Sin A + Sin B Formula

Sin A + Sin B Formula is a very significant formula in trigonometry, enabling the calculation of the sum of sine values for angles A and B. Sin A + Sin B Formula provides a way to express the sum of two sine functions in terms of the product of sine and cosine functions. It is given as:

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

This formula is used in various problems in both theoretical and practical trigonometry. It is also referred to as the Sum to Product Formula for sine. In this article, we will discuss the formula, its derivation, and some solved examples as well.

Trigonometry Identities

Trigonometric identities are equations involving trigonometric functions that are true for all possible values of the variables within their domains.

These identities are fundamental in trigonometry and are used to simplify expressions, solve equations, and establish relationships between different trigonometric functions.

Read More: Trigonometry Identities.

Sin A + Sin B Formula

Trigonometric identity i.e. Sin A + Sin B represents the sum of sine of angles A and B. Sin A + Sin B formula can be applied to represent the sum of sine of angles A and B in the product form of sine of (A + B) and cosine of (A – B) . Given below is the Sin A + Sin B Formula :

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

Some Similar Formulas

Some similar formulas to Sin A + Sin B Formula are:

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

Read More about Sum to Product Formula.

Sin A + Sin B Formula Proof

Proof of Sin A + Sin B can be explained very easily using the expansion of simple trigonometric identities that is sin(A + B) and sin(A – B) formula:

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

Adding equation (i) and (ii)

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

Assume angles X and Y, such that X = A + B and Y = A – B.

Solving X = A + B and Y = A – B , we found

A = (X + Y)/2 and B = (X – Y)/2

Replace A and B in eq (iii)

⇒ sin X + sin Y = 2sin{(X + Y)/2} .cos {(X – Y)/2}

or we can say Sin A + Sin B = 2 {sin(A + B)/2 }.cos {(A – B)/2}

Hence proved.

How to Apply Sin A + Sin B Formula?

Sin A + Sin B formula is used to solve various trigonometric problems, to apply the formula we can use the following steps:

Step 1: Determine the values of A and B in the expression sin A + sin B.

Step 2: Find the average (A + B)/2 and the difference (A – B)/2.

Step 3: Substitute the values in formula.

Step 4: Simplify

Let’s consider an example for the same.

Example: Simplify Sin 60° + Sin 30°.

Solution:

For Sin 60° + Sin 30°

A = 60° and B = 30°

• (A + B)/2 = (60 + 30)/2 = 45°, and
• (A – B)/2 = (60 – 30)/2 = 15°, and

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

= 2 sin 45°.cos 15°

= 2 (1/√2) ((√3 + 1)/2√2)

= (√3 + 1)/2

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

Sin A + Sin B + Sin C Formula

The sin A + sin B + sin C is a sum to product formula in trigonometry for three angles A ,B , and C given as,

Sin A + Sin B + Sin C= 4cos(A/2)cos(B/2)cos(C/2)

Let’s consider an example for the same.

Example : Simplify: sin 60° + sin 90° + sin 120°.

Solution:

A = 60°, B = 90°, and C = 120°

Sin 60° + sin 90° + sin 120°

= 4 cos(60/2)cos(90/2)cos(120/2)

= 4cos 30°.cos 45°.cos 60°

= 4.(√3/2).(√2/2).(1/2) = √6/2

Thus, Sin 60° + sin 90° + sin 120° = √6/2.

Conclusion: Sin A + Sin B Formula

In conclusion, Sin A + Sin B Formula gives the relationship between sum of two different values of sin and convert it into product of sin and cos with operations of the input angles. Formula Sin A + Sin B = 2 {sin(A + B)/2 }.cos {(A – B)/2} is used to solve various problems.

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Solved Examples on Sin A + Sin B Formula

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

Solution:

We know, Sin A + Sin B = 2 sin½ (A + B) cos ½ (A – B)

Here, A = 145°, B = 35°

sin 145° + sin 35° = 2 sin½ (145° + 35°) cos ½ (65° – 35°)

⇒ sin 145° + sin 35° = 2 sin 90° cos 15°

⇒ sin 145° + sin 35° = 2 x 1 x ((√3 + 1)/2√2)

⇒ sin 145° + sin 35° =((√3 + 1)/√2)

Example 2: Verify the given expression using expansion of Sin A + Sin B: sin 70° + cos 70° = √2 cos 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 sin½ (A + B) cos ½ (A – B)

L.H.S. = sin 70° + sin 20° = 2 sin½ (70° + 20°) cos ½ (70° – 20°)

⇒ L.H.S. = 2 sin 45° cos 25°

⇒ L.H.S. = 2.(1/√2).cos 25° = √2 cos 25°

⇒ L.H.S. = R.H.S.

Hence, verified.

Practice Problems on Sin A + Sin B Formula

Problem 1: Find the value of sin(2π/6) + sin(3π/3).

Problem 2: Determine the sum of sin(θ) + sin(2θ).

Problem 3: Calculate the value of sin(2α) and sin(3α).

Problem 4: Find the sum of sin(45°) + sin(120°).

Problem 5: Evaluate sin(π/4) + sin(π/6).

FAQs on Sin A + Sin B Formula

Define Sine Function.

Sine function, denoted as sin, is a fundamental trigonometric function that relates the ratio of the length of the side opposite to an angle in a right triangle to the length of the hypotenuse.

What is Sin A + Sin B Formula in Trigonometry?

The trigonometric identity sin A + sin B is used to represent the sum of sine of angles A and B in the product form using the compound angles (A + B) and (A – B). It says sin A + sin B = 2 sin [(A + B)/2] cos [(A – B)/2].

How to use sin A + sin B identity in a given expression?

To use sin A + sin B identity in a given expression, compare the sin A + sin B formula, sin A + sin B = 2 sin ½ (A + B) cos ½ (A – B), with the given expression and substitute the values of angles A and B.

Why it is Called Sum to Product Formula?

Sum-to-Product formula in trigonometry is called so because it involves converting the sum or subtraction of two trigonometric functions into a product.

How can the Sin A + Sin B formula be proved?

Proof of the sin A + sin B formula [sin A + sin B = 2 sin {(A + B)/2} cos {(A – B)/2}] can be demonstrated using the expansion of sin (A + B) and sin (A – B) formulas.