Sin a cos b is an important trigonometric identity that is used to solve complicated problems in trigonometry. Sin a cos b is used to obtain the product of the sine function of angle a and cosine function of angle b. It can be obtained from angle sum and angle difference identities of the sine function. Sin a cos b formula is written as (1/2).

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In this article, we will explore the sin a cos b formula, its proof, and learn its application to solve various trigonometric problems with the help of solved examples.

 1 What is Sin a Cos b Identity? 2 Proof of Sin a Cos b Formula 3 Application of Sin a Cos b Identity 4 FAQs on Sin a Cos b

## What is Sin a Cos b Identity?

Sin a cos b is a trigonometric identity used to solve various problems in trigonometry. Sin a cos b is equal to half the sum of sine of the sum of angles a and b, and sine of difference of angles a and b. Mathematically, it is written as sin a cos b = (1/2), that is, it can be derived using the trigonometric identities sin (a + b) and sin(a - b). Sin a cos b formula can be applied when the sum and difference of angles a and b are known, or when two angles a and b are known.

### Sin a Cos b Formula

The formula for sin a cos b is given by, sin a cos b = (1/2). The formula for sin a cos b can be applied when the compound angles (a + b) and (a - b) are known, or when values of angles a and b are known. ## Proof of Sin a Cos b Formula

Now that we know the formula of sin a cos b, which is sin a cos b = (1/2), we will derive this formula using the trigonometric formulas and identities. Sin a cos b formula can be derived using the angle sum and angle difference formulas of the sine function. We will use the following trigonometric formulas:

sin (a + b) = sin a cos b + cos a sin b --- (1)sin (a - b) = sin a cos b - cos a sin b --- (2)

Adding equations (1) and (2), we have

sin (a + b) + sin (a - b) = (sin a cos b + cos a sin b) + (sin a cos b - cos a sin b) (From (1) and (2))

⇒ sin (a + b) + sin (a - b) = sin a cos b + cos a sin b + sin a cos b - cos a sin b

⇒ sin (a + b) + sin (a - b) = (sin a cos b + sin a cos b) + (cos a sin b - cos a sin b)

⇒ sin (a + b) + sin (a - b) = 2 sin a cos b + 0

⇒ sin (a + b) + sin (a - b) = 2 sin a cos b

⇒ sin a cos b = (1/2)

Hence, we have obtained the sin a cos b formula using the sin (a + b) and sin (a - b) identities.

## Application of Sin a Cos b Identity

Since we have derived the sin a cos b formula, now we will learn how to apply the formula to solve simple trigonometric and integration problems. We will consider some examples based on sin a cos b identity and solve them step-wise. Let us understand the application of the sin a cos b formula by following the given steps:

Example 1: Express the trigonometric function sin 7x cos 3x as a sum of the sine function.

Step 1: We will use the sin a cos b formula: sin a cos b = (1/2) . Identify the values of a and b in the formula. We have sin 7x cos 3x, here a = 7x, b = 3x.

Step 2: Substitute the values of a and b in the formula sin a cos b = (1/2)

sin 7x cos 3x = (1/2)

⇒ sin 7x cos 3x = (1/2)

⇒ sin 7x cos 3x = (1/2) sin (10x) + (1/2) sin (4x)

Hence, we can write sin 7x cos 3x as (1/2) sin (10x) + (1/2) sin (4x) as a sum of sine function.

Example 2: Evaluate the integral ∫sin 2x cos 4x dx using the sin a cos b formula.

Step 1: First, we will express sin 2x cos 4x as a sum of sine function using the formula sin a cos b = sin a cos b = (1/2) . Identify a and b in sin 2x cos 4x. We have a = 2x, b = 4x.

Step 2: Substitute the values of a and b in the formula sin a cos b = (1/2)

sin 2x cos 4x = (1/2)

⇒ sin 2x cos 4x = (1/2)

⇒ sin 2x cos 4x = (1/2) sin (6x) - (1/2) sin (2x)

Step 3: Substitute sin 2x cos 4x = (1/2) sin (6x) - (1/2) sin (2x) into the integral ∫sin 2x cos 4x dx.

∫sin 2x cos 4x dx = ∫ <(1/2) sin (6x) - (1/2) sin (2x)> dx

⇒ ∫sin 2x cos 4x dx = (1/2) ∫sin(6x) dx - (1/2) ∫sin(2x) dx

⇒ ∫sin 2x cos 4x dx = (1/2)<-cos(6x)>/6 - (1/2)<-cos(2x)>/2 + C

⇒ ∫sin 2x cos 4x dx = (-1/12) cos (6x) + (1/4) cos (2x) + C

Hence, we have solved the integral ∫sin 2x cos 4x dx using sin a cos b formula and is equal to (-1/12) cos (6x) + (1/4) cos (2x) + C.

Important Notes on Sin a Cos b

sin a cos b = (1/2)sin a cos b formula is applied when angles a and b are known, or when the sum and difference of angles a and b are known.sin a cos b formula is used to solve simple and complex trigonometric problems.Sin a cos b is equal to half the sum of sine of the sum of angles a and b, and sine of difference of angles a and b.

Related Topics on Sin a Cos b

Example 1: Evaluate the integral ∫sin 12x cos 3x dx using the sin a cos b formula.

Solution: First, we will express sin 12x cos 3x as a sum of sine function using the formula sin a cos b = sin a cos b = (1/2) . Identify a and b in sin 12x cos 3x. We have a = 12x, b = 3x.

Substitute the values of a and b in the formula sin a cos b = (1/2)

sin 12x cos 3x = (1/2)

⇒ sin 12x cos 3x = (1/2)

⇒ sin 12x cos 3x = (1/2) sin (15x) + (1/2) sin (9x)

Substitute sin 12x cos 3x = (1/2) sin (15x) + (1/2) sin (9x) into the integral ∫sin 12x cos 3x dx.

∫sin 12x cos 3x dx = ∫ <(1/2) sin (15x) + (1/2) sin (9x)> dx

⇒ ∫sin 12x cos 3x dx = (1/2) ∫sin(15x) dx + (1/2) ∫sin(9x) dx

⇒ ∫sin 12x cos 3x dx = (1/2)<-cos(15x)>/15 + (1/2)<-cos(9x)>/9 + C

⇒ ∫sin 12x cos 3x dx = (-1/30) cos (15x) + (-1/18) cos (9x) + C

Hence, we have solved the integral ∫sin 12x cos 3x dx using sin a cos b formula and is equal to (-1/30) cos (15x) + (-1/18) cos (9x) + C.

Answer: ∫sin 12x cos 3x dx = (-1/30) cos (15x) + (-1/18) cos (9x) + C

Example 2: Express the trigonometric function sin 3x cos 9x as a sum of the sine function using sin a cos b formula.

Solution: We will use the sin a cos b formula: sin a cos b = (1/2) . Identify the values of a and b in the formula. We have sin 3x cos 9x, here a = 3x, b = 9x.

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Substitute the values of a and b in the formula sin a cos b = (1/2)

sin 3x cos 9x = (1/2)

⇒ sin 3x cos 9x = (1/2)

⇒ sin 3x cos 9x = (1/2) sin (12x) - (1/2) sin (6x)

Hence, we can write sin 3x cos 9x as (1/2) sin (12x) - (1/2) sin (6x) as a sum of sine function using the sin a cos b formula.