Sin A + Sin B

Sin A + Sin B, an important identity in trigonometry, is used to find the sum of values of sine function for angles A and B. It is one of the sum to product formulas used to represent the sum of sine function for angles A and B into their product form. The result for sin A + sin B is given as 2 sin ½ (A + B) cos ½ (A - B).

Let us understand the sin A + sin B formula and its proof in detail using solved examples.

The trigonometric identity sinA + sinB is used to represent the sum of sine of angles A and B, sin A + sin 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] cWe will study the sin A + sin B formula in detail in the following sections.

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

Sin A + Sin B = 2 sin [½ (A + B)] cos [½ (A - B)]

Here, A and B are angles, and (A + B) and (A - B) are their compound angles.

We can give the proof of sin A + sin B formula (sin A + sin B = 2 sin ½ (A + B) cos ½ (A - B)) using the expansion of sin(A + B) and sin(A - B) formula. We know, using trigonometric identities, ½ [sin(α + β) + sin(α - β)] = sin α cos β, for any angles α and β. From this,

[sin(α + β) + sin(α - β)] = 2 sin α cos β ... (1)

Let us assume that (α + β) = A and (α - β) = B.

⇒ 2α = A + B
⇒ α = (A + B)/2

⇒ 2β = A - B
⇒ β = (A - B)/2

Substituting all these values in (1)

⇒ sinA + sinB = 2 sin ½(A + B) cos ½(A - B)

Hence, proved.

We can apply the sin A + sin B formula as a sum to the product identity. Let us understand its application using an example of sin 60º + sin 30º. We will solve the value of the given expression by 2 methods, using the formula and by directly applying the values, and compare the results. Have a look at the below-given steps.

• Compare the angles A and B with the given expression, sin 60º + sin 30º. Here, A = 60º, B = 30º.
• Solving using the expansion of the formula sin A + sin B, given as, sin A + sin B = 2 sin ½ (A + B) cos ½ (A - B), we get,
  Sin 60º + Sin 30º = 2 sin ½ (60º + 30º) cos ½ (60º - 30º) = 2 sin 45º cos 15º = 2 (1/√2) ((√3 + 1)/2√2) = (√3 + 1)/2.
• Also, we know that sin 60º + sin 30º = (√3/2 + 1/2) = (√3 + 1)/2 (from trig table).

Hence, the result is verified.

☛ Related Topics:

• Trigonometric Chart
• Trigonometric Functions
• sin cos tan
• Law of Sines

Let us have a look at a few examples to understand the concept of sin A + sin B better.

What is the Value of Sin A Plus Sin B?

Sin A plus Sin B is an identity or trigonometric formula, used in representing the sum of sine of angles A and B, Sin A + Sin B in the product form using the compound angles (A + B) and (A - B). Here, A and B are angles.

What is the Formula of SinA + SinB?

SinA + SinB formula, for two angles A and B, can be given as sinA + sinB = 2 sin ½ (A + B) cos ½ (A - B). Here, (A + B) and (A - B) are compound angles.

What is the Product Form of Sin A + Sin B in Trigonometry?

The product form of sin A + sin b formula is given as, sin A + sin B = 2 sin ½ (A + B) cos ½ (A - B), where A and B are any given angles.

How to Prove the Expansion of SinA + SinB Formula?

The expansion of sin A + sin B, given as sinA + sinB = 2 sin ½ (A + B) cos ½ (A - B), can be proved using the 2 sin α cos β product identity in trigonometry. Click here to check the detailed proof of the formula.

How to Use Sin A + Sin B Formula?

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 given expression and substitute the values of angles A and B.

What is the Application of SinA + SinB Formula?

SinA + SinB 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), using the formula, sin A + sin B = 2 sin ½ (A + B) cos ½ (A - B).