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2 Cos A Cos B formula is a product to sum up the trigonometric formulas used to rewrite the product of cosines as sum or difference. The 2 cos A cos B formula can be used to solve integration problems that use the product of a trigonometric ratio like cosine.
- 2 Cos A Cos B Formula turns out to be very useful in simplifying the trigonometric expression by considering the product term such as Cos A Cos B and converting it into a sum.
- Also, Cos a cos b formula can be represented by, cos a cos b = (1/2)[cos(a + b) + cos(a - b)].
- Trigonometry is the branch of mathematics concerned with the relationship between right-angle triangles, heights, and lengths.
- Trigonometric ratios are proportions between the sides of a right triangle. Sin, cos, tan, cot, sec, and cosec are the six basic trigonometric ratios.
- The formulas for each of these ratios are different. It makes use of a right-angled triangle's three sides and angles.
Read More: NCERT Solutions for Class 11 Mathematics Trigonometric Functions
| Table of Content |
Key Terms: Cosines, Sines, Integration, Compound angles, Right-angle triangle, Secant, Cosecant, Tangent, Cotangent, Perpendicular, Hypotenuse, Integrals, Trigonometric Ratio, Trigonometry, Cosec Cot Formula
What is 2 Cos A Cos B?
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2 Cos A Cos B can be expressed as the product of trigonometric sum mathematical statements that are used to rewrite the product of cosines into difference or sum. The 2 cos a cos b expression helps to evaluate the integration mathematical statements that refer to the product of trigonometric ratios, like cosine.
The product-to-sum trigonometric formula is used to rephrase the product of cosines as a sum or difference is 2 Cos A Cos B. This formula is one of the de-factorization formulas used in trigonometry.
- Defactorization means a product gets converted into a sum or difference.
- The 2 cos A cos B formula can be used to solve integration calculations that use the product of trigonometric ratios like cosine.
- By considering the product term such as Cos A Cos B and turning it to a sum, the formula 2 Cos A Cos B can also be very useful in reducing the trigonometric statement.
2 Cos a Cos b Formula
2 Cos a Cos b formula can be expressed as one of the product-to-sum formulas since it is used to convert a product into a sum. 2 Cos A Cos B Formula is:
| 2 Cos A Cos B = Cos (A + B) + Cos (A - B) |
- Trigonometry is a field which is known to deal with the relationship between angles, heights, and lengths of all right triangles.
- The ratios of the sides of a right triangle are 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.
Trigonometric Functions Detailed Explanation Video
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2 Cos A Cos B Formula Derivation
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The sum and difference formula for cosine can be used to derive the 2 Cos A Cos B formula.
We know that
- cosθ = Adjacent Side/Hypotenuse
- sinθ = Opposite Side/Hypotenuse
We know that
Cos ( A + B ) = Cos A Cos B - Sin A Sin B ... (eq. 1)
Cos ( A- B ) = Cos A Cos B + Sin A Sin B ... (eq. 2)
When we combine the above-given equations (1) and (2), we get
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) = 2 Cos A Cos B
(Because of the opposite sign, the term Sin A Sin B is nullified.)
Right-Angle Triangle
As a result, the formula for 2 Cos A Cos B is:
| 2 Cos A Cos B = Cos (A + B) + Cos (A - B) |
The left-hand side of the above 2 Cos A Cos B formula is the product of cosine, whereas the right-hand side is the sum of cosine.
Also Read: Tangent 3 Theta formula
2 Cos A Cos B Application
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Some applications of the 2 Cos a Cos b Formula include:
- The sum and difference trigonometric identities for cosine are used to derive the 2 Cos A Cos B formula.
- In order to solve integration difficulties, the Cos A Cos B formula is utilised.
- The trigonometric identity Cos A + Cos B is used to represent the sum of the cosine of angles A and B in the product form using the compound angles (A + B) and (A - B).
What do you mean by 2 Cos a Cos b?We are aware that the formula of 2 Cos a Cos b is – Cos (a + b) + Cos (a – b) = 2 Cos a Cos b. This mathematical formula helps to convert the product of two functions of the Cos function as the sum. For the same, the following examples can be represented:
|
Also Read: Sin Squared X Formula
Things to Remember
- 2 Cos A Cos B is a product to sum trigonometric formula for rewriting the product of cosines into sum or difference.
- The sum and difference trigonometric identities for cosine are used to generate the 2 Cos A Cos B formula.
- 2 Cos A Cos B = Cos (A + B) + Cos (A - B) is the formula for 2 Cos a Cos b.
- The 2 cos A cos B formula can aid in the solution of integration formulas using the product of a trigonometric ratio, such as cosine.
- By considering the product term such as Cos A Cos B and turning it to sum, the formula 2 Cos A Cos B can also be very useful in reducing the trigonometric statement.
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Solved Questions
Ques. Express 20 cos x cos 4x in terms of sum function. (3 marks)
Ans. Given term 20 cos x cos 4x = 10 [2 cos x cos 4x]
By using the 2 Cos A Cos B Formula, it will be
2 cos A cos B = cos (A + B) + cos (A – B)
Hence, 20 cos x cos 4x
= 10 [2 cos x cos 4x]
= 10 [cos (x + 4x) + cos (x – 4x)]
= 10 [cos 5x + cos (-3x)]
= 10 [cos 5x + cos 3x]
Thus, 20 cos x cos 3x in terms of sum function will be calculated as 10 [cos 5x + cos 3x].
