Sin A Sin B Adalah

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.

1. What is Sin A + Sin B Identity in Trigonometry?
2. Sin A + Sin B Sum to Product Formula
3. Proof of Sin A + Sin B Formula
4. How to Apply Sin A + Sin B?
5. FAQs on Sin A + Sin B

What is SinA + SinB Identity in Trigonometry?

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.

Sin A + Sin B Sum to Product Formula

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.

expansion of sin A + sin B formula

Proof of SinA + SinB Formula

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.

How to Apply Sin A + Sin B?

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.

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FAQs on Sin A + Sin B

What is the Value of Sin A Berlebih 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).

Source: https://www.cuemath.com/trigonometry/sin-a-plus-sin-b/