In this blog, let’s understand how to find all six trigonometric ratios when sin A = 1/2 using Triangle and Identity method

1^st Method : Triangle Method 

We are given:

sin A = 1/2

By definition,
sin A = Perpendicular / Hypotenuse

So, we can assume:

Perpendicular (P) = 1

Hypotenuse (H) = 2

We assume the simplest numbers possible to make calculations easy.

📌 Step 1: Find the Base

To find other ratios like cos A and tan A, we need to find the Base (B) of the triangle.

We know, by Pythagoras theorem:

H² = P² + B²

Substitute:

2² = 1² + B²

4 = 1 + B²

B² = 4 − 1 = 3

So, B = √3

So,

Base (B) = √3

📌 Step 2: Find cos A

The cosine of an angle is:

cos A = Base / Hypotenuse

Substitute the values:

cos A = √3 / 2

So,

cos A = √3 / 2

📌 Step 3: Find tan A

Tangent is:

tan A = Perpendicular / Base

So,

tan A = 1 / √3

We normally rationalize the denominator:

1 / √3 = √3 / 3

So,

tan A = √3 / 3

📌 Step 4: Find cot A

Cotangent is the reciprocal of tangent:

cot A = 1 / tan A

cot A = √3

So,

cot A = √3

📌 Step 5: Find sec A

Secant is the reciprocal of cosine:

sec A = 1 / cos A

sec A = 2 / √3

Again, rationalize:

2 / √3 = (2√3) / 3

So,

sec A = 2√3 / 3

📌 Step 6: Find cosec A

Cosecant is the reciprocal of sine:

cosec A = 1 / sin A

cosec A = 1 / (1/2)

cosec A = 2

So,

cosec A = 2

✅ All Six Ratios Summary

Let’s summarise all the ratios in one place:

• sin A = 1/2
• cos A = √3 / 2
• tan A = √3 / 3
• cot A = √3
• sec A = 2√3 / 3
• cosec A = 2

📌 Why These Steps Work

When one trigonometric ratio is given, it means you already know two sides of a right triangle. Using the Pythagoras theorem, you can always find the third side. Then you can write any other ratio using the basic definitions:

• sin = P/H
• cos = B/H
• tan = P/B
• cot = B/P
• sec = H/B
• cosec = H/P

These definitions are the backbone of trigonometry.

📌 Extra Tip: Always Rationalize

Whenever your answer has a square root in the denominator, rationalize it. This means multiply the numerator and denominator by the square root to make the denominator a whole number.

Example:

1 / √3 = √3 / 3

This makes your answers neater and easier to read.

2^nd Method : Trigonometric Identities Method 

When one ratio is given, you don’t always need to draw a triangle and use Pythagoras.
You can directly use the basic identity:

sin²A + cos²A = 1

✅ Step 1: Given

sin A = 1/2

So,

sin²A = (1/2)² = 1/4

✅ Step 2: Find cos A

From the identity:

sin²A + cos²A = 1

Substitute:

1/4 + cos²A = 1

cos²A = 1 − 1/4

cos²A = 3/4

So, cos A = √(3/4) = √3 / 2

We take the positive value because the question usually assumes acute angle A (0° < A < 90°).

✅ Step 3: Find tan A

By definition:

tan A = sin A / cos A

So,

tan A = (1/2) / (√3 / 2)

= 1 / √3

= √3 / 3   (after rationalizing)

✅ Step 4: Find cot A

cot A = 1 / tan A = √3

✅ Step 5: Find sec A

Another identity:

sec A = 1 / cos A

= 1 / (√3 / 2)

= 2 / √3

= 2√3 / 3  (rationalized)

✅ Step 6: Find cosec A

cosec A = 1 / sin A = 1 / (1/2) = 2

✅ Same Answer — Quicker Method

Using this identity method, you get the same answers as with the triangle method:

• sin A = 1/2
• cos A = √3 / 2
• tan A = √3 / 3
• cot A = √3
• sec A = 2√3 / 3
• cosec A = 2

📌 Conclusion

So, there are two ways to solve:

1️⃣ Triangle method → draw triangle, use Pythagoras
2️⃣ Identity method → use sin²A + cos²A = 1 and other formulas

Both give you the same answer.

Similar Question 

Example : If sin A = 1/3 , find all the Trigonometric Ratios.

Solution : Check the Answer in the Video given below 👇👇