Cos a Cos b

Cos a Cos b is a trigonometric formula that is used in trigonometry. Cos a cos b formula is given by, cos a cos b =  (1/2)[cos(a + b) + cos(a - b)]. We use the cos a cos b formula to find the value of the product of cosine of two different angles. cos a cos b formula can be obtained from the cosine trigonometric identity for sum of angles and difference of angles.

The cos a cos b formula helps in solving integration formulas and problems involving the product of trigonometric ratio such as cosine. Let us understand the cos a cos b formula and its derivation in detail in the following sections.

Cos a Cos b 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 cos (a + b) and cos (a - b) trigonometry identities which are some of the important trigonometric identities. The cos a cos b identity is half the sum of the cosines of the sum and difference of the angles a and b, that is, cos a cos b = (1/2)[cos(a + b) + cos(a - b)].

The formula for cos a cos b can be derived using the sum and 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 (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 [The term sin a sin b got cancelled because of opposite signs]

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

⇒ cos a cos b = (1/2)[cos (a + b) + cos (a - b)]

Hence the cos a cos b formula has been derived.

Thus, cos a cos b = (1/2)[cos (a + b) + cos (a - b)]

Now that we know the cos a cos b formula, we will understand its application in solving various problems. This identity can be used to solve simple trigonometric problems and complex integration problems. We can follow the steps given below to learn to apply cos a cos b identity. Let us go through some examples to understand the concept 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)[cos (a + b) + cos (a - b)]

Identify a and b in the given expression. Here a = 2x, b = 5x. Using the above formula, we will process to the second step.

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

cos 2x cos 5x = (1/2)[cos (2x + 5x) + cos (2x - 5x)]

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

⇒ cos 2x cos 5x = (1/2)cos (7x) + (1/2)cos (3x) [Because cos(-x) = cos x]

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

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

To solve 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)[cos (a + b) + cos (a - b)]

Identify a and b in the given expression. Here a = x, b = 3x. Using the above 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 + 3x) + cos (x - 3x)]

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

⇒ cos x cos 3x = (1/2)cos (4x) + (1/2)cos (x) [Because cos(-x) = cos x]

Step 3: Now, substitute 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 of the cosine function ∫ 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) [sin (4x)]/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 using the cos a cos b formula.

Important Notes on cos a cos b

• cos a cos b = (1/2)[cos (a + b) + cos (a - b)]
• It is applied when either the two angles a and b are known or when the sum and difference of angles are known.
• The cos a cos b formula helps in solving integration formulas and problems involving the product of trigonometric ratio such as cosine

Related Topics on cos a cos b

• sin(a + b)
• sin of 2 pi
• cos (a + b)
• sin (a - b)

What is cos a cos b Formula in Trigonometry?

Cos a Cos b is the trigonometry identity for two different angles whose sum and difference are known. The cos a cos b identity is half the sum of the cosines of the sum and difference of the angles a and b, that is, cos a cos b = (1/2)[cos(a + b) + cos(a - b)].

How Do you Derive cos a cos b Identity?

cos a cos b can be derived using the sum and difference identities of the cosine function. It can be derived by adding the cos (a + b) and cos (a - b) formulas.

What is the Formula for 2 cos a cos b?

We know that cos a cos b = (1/2)[cos (a + b) + cos (a - b)]. Multiply both sides of the equation cos a cos b = (1/2)[cos (a + b) + cos (a - b)] by 2, we have 2 cos a cos b = cos (a + b) + cos (a - b)]. Hence, the formula for 2 cos a cos b is cos (a + b) + cos (a - b).

What is the Formula for cos a cos b?

cos a cos b is one of the important trigonometric formulas used in trigonometry. The formula for cos a cos b is cos a cos b = (1/2)[cos(a + b) + cos(a - b)].

How to Prove cos a cos b Formula?

cos a cos b can be proved using the sum and difference identities of the cosine function. It can be proved by adding the cos (a + b) and cos (a - b) formulas.

What are the Applications of cos a cos b Formula?

The cos a cos b formula helps in solving integration formulas and problems involving the product of trigonometric ratio such as cosine. This identity can be used to solve simple trigonometric problems and complex integration problems.