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A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.

2cosacosb Formula

2cosacosb formula is one of the product-to-sum formulas as this formula is used to convert a product into a sum. Trigonometry is the field of study that deals with the relationship between angles, heights, and lengths of right triangles. The ratios of the sides of a right triangle is known as trigonometric ratios. Trigonometry has six main ratios namely sin, cos, tan, cot, sec, and cosec. All these ratios have different formulas. It uses the three sides and angles of a right-angled triangle. Let's look into the 2cosacosb formula in detail.

What is 2 cos a cos b?

The 2cosacosb formula is 2 cos A cos B = cos (A + B) + cos (A – B). This formula converts the product of two cos functions as the sum of two other cos functions. For example:

  • 2 cos (2x) cos (2y) = cos (2x + 2y) + cos (2x - 2y)
  • 2 cos (x/2) cos (y/2) = cos (x/2 + y/2) + cos (x/2 - y/2)

2cosacosb Formula Derivation

Let us see how to derive this formula. By the sum and difference formulas of trigonometry, we know that,

  • 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 (1) and (2), the term of sin A sin B gets canceled in both equations. Then we get:

cos (A + B) + cos (A – B) = 2 cos A cos B

Thus, the 2 cos A cos B formula is,

  • 2 cos A cos B = cos (A + B) + cos (A – B)

For any two angles A and B in a right triangle, the 2cosacosb formula is given by

Formula to calculate 2 cosa cosb

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Solved Examples using 2 cos a cos b Formula

  1. Example 1: Express 8 cos y cos 2y in terms of sum function.

    Solution: 8 cos y cos 2y

    = 4 [2 cos y cos 2y]

    Using the 2cosa cosb Formula,

    2 cos A cos B = cos (A + B) + cos (A – B)

    = 4[cos (y + 2y) + cos (y – 2y)]

    = 4[cos 3y + cos (-y)]

    = 4 [cos 3y + cos y]

    Answer: Thus, 8 cos y cos 2y in terms of sum function is 4 [cos 3y + cos y].

  2. Example 2: Write 20 cos x cos 4x as sum.

    Solution: 20 cos x cos 4x

    = 10 [2 cos x cos 4x]

    Using the 2cosacosb Formula,

    2 cos A cos B = cos (A + B) + cos (A – B)

    = 10 [cos (x + 4x) + cos (x – 4x)]

    = 10 [cos 5x + cos (-3x)]

    = 10 [cos 5x + cos 3x]

    Answer: Thus, 20 cos x cos 3x in terms of sum function is 10 [cos 5x + cos 3x].

  3. Example 3: What is the value of the integral ∫ 2 cos x cos 3x dx?

    Solution:

    By 2cosAcosB formula,

    2 cos A cos B = cos (A + B) + cos (A – B)

    2 cos x cos 3x = cos (x + 3x) + cos (x - 3x)
    = cos 4x + cos (-2x)
    = cos 4x + cos 2x (∵ cos(-θ) = cos θ)

    Then the given integral becomes:

    ∫ 2 cos x cos 3x dx = ∫ (cos 4x + cos 2x) dx
    = (1/4) sin 4x + (1/2) sin 2x + C (Using the substitution method of integration)

    Answer: ∫ 2 cos x cos 3x dx = (1/4) sin 4x + (1/2) sin 2x + C.

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FAQs on 2cosAcosB

What is the Formula of 2cosAcosB?

2cosAcosB formula is cos (A + B) + cos (A - B). i.e., 2 cos A cos B is the sum of two cosines where one cosine is with the sum of two angles and the other cosine is with the difference of two angles.

How to Derive the Formula of 2 cos A cos B?

It is one of the product to sum formulas of trigonometry. To derive this, we use the sum and difference formulas of cos. The sum formula of cosine is cos (A + B) = cos A cos B – sin A sin B. The difference formula of cosine is cos (A – B) = cos A cos B + sin A sin B. Adding these two we get 2 cos A cos B = cos (A + B) + cos (A - B).

Write an Example of 2 cos A cos B Formula.

Let A = 3x and B = 2y then 2 cos (3x) cos (2y) = cos (3x + 2y) + cos (3x - 2y). This is because we have a formula in trigonometry: 2 cosA cosB = cos (A + B) + cos (A - B).

What are the Applications of the 2cosAcosB Formula?

The 2cosAcosB formula has its main application in integration. To integrate a function of the form 2cosAcosB, we first convert it into the sum using 2 cosA cosB = cos (A + B) + cos (A - B) and then integrate it easily.

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