Ques. In terms of the sum function, calculate 6 cos x cos 2x. (3 marks)
Ans. Let 6 cos x cos 2x = 3 [2 cos x cos 2x]
Given formula that 2 Cos A Cos B = Cos (A + B) + Cos (A - B)
Using the above-mentioned formula
6 cos x cos 2x = 3[cos (x + 2x) + cos (x – 2x)]
= 3[cos 3x + cos (-x)]
= 3 [cos 3x + cos x]
Ques. As a sum, calculate 2 Cos 7x Cos 3y. (3 marks)
Ans. Consider A = 7x and B = 3y
Given formula that 2 Cos A Cos B = Cos (A + B) + Cos (A - B)
By substituting the given values of A and B in the above formula respectively, we get
2 Cos A Cos B = Cos (7x + 3y) + Cos (7x - 3y)
So, 2 Cos A Cos B will be = Cos 10x + Cos 4y
Hence, 2 Cos 7x Cos 3y will be calculated as = Cos 10x + Cos 4y
Ques. In terms of the sum function, calculate 8 cos y cos 2y (3 marks)
Ans. Given expression 8 cos y cos 2y = 4 [2 cos y cos 2y]
By using the 2 Cos A Cos B Formula, it will be
2 cos A cos B = cos (A + B) + cos (A – B)
Hence,
8 cos y cos 2y
= 4 [2 cos y cos 2y]
= 4[cos (y + 2y) + cos (y – 2y)]
= 4[cos 3y + cos (-y)]
= 4 [cos 3y + cos y]
Thus, 8 cos y cos 2y in terms of sum function will be calculated as 4 [cos 3y + cos y].
Ques. Solve the following expression- 2 cos (π/13) cos (9π/13) + cos (3π/13) + cos (5π/13) (3 marks)
Ans.
By using the 2 Cos A Cos B Formula, it will be
2 cos A cos B = cos (A + B) + cos (A – B)
2 cos (π/13) cos (9π/13)
= cos ((9π/13) + (π/13) + cos ((9π/13) - (π/13)) + cos (3π/13) + cos (5π/13)
= cos (10π/13) + cos (8π/13) + cos (3π/13) + cos (5π/13)
= cos (π - (3π/13)) + cos (π - (5π/13)) + cos (3π/13) + cos (5π/13)
= - cos (3π/13) - cos (5π/13) + cos (3π/13) + cos (5π/13)
So, 2 cos (π/13) cos (9π/13) + cos (3π/13) + cos (5π/13) will be = 0
Ques. Calculate the given expression into sum or difference- cos 75° cos 15° (3 marks)
Ans. Given values, A = 75° and B = 15°
Now putting all these values in the 2 cos A cos B = cos (A + B) + cos (A – B) formula,
We will get
cos 75° cos 15°
= ½ (cos (75+15) + cos (75-15))
= ½ (cos (90°) + cos (60°))
So cos 75° cos 15° will be calculated as ½ (cos (90°) + cos (60°))
Ques. Prove the following expression- cos 2x cos (x/2) - cos 3x cos (9x/2) = sin 5x sin (5x/2) (5 marks)
Ans. On the LHS side,
cos 2x cos (x/2) - cos 3x cos (9x/2) = ½ [ 2 cos 2x cos (x/2) - 2 cos 3x cos (9x/2)]
Using the 2 Cos A Cos B Formula, it will be
2 cos A cos B = cos (A + B) + cos (A – B)
= ½ [cos (2x + (x/2)) + cos (2x - (x/2)) - cos ((9x/2) + 3x) - cos ((9x/2) - 3x)]
= ½ [cos (5x/2) + cos (3x/2) - cos (15x/2) - cos (3x/2)]
= ½ [cos (5x/2) - cos (15x/2)]
Using the cos x - cos y formula,
= ½ [- 2 sin ((5x/2) + (15x/2))/2 sin ((5x/2) - (15x/2))/2]
= - sin 5x sin (-5x/2) = sin 5x sin (5x/2) = RHS
As, LHS = RHS, hence the above expression is proved.
Ques. How to express 4 cos A. cos B. cos C as the sum of four cosines? (3 marks)
Ans. Given the above expression 4 cos A. cos B. cos C = 2 cos A * 2 cos B cos C
{cos (A + B) + cos (A - B) = 2 cos A. cos B}
= 2 cos A {cos (B + C) + cos (B - C)}
= 2 cos (B + C) cos A + 2 cos (B - C) . cos A
= cos (B + C + A) + cos (B + C - A) + cos (B - C + A) + cos (B - C - A)
Hence the above-given term is the expression of the sum of four cosines.
Ques. Prove the given equation- 2cos(a+b)cos(a-b) = Cos2a+Cos2b. (3 marks)
Ans. 2 cosx cosy= cos(x+y)+ cos(x-y)........(1)
Let x= a+b and y= a-b
Therefore, x+y= 2a and x-y=2b
Using these in eq (1),
2cos(a+b)cos(a-b)= cos 2a+cos 2b.
Ques. Solve 2 cos 75 cos 15. (3 marks)
Ans. We know that
2 cos A cos B is cos (A + B) + cos (A – B)
2 cos 75 cos 15 = cos (75 + 15) + cos (75 – 15)
= cos 90 + cos 60
= 0 + 1/2
= 1/2
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