2 Cos A Cos B Formula – The 2cosacosb formula focuses on the cosine (cos) function and its interaction with angles. This formula expresses the product of two cosine functions as a sum of trigonometric functions. It is frequently employed in trigonometric identities and equations to simplify expressions or solve trigonometric problems.
To fully grasp this formula, apply it to a specific trigonometric problem or identity. Depending on the situation, you may need to alter trigonometric expressions using this formula to simplify or solve equations with cosine functions.
2 Cos A Cos B Formula
The 2cosacosb formula is a product-to-sum formula, meaning it converts a product into a sum. Trigonometry is the study of the angles, heights, and lengths of right triangles. Trigonometric ratios refer to the ratios of a right triangle’s sides. Trigonometry contains six major ratios: sin, cos, tan, cot, sec, and cosec. Each of these ratios has a separate formula. It employs the three sides and angles of a right-angled triangle. Let’s look at the 2cosacosb formula in detail.
Trigonometry is the branch of Mathematics that deals with particular angles’ functions and how to use those functions in calculations. There are six popular trigonometric functions for an angle. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their respective names and acronyms (csc).
2 Cos A Cos B Formula Derivation
The 2 Cos A Cos B formula can be derived by observing the addition and substraction formula for cosine.
As we know,
Adding the equation (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 (The term Sin A Sin B is cancelled due to the opposite sign).
Therefore, the formula of 2 Cos A Cos B is given as:
2 Cos A Cos B = cos (A + B) + cos (A – B)
In the above 2 Cos A Cos B formula, the left-hand side is the product of cosine whereas the right-hand side is the sum of the cosine.
2Cos A Cos B Solved Examples
Example 1: Find the value of 2 Cos 8x Cos 2x.
Solution: Let A = 8x and B = 2x
Using the formula:
2 Cos A Cos B = Cos (A + B) + Cos (A – B)
Substituting the values of A and B in the above formula, we get
2 Cos A Cos B = Cos (8x + 2x) + Cos (8x – 2x)
2 Cos A Cos B = Cos 10x + Cos 6x
Hence, 2 Cos 8x Cos 3x = Cos 10x + Cos 6x
Example 2: Express 8 cos 6y cos 2y in terms of sum function.
Solution: 8 cos 6y cos 2y
= 4 [2 cos 6y cos 2y]
Using the 2cosa cosb Formula,
2 cos A cos B = cos (A + B) + cos (A – B)
= 4[cos (6y + 2y) + cos (6y – 2y)]
= 4[cos 8y + cos 4y]