Cos a Cos b is a trigonometric formula the isused in trigonometry. Cos a cos b formula is given by, cos a cos b =(1/2).We usage the cos a cos b formula to discover the worth of the product of cosine of two different angles. Cos a cos b formula can be acquired from the cosine trigonometric identification for sum of angles and difference that angles.

You are watching: Cos(a) + cos(b)

The cos a cos b formula helps in fixing integration formulas and problems involving the product the trigonometric proportion such together cosine. Allow us know the cos a cos b formula and also its derivation in detail in the following sections.

1.What is Cos a Cos b in Trigonometry?
2.Derivation that Cos a Cos b Formula
3.How to apply 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 are known. The is used when one of two people the two angles a and also b are recognized or once the sum and difference the angles are known. It have the right to be obtained using cos (a + b) and also cos (a - b) trigonometry identities i m sorry are some of the vital trigonometric identities. The cos a cos b identification is half the sum of the cosines the the sum and difference that the angle a and b, the is, cos a cos b = (1/2).

*


Derivation that Cos a Cos b Formula


The formula because that cos a cos b deserve to be obtained using the sum and difference identities that 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), us 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 apply Cos a Cos b Formula?


Now the we recognize the cos a cos b formula, us will understand its applications in solving assorted problems. This identity deserve to be offered to solve an easy trigonometric difficulties and facility integration problems. We deserve to follow the measures given listed below to find out to use cos a cos b identity. Let united state go with some instances to know the principle clearly:

Example 1: Express cos 2x cos 5x as a amount of the cosine function.

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

Identify a and b in the given expression. Right here a = 2x, b = 5x. Using the above 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 sum of the cosine function.

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

To fix 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 also b in the provided expression. Right here a = x, b = 3x. Utilizing the over formula, us 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) right into the intergral ∫ cos x cos 3x dx. We will use the integral formula of the cosine duty ∫ 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.

See more: Flat Character In Romeo And Juliet, What Characters Are Flat In Romeo And Juliet

Important note on cos a cos b

cos a cos b = (1/2)It is used when one of two people the two angles a and b are recognized or once the sum and also difference the angles are known.The cos a cos b formula help in fixing integration formulas and also problems involving 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 utilizing cos a cos b identity.

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

Identify a and b in the given expression. Below a = 2x, b = 4x. Utilizing 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, substitute cos 2x cos 4x = (1/2)cos (6x) + (1/2)cos (2x) right into the intergral ∫ cos 2x cos 4x dx. Us will use the integral formula that the cosine role ∫ 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