Understanding sin(a + b) Formula in Trigonometry with Example

What is sin(a + b) Formula?

In trigonometry, sin(a + b) is a standard identity that helps us calculate the sine of the sum of two angles.

The formula is:

sin(a + b) = sin a · cos b + cos a · sin b

This formula is useful when we know the values of sine and cosine for two angles a and b, and we want to find sin(a + b).

Example of sin(a + b)

Let’s calculate:

Find sin(45° + 30°)

We know:

sin 45° = √2/2

cos 45° = √2/2

sin 30° = 1/2

cos 30° = √3/2

Using the formula:

sin(a + b) = sin a · cos b + cos a · sin b

Substitute the values:

sin(45° + 30°)

= sin 45° · cos 30° + cos 45° · sin 30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= (√6/4) + (√2/4)

= (√6 + √2)/4

Why is sin(a + b) Important?

Helps in simplifying trigonometric expressions

Useful in competitive exams like JEE, NEET, and board exams

Foundation for advanced topics like calculus and physics

sin(a + b) vs sin(a - b)

Also remember:

sin(a - b) = sin a · cos b - cos a · sin b

Just like sin(a + b), but with a minus sign between the terms.

Practice Questions on sin(a + b) Formula

Formula Reminder:

sin(a + b) = sin a · cos b + cos a · sin b

1. Find sin(30° + 45°)

Solution:

sin 30° = 1/2

cos 45° = √2/2

cos 30° = √3/2

sin 45° = √2/2

sin(30° + 45°) = sin 30° · cos 45° + cos 30° · sin 45°

= (1/2)(√2/2) + (√3/2)(√2/2)

= (√2/4) + (√6/4)

= (√2 + √6)/4

2. Find sin(60° + 30°)

sin 60° = √3/2

cos 30° = √3/2

cos 60° = 1/2

sin 30° = 1/2

sin(60° + 30°) = (√3/2)(√3/2) + (1/2)(1/2)

= (3/4) + (1/4) = 1

3. Find sin(45° + 45°)

sin 45° = √2/2

cos 45° = √2/2

sin(45° + 45°) = (√2/2)(√2/2) + (√2/2)(√2/2)

= (2/4) + (2/4) = 1

4. Find sin(90°) using sin(60° + 30°)

Already calculated above.

Answer: 1

5. Find sin(15° + 30°)

sin 15° = (√6 - √2)/4

cos 30° = √3/2

cos 15° = (√6 + √2)/4

sin 30° = 1/2

sin(15° + 30°) = sin 15° · cos 30° + cos 15° · sin 30°

= (√6 - √2)/4 + (√6 + √2)/4

= (√18 - √6)/8 + (√6 + √2)/8

= Simplified form = (√18 + √2)/8

6. Find sin(0° + 90°)

sin 0° = 0

cos 90° = 0

cos 0° = 1

sin 90° = 1

sin(0° + 90°) = 0·0 + 1·1 = 1

7. Find sin(60° + 45°)

sin 60° = √3/2

cos 45° = √2/2

cos 60° = 1/2

sin 45° = √2/2

sin(60° + 45°) = (√3/2)(√2/2) + (1/2)(√2/2)

= (√6/4) + (√2/4) = (√6 + √2)/4

8. Find sin(90° + 0°)

Same as Question 6

Answer: 1

9. Find sin(30° + 60°)

Already solved in Q2

Answer: 1

10. Find sin(0° + 45°)

sin 0° = 0

cos 45° = √2/2

cos 0° = 1

sin 45° = √2/2

sin(0° + 45°) = 0·(√2/2) + 1·(√2/2) = √2/2