Cos a Cos b is a trigonometric formula that isused in trigonometry. Cos a cos b formula is provided by, cos a cos b =(1/2).We use the cos a cos b formula to discover the worth of the product the cosine the two different angles. Cos a cos b formula have the right to be acquired from the cosine trigonometric identification for amount of angles and also difference that angles.

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The cos a cos b formula help in solving integration formulas and problems entailing the product the trigonometric ratio such as cosine. Let us recognize the cos a cos b formula and its derivation in information in the complying with sections.

1.What is Cos a Cos b in Trigonometry?
2.Derivation that Cos a Cos b Formula
3.How to use cos a cos b Formula?
4.FAQs on Cos a Cos b

What is Cos a Cos bin Trigonometry?


Cos a Cos b is the trigonometry identification for two various angles whose sum and also difference space known. It is applied when one of two people the 2 angles a and also b are recognized or as soon as the sum and also difference of angles room known. It have the right to be derived using cos (a + b) and also cos (a - b) trigonometry identities i m sorry are some of the important trigonometric identities. The cos a cos b identity is fifty percent the amount of the cosines of the sum and difference that the angles a and b, that is, cos a cos b = (1/2).

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Derivation that Cos a Cos b Formula


The formula for cos a cos b have the right to be acquired using the sum and also difference identities that the cosine function. Us will usage the complying with cosine identities to have the cos a cos b formula:

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

Adding equations (1) and (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) = cos a cos b + cos a cos b

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

⇒ cos a cos b = (1/2)

Hence the cos a cos b formula has been derived.

Thus, cos a cos b = (1/2)


How to use Cos a Cos b Formula?


Now that we recognize the cos a cos b formula, we will recognize its applications in solving miscellaneous problems. This identity deserve to be used to solve an easy trigonometric troubles and complicated integration problems. We deserve to follow the steps given listed below to discover to use cos a cos b identity. Let us go v some examples to recognize the concept clearly:

Example 1: Express cos 2x cos 5x together a sum of the cosine function.

Step 1: We know that cos a cos b = (1/2)

Identify a and b in the offered expression. Below a = 2x, b = 5x. Utilizing the above formula, we will procedure to the second step.

Step 2: Substitute the worths of a and b in the formula.

cos 2x cos 5x = (1/2)

⇒ cos 2x cos 5x = (1/2)

⇒ cos 2x cos 5x = (1/2)cos (7x) + (1/2)cos (3x)

Hence, cos 2x cos 5x can be expressed as (1/2)cos (7x) + (1/2)cos (3x) together a sum of the cosine function.

Example 2: Solve the integral ∫ cos x cos 3x dx.

To settle the integral ∫ cos x cos 3x dx, we will use the cos a cos b formula.

Step 1: We know that cos a cos b = (1/2)

Identify a and also b in the offered expression. Right here a = x, b = 3x. Using the over formula, we have

Step 2: Substitute the values of a and b in the formula and solve the integral.

cos x cos 3x = (1/2)

⇒ cos x cos 3x = (1/2)

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

Step 3: Now, instead of cos x cos 3x = (1/2)cos (4x) + (1/2)cos (x) into the intergral ∫ cos x cos 3x dx. We will use the integral formula that the cosine role ∫ cos x dx = sin x + C

∫ cos x cos 3x dx = ∫ <(1/2)cos (4x) + (1/2)cos (x)> dx

⇒ ∫ cos x cos 3x dx = (1/2) ∫ cos (4x) dx + (1/2) ∫ cos (x) dx

⇒ ∫ cos x cos 3x dx = (1/2) /4 + (1/2) sin (x) + C

⇒ ∫ cos x cos 3x dx = (1/8) sin (4x) + (1/2) sin (x) + C

Hence, the integral ∫ cos x cos 3x dx = (1/8) sin (4x) + (1/2) sin (x) + C utilizing the cos a cos b formula.

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Important notes on cos a cos b

cos a cos b = (1/2)It is used when either the two angles a and also b are known or when the sum and difference the angles space known.The cos a cos b formula helps in resolving integration formulas and problems including the product of trigonometric proportion such as cosine

Related object on cos a cos b


Example 2: Solve the integral ∫ cos 2x cos 4x dx using cos a cos b identity.

Solution: We recognize that cos a cos b = (1/2)

Identify a and also b in the given expression. Below a = 2x, b = 4x. Making use of the above formula, we have

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

⇒ cos 2x cos 4x = (1/2)

⇒ cos 2x cos 4x = (1/2)cos (6x) + (1/2)cos (2x)

Now, instead of cos 2x cos 4x = (1/2)cos (6x) + (1/2)cos (2x) right into the intergral ∫ cos 2x cos 4x dx. Us will usage the integral formula that the cosine duty ∫ cos x dx = sin x + C

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

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