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

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The cos a cos b formula helps in resolving integration formulas and also problems entailing the product of trigonometric ratio such as cosine. Permit us understand the cos a cos b formula and its source in detail in the adhering to sections.

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

What is Cos a Cos bin Trigonometry?


Cos a Cos b is the trigonometry identification for two different angles who sum and also difference space known. The is used when either the 2 angles a and also b are known or when the sum and difference the angles space known. It deserve to be obtained using cos (a + b) and cos (a - b) trigonometry identities i beg your pardon are several of the essential trigonometric identities. The cos a cos b identity is fifty percent the sum of the cosines of the sum and difference the the angle a and also b, the is, cos a cos b = (1/2).

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Derivation that Cos a Cos b Formula


The formula because that cos a cos b can be acquired using the sum and also difference identities of the cosine function. We will use the following cosine identities to derive 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 also (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 actually been derived.

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


How to use Cos a Cos b Formula?


Now the we recognize the cos a cos b formula, we will understand its application in solving various problems. This identity can be provided to solve straightforward trigonometric troubles and facility integration problems. We deserve to follow the actions given below to find out to apply cos a cos b identity. Let us go v some instances to recognize the ide clearly:

Example 1: Express cos 2x cos 5x as 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 over formula, us will procedure to the 2nd step.

Step 2: Substitute the values of a and also 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 amount of the cosine function.

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

To resolve the integral ∫ cos x cos 3x dx, we will usage the cos a cos b formula.

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

Identify a and b in the provided expression. Below a = x, b = 3x. Making use of the over formula, us have

Step 2: Substitute the worths of a and also 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) right into the intergral ∫ cos x cos 3x dx. Us will usage the integral formula of 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 once the sum and also difference the angles are known.The cos a cos b formula helps in addressing integration formulas and problems including the product of trigonometric ratio such as cosine

Related object on cos a cos b


Example 2: Solve the integral ∫ cos 2x cos 4x dx making use of cos a cos b identity.

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

Identify a and b in the given expression. Here a = 2x, b = 4x. Making use of the above formula, us 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. We will usage the integral formula of the cosine function ∫ 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