Sin A minus Sin B is an important trigonometric formula. Sin A – Sin B formula, for two angles A and B, is given as, Sin A – Sin B = 2 cos (A + B)/2 sin (A – B)/2. This formula is also called the difference to product formula for sine.

In this article, we will learn about, Sin A – Sin B identity, Sin A – Sin B Formula, Proof of Sin A – Sin B Formula, related examples and others in detail.

Sin A – Sin B Identity

Trigonometric identity Sin A – Sin B represents the difference of sine of angles A and B, Sin A – Sin B is also called the difference to product formula of sine.

Sin A – Sin B Difference to Product Formula

Difference to the product formula of sine is defined as the difference of sine function found using the product of two trigonometric functions. This is added in the form of the image added below:

Sin A – Sin B Formula

Trigonometric identity Sin A – Sin B represents the difference of sine of angles A and B. Sin A – Sin B formula can be applied to represent the difference of sine of angles A and B in the product form of sine of (A – B) and cosine of (A + B),

sin(A) – sin(B) = 2cos{(A + B)/2}.sin{(A – B/2}

Proof of Sin A – Sin B Formula

Proof of Sin A – Sin B is explained using the expansion of sin(A + B) and sin(A – B) formula. Let take angles X and Y, such that X = A + B and Y = A – B.

Now we know that,

• sin(A + B) = sin A cos B + cos A sin B…(i)
• sin(A – B) = sin A cos B – cos A sin B…(ii)

Eq(i) – Eq(ii)

sin(A + B) – sin(A – B) = 2 sin B cos A…(iii)

Now, X = A + B and Y = A – B

X + Y = 2A

∴ A = (X + Y)/2

X – Y = 2B

∴ B = (X – Y)/2

Replace A and B in eq(iii)

∴ sin X – sin Y = 2cos((A+B)/2)sin((A-B)/2)

We can also say that,

sin A – sin B = 2 sin ½ (A – B) cos ½ (A + B)

Hence, proved.

How to Apply Sin A – Sin B?

Sin A – Sin B formula is used to solve various trigonometric problems, this is explained by the example added below:

Simplify: Sin 60° – Sin 30°

= Sin 60° – Sin 30°

= 2 cos {60° + 30°}/2.sin{60° – 30°}/2

= 2 cos 45°.sin 15°

= 2 (1/√2) ((√3 – 1)/2√2)

= (√3 – 1)/2

Thus, Sin 60° – Sin 30° = (√3/2 – 1/2) = (√3 – 1)/2