Sin a Cos b Formula, Proof, Examples

The sin a cos b formula is an essential trigonometric identity that is implemented to solve many tricky problems. Trigonometric identities hold for all values of variables on either side of an equation. The major trigonometry functions are sine, cosine, and tangent, while the other three functions are cotangent, secant, and cosecant. here Sin a cos b is used to calculate the product of the sine and cosine functions of angles a and b.

Sin a Cos b Formula

There are numerous trigonometric identities relating to the side length and angle of a triangle. Only the right-angle triangle has the trigonometric identities. The sin a cos b formula is expressed as [sin(a+b) + sin(a-b)]/2. It is derived from the sine function’s angle sum and angle difference identities. In this post, we will look at the sin a cos b formula, its derivation, how to use it to solve various trigonometric problems, the significance of the formula of sin a Cos b, and solved examples.

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What is Sin a Cos b Identity?

Trigonometric Identities: Trigonometric Identities represent equalities that utilize trigonometry functions and hold true for all variables in the equation. In terms of geometry, these identities include one or more trigonometric functions (such as sine, cosine, and tangent).

• The six trigonometric ratios serve as the foundation for all trigonometric identities. Sine, cosine, tangent, cosecant, secant, and cotangent are the functions. Sin a cos b is a trigonometric identity that can be used to solve a variety of trigonometric problems.
• All of these trigonometric ratios can be defined using the sides of a right triangle, specifically the adjacent, opposite, and hypotenuse sides.
• Now, Sin a cos b equals half the sum of the sine of the sum of angles a and b and the sine of variance of angles a and b. It is expressed mathematically as sin a cos b = (1/2)[sin(a + b) + sin(a – b)], which can be determined by applying the trigonometric values sin (a + b) and sin(a – b).
• Whenever the sum and difference of angles a and b are established, or when two angles a and b are recognized, the sin a cos b formula can be used.

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Sin a Cos b ka Formula

The sin a cos b formula can be utilized in situations when the parameters of angles a and b, as well as the combined angles (a + b) and (a – b), are known.

Sin a cos b is defined as –

sin a cos b = (1/2)[sin(a + b) + sin(a – b)].

Derivation of Sin a Cos b Formula

Since we already know the formula for sin a cos b,
sin a cos b = (1/2)[sin(a + b) + sin(a – b)],
we can use the trigonometric identities and formulae to get this value.

• The sine function’s angle sum and angle difference formulas can be used to construct the sin a cos b formula. The following trigonometric formulas will be used:

sin (a + b) = sin a cos b + cos a sin b — (i)
sin (a – b) = sin a cos b – cos a sin b — (ii)

• By combining mathematical formulas (i) and (ii), we get
  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 + 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) [sin (a + b) + sin (a – b)]

As a result, we produced the sin a cos b formula by combining the sin (a + b) with sin (a – b) identities.

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Significance of Sin a Cos b Identity

The sin a Cos b formula is – sin a cos b = (1/2)[sin(a+b) + sin(a-b)]. Let’s discuss some key uses of sin a Cos b identity here.

1. To solve simple and difficult trigonometric problems, the sin a cos b formula is utilized.
2. When angles a and b have been determined, or when the sum and difference of angles a and b are known, the sin a cos b formula is used.
3. Sin a cos b is equal to half the sum of the sines of angles a and b and the sines of angles a and b difference.

Solving Problem Using Sin a Cos b Formula

The following solved example demonstrates how to apply the sin a Cos b formula to every trigonometric problem.

Sin a Cos b Formula Examples

Example 1: Applying the sin a cos b formula, describe the trigonometric function sin 3x cos 9x as a sum of the sine function.

Answer: The sin a cos b formula will be used: sin a cos b = (1/2) [sin (a + b) + sin (a – b)]. Determine the a and b values in the formula. Sin 3x cos 9x, where a = 3x and b = 9x.

Substitute a and b values in the calculation sin a cos b = (1/2). [a + b sin + a – b sin]

sin 3x cos 9x = (1/2) [sin (3x + 9x) + sin (3x – 9x)]
⇒ sin 3x cos 9x = (1/2) [sin (12x) + sin (-6x)]
⇒ sin 3x cos 9x = (1/2) sin (12x) – (1/2) sin (6x) [Due to the fact that sin(-a) = -sin a]

As a result, using the sin a cos b formula, we may express sin 3x cos 9x as (1/2) sin (12x) – (1/2) sin (6x) as a sum of sine functions.

Solution: sin 3x cos 9x = (1/2) (12x) sin – (1/2) sin (6x)

More examples will be updated later.

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