Sin A minus Sin B

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

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

Examples Using Sin A – Sin B

Example 1: Find the value of sin 145° – sin 35° using sin A + sin B identity.

Solution:

We know,

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

Here, A = 145°, B = 35°

sin 145° – sin 35° = 2 cos ½ (145° + 35°) sin ½ (65° – 35°)

= 2 cos 90° sin 15°

= 0 [∵cos 90º = 0]

Example 2: Verify the given expression using expansion of Sin A – Sin B: sin 70° – cos 70° = √2 sin 25°

Solution:

L.H.S. = sin 70° – cos 70°

Since, cos 70° = cos(90° – 20°) = sin 20°

⇒ sin 70° – cos 70° = sin 70° – sin 20°

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

⇒ sin 70° – sin 20° = 2 cos ½ (70° + 20°) sin ½ (70° – 20°)

= 2 cos 45° sin 25°

= √2 sin 25°

= R.H.S.

Hence, verified.

Practice Problem on Sin A – Sin B Formula

Problem 1: Simplify sin (45) – sin(15)

Problem 2: Simplify sin(2x) – sin(x)

Problem 3: Simplify sin(t) – sin(1/t)

Problem 4: Simplify sin(36) – sin(18)

FAQs on Sin A – Sin B

What is Sin(A) – Sin(B) in Trigonometry?

Sin(A) – Sin(B) is a mathematical expression representing the difference between the sine of angle A and the sine of angle B.

What is the Formula of Sin A – Sin B?

Sin A – Sin B formula, is given as: Sin A – Sin B = 2 cos ½ (A + B) sin ½ (A – B).

Can Sin(A) – Sin(B) be negative?

Yes, Sin(A) – Sin(B) can be negative if the sine (A) is smaller than the sine (B).

What is the application of Sin A – Sin B formula?

Sin A – Sin B formula is the used for solving various mathematical formulas.

What is sin A minus B?

sin(a – b) formula is given by: sin (a – b) = sin a cos b – cos a sin b