units

So the hypotenuse is clearly the biggest side.

This confirms that when you divide by the hypotenuse, the fraction stays under 1.

📌 How This Helps in Real Questions

This simple fact helps you:

• Check your work: If sin A or cos A is ever more than 1, there’s a mistake.
• Know the limits:
  • Minimum value: 0
  • Maximum value: 1

📌 Everyday Example

Think of a ladder:

• The ladder is leaning against a wall, making a right triangle with the ground.
• The ladder is the hypotenuse.
• The height up the wall is the perpendicular.
• The distance along the ground is the base.

So,

• sin (angle) = height / ladder length (Perpendicular / Hypotenuse)
• cos (angle) = ground distance / ladder length (Base / Hypotenuse)

Since the ladder is the longest side, these ratios are always less than or equal to 1.

📌 Another Example with Steps

Suppose:

• Hypotenuse = 13 units
• Perpendicular = 5 units
• Base = 12 units

Let’s calculate sin A and cos A.

Step 1: Find sin A

sin A = Perpendicular / Hypotenuse
sin A = 5/13 ≈ 0.3846

Step 2: Find cos A

cos A = Base / Hypotenuse
cos A = 12/13 ≈ 0.9231

Again, both are less than 1.

📌 Why This Is Always True

• The hypotenuse is never smaller than the perpendicular or base.
• So, Perpendicular < Hypotenuse and Base < Hypotenuse.
• So, (Perpendicular / Hypotenuse) and (Base / Hypotenuse) are always between 0 and 1.

📌 Can tan A Be More Than 1?

Yes! Because tan A = Perpendicular / Base.
Both are legs — so either one can be larger than the other.
That’s why tan A can be greater than 1, but sin A and cos A cannot.

📌 Why It Matters in Exams

• If you ever find sin A = 1.3 — there’s a calculation error.
• This fact helps you check your answers.
• It’s super useful for questions on heights and distances.

📌 Another Example for Practice

Given:

• Hypotenuse = 17 units
• Perpendicular = 8 units
• Base = 15 units

Check:

• sin A = 8/17 ≈ 0.4705
• cos A = 15/17 ≈ 0.8823

Both values are less than 1.

📌 Easy Tips to Remember

• Hypotenuse = longest side
• sin A = Perpendicular / Hypotenuse
• cos A = Base / Hypotenuse
• Always between 0 and 1
• Equals 1 only at special angles

Final Takeaway

Whenever you work with sin A or cos A in right triangles, remember:
✅ They can never be more than 1.
✅ If you get a value above 1, double-check your calculation.
✅ This fact keeps your trigonometry answers accurate and reliable.

About the Author

Aman Sir is a dedicated Maths teacher and the founder of Maths Vidya Institute. With 14+ years of experience, he has guided thousands of CBSE students from Class 9 to Class 12, helping them build a strong foundation in Mathematics with simple and clear explanations.