Sina Sinb is an important formula in trigonometry that is used to simplify various problems in trigonometry. Sina Sinb formula can be derived using addition and subtraction formulas of the cosine function. It is used to find the product of the sine function for angles a and b. The result of sina sinb formula is given as (1/2)[cos(a – b) – cos(a + b)].
Let us understand the sin a sin b formula and its derivation in detail in the following sections along with its application in solving various mathematical problems.
1. What is Sina Sinb in Trigonometry? 2. Sina Sinb Formula 3. Proof of Sina Sinb Formula 4. How to Apply Sina Sinb Formula? 5. FAQs on Sina Sinb
Sina Sinb is the trigonometry identity for two different angles whose sum and difference are known. It is applied when either the two angles a and b are known or when the sum and difference of angles are known. It can be derived using angle sum and difference identities of the cosine function cos (a + b) and cos (a – b) trigonometry identities which are some of the important trigonometric identities.
Sina Sinb formula is used to determine the product of sine function for angles a and b separately. The sina sinb formula is half the difference of the cosines of the difference and sum of the angles a and b, that is, sina sinb = (1/2)[cos(a – b) – cos(a + b)].
The sina sinb product to difference formula in trigonometry for angles a and b is given as, sina sinb = (1/2)[cos(a – b) – cos(a + b)]. Here, a and b are angles, and (a + b) and (a – b) are their compound angles. Sina Sinb formula is used when either angles a and b are given or their sum and difference are given.
Now, that we know the sina sinb formula, we will now derive the formula using angle sum and difference identities of the cosine function. The trigonometric identities which we will use to derive the sin a sin b formula are:
- cos (a + b) = cos a cos b – sin a sin b – (1)
- cos (a – b) = cos a cos b + sin a sin b – (2)
Subtracting equation (1) from (2), we have
cos (a – b) – cos (a + b) = (cos a cos b + sin a sin b) – (cos a cos b – sin a sin b)
⇒ cos (a – b) – cos (a + b) = cos a cos b + sin a sin b – cos a cos b + sin a sin b
⇒ cos (a – b) – cos (a + b) = cos a cos b – cos a cos b + sin a sin b + sin a sin b
⇒ cos (a – b) – cos (a + b) = sin a sin b + sin a sin b [The term cos a cos b got cancelled because of opposite signs]
⇒ cos (a – b) – cos (a + b) = 2 sin a sin b
⇒ sin a sin b = (1/2)[cos (a – b) – cos (a + b)]
Hence the sina sinb formula has been derived.
Thus, sina sinb = (1/2)[cos(a – b) – cos(a + b)]
Next, we will understand the application of sina sinb formula in solving various problems since we have derived the formula. The sin a sin b identity can be used to solve simple trigonometric problems and complex integration problems. Let us go through some examples to understand the concept clearly and follow the steps given below to learn to apply sin a sin b identity:
Example 1: Express sin x sin 7x as a difference of the cosine function using sina sinb formula.
Step 1: We know that sin a sin b = (1/2)[cos(a – b) – cos(a + b)].
Identify a and b in the given expression. Here a = x, b = 7x. Using the above formula, we will proceed to the second step.
Step 2: Substitute the values of a and b in the formula.
sin x sin 7x = (1/2)[cos (x – 7x) – cos (x + 7x)]
⇒ sin x sin 7x = (1/2)[cos (-6x) – cos (8x)]
⇒ sin x sin 7x = (1/2) cos (6x) – (1/2) cos (8x) [Because cos(-a) = cos a]
Hence, sin x sin 7x can be expressed as (1/2) cos (6x) – (1/2) cos (8x) as a difference of the cosine function.
Example 2: Solve the integral ∫ sin 2x sin 5x dx.
To solve the integral ∫ sin 2x sin 5x dx, we will use the sin a sin b formula.
Step 1: We know that sin a sin b = (1/2)[cos(a – b) – cos(a + b)]
Identify a and b in the given expression. Here a = 2x, b = 5x. Using the above formula, we have
Step 2: Substitute the values of a and b in the formula and solve the integral.
sin 2x sin 5x = (1/2)[cos (2x – 5x) – cos (2x + 5x)]
⇒ sin 2x sin 5x = (1/2)[cos (-3x) – cos (7x)]
⇒ sin 2x sin 5x = (1/2)cos (3x) – (1/2)cos (7x) [Because cos(-a) = cos a]
Step 3: Now, substitute sin 2x sin 5x = (1/2)cos (3x) – (1/2)cos (7x) into the intergral ∫ sin 2x sin 5x dx. We will use the integral formula of the cosine function ∫ cos x = sin x + C
∫ sin 2x sin 5x dx = ∫ [(1/2)cos (3x) – (1/2)cos (7x)] dx
⇒ ∫ sin 2x sin 5x dx = (1/2) ∫ cos (3x) dx – (1/2) ∫ cos (7x) dx
⇒ ∫ sin 2x sin 5x dx = (1/2) [sin (3x)]/3 – (1/2) [sin (7x)]/7 + C
⇒ ∫ sin 2x sin 5x dx = (1/6) sin (3x) – (1/14) sin (7x) + C
Hence, the integral ∫ sin 2x sin 5x dx = (1/6) sin (3x) – (1/14) sin (7x) + C using the sin a sin b formula.
Important Notes on sina sinb Formula
- sin a sin b is applied when either the two angles a and b are known or when the sum and difference of angles are known.
- sin a sin b = (1/2)[cos(a – b) – cos(a + b)]
- It can be derived using angle sum and difference identities of the cosine function
Topics Related to sina sinb:
- cos a cos b
- cos 2pi
- cos (a – b)
