2cosacosb
Trigonometry is the field of study which 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 2cosacosb formula in detail.
As we know that there are six Trigonometric functions of angles and their names are:
- Sine
- Cosine
- Tangent
- Cotangent
- Secant
- Cosecant
These functions are in relation to the right triangle in the following way:
In any right triangle ABC,
| Sin A = Perpendicular/ Hypotenuse |
| Cos A = Base/ Hypotenuse |
| Tan A = Perpendicular/ Base |
| Cot A = Base/ Perpendicular |
| Cosec A = Hypotenuse/Perpendicular |
| Sec A = Hypotenuse/ Base |
What is 2cosacosb Formula?
Consider a right triangle ABC as shown below,
We know that,
cosθ = Adjacent/Hypotenuse
sinθ = Opposite/Hypotenuse
- cos A = AB/AC
- cos B = BC/AC
- sin A = BC/AC
- sin B = AB/AC
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), we get
cos (A + B) + cos (A – B) = 2 cos A cos B
For any two acute angles A and B in a right triangle the 2cosacosb formula is given by
Formula of 2cosacosb
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), we get
cos (A + B) + cos (A – B) = 2 cos A cos B
Solved Examples using 2cosacosb Formula
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]
Thus, 8 cos y cos 2y in terms of sum function is 4 [cos 3y + cos y].
Example 2: Express 20 cos x cos 4x in terms of sum function.
Solution: 20 cos x cos 4x
= 10 [2 cos x cos 4x]
Using the 2cosa cosb 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]
Thus, 20 cos x cos 3x in terms of sum function is 10 [cos 5x + cos 3x].
Example 2: Express 6 cos x cos 2x in terms of sum function.
Solution:
Consider,
6 cos x cos 2x
= 3 [2 cos x cos 2x]
Using the formula 2 cos A cos B = cos (A + B) + cos (A – B),
= 3[cos (x + 2x) + cos (x – 2x)] = 3[cos 3x + cos (-x)] = 3 [cos 3x + cos x]
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2 Cos A Cos B Formula Application
Express 2 Cos 7x Cos 3y as a Sum
Solution:
Let A = 7x and B = 3y
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 (7x + 3y) + Cos (7x – 3y)
2 Cos A Cos B = Cos 10x + Cos 4y
Hence, 2 Cos 7x Cos 3y = Cos 10x + Cos 4y
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