Sin a Cos b
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)[sin(a+b) + sin(a-b)].
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)[sin(a + b) + sin(a - b)], 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)[sin(a + b) + sin(a - b)]. 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)[sin(a + b) + sin(a - b)], 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) [sin (a + b) + sin (a - b)]
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) [sin (a + b) + sin (a - b)]. 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 (a + b) + sin (a - b)]
sin 7x cos 3x = (1/2) [sin (7x + 3x) + sin (7x - 3x)]
⇒ sin 7x cos 3x = (1/2) [sin (10x) + sin (4x)]
⇒ 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) [sin (a + b) + sin (a - b)]. 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 (a + b) + sin (a - b)]
sin 2x cos 4x = (1/2) [sin (2x + 4x) + sin (2x - 4x)]
⇒ sin 2x cos 4x = (1/2) [sin (6x) + sin (-2x)]
⇒ sin 2x cos 4x = (1/2) sin (6x) - (1/2) sin (2x) [Because sin(-a) = -sin a]
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+b) + sin(a-b)]
- 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
FAQs on Sin a Cos b
What is Sin a Cos b in Trigonometry?
Sin a cos b is an important trigonometric identity that is used to solve complicated problems in trigonometry given by sin a cos b = (1/2) [sin (a + b) + sin (a - b)]
What is the Formula of Sin a Cos b?
The formula of sin a cos b is sin a cos b = (1/2) [sin (a + b) + sin (a - b)]
What is the Formula of 2 sin a cos b?
The formula for 2 sin a cos b is given by, 2 sin a cos b = sin (a + b) + sin (a - b)
Find the Exact Value of sin a cos b when a = 90° and b = 180°.
Substitute a = 90° and b = 180° in sin a cos b = (1/2) [sin (a + b) + sin (a - b)]. sin 90° cos 180° = (1/2) [sin (90° + 180°) + sin (90° - 180°)] = (1/2) [sin 270° + sin(-90°)] = (1/2)(-1-1) = -1. Hence, sin a cos b = -1 when a = 90° and b = 180°
How to Find sin a cos b formula?
Sin a Cos b formula can be calculated using sin(a + b) and sin (a - b) trigonometric identities.
When is sin a cos b equal to (1/2) sin 2a?
sin a cos b is equal to (1/2) sin 2a when a = b. When a = b in sin a cos b = (1/2) [sin (a + b) + sin (a - b)], we have sin a cos b = (1/2) [sin (a + a) + sin (a - a)] = (1/2) [sin 2a + 0] = (1/2) sin 2a.
How to Prove sin a cos b Identity?
Sin a cos b formula can be proved using the angle sum and angle difference formulas of the sine function.
What is the Expansion of Sin a Cos b?
The expansion of sin a cos b is given by sin a cos b = (1/2) [sin (a + b) + sin (a - b)].
What is the Difference Between Sin a Cos b Formula and Cos a Sin b Formula?
Sin a cos b formula is the sum of sin (a + b) and sin (a - b) trigonometric identities, whereas cos a sin b formula is the difference of sin (a + b) and sin (a - b) trigonometric identities, that is, sin a cos b = (1/2) [sin (a + b) + sin (a - b)] and cos a sin b = (1/2) [sin (a + b) - sin (a - b)].