Ratios & Identities: Product of Cosine Where Angles are in Geometric Progression (GP) with r = 2 or 1/2

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the key concepts in trigonometry is the use of trigonometric identities and ratios to simplify and solve problems. In this content, we will explore a specific case of trigonometric identities involving the product of cosines where the angles are in geometric progression (GP) with a common ratio (r) of 2 or 1/2.

Understanding Geometric Progression (GP)

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, if we have a GP with a first term a and a common ratio r, the sequence is:

a, ar, ar^2, ar^3, ...

In the context of angles in trigonometry, if we have an initial angle α, and the angles are in GP with a common ratio r, the angles would be:

α, αr, αr^2, αr^3, ...

Product of Cosines in GP

When dealing with the product of cosines where the angles are in GP, we can use trigonometric identities to simplify the expression. The product of cosines for angles in GP can be represented as:

cos(α) * cos(αr) * cos(αr^2) * ... * cos(αr^(n-1))

where n is the number of terms.

Case 1: Common Ratio r = 2

When the common ratio r is 2, the angles double each time. The product of cosines for the first few terms would look like this:

cos(α) * cos(2α) * cos(4α) * ...

Case 2: Common Ratio r = 1/2

Conversely, when the common ratio r is 1/2, the angles are halved each time. The product of cosines for the first few terms would be:

cos(α) * cos(α/2) * cos(α/4) * ...

Formulas and Identities

To simplify the product of cosines, we can use the following trigonometric identity:

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

By applying this identity iteratively, we can simplify the product of cosines in GP.

Examples

Let's consider an example where α = π/6 and we have three terms with a common ratio r = 2.

cos(π/6) * cos(π/3) * cos(2π/3)

Using the identity mentioned above, we can simplify the product step by step:

cos(π/6)cos(π/3) = 1/2[cos(π/6 + π/3) + cos(π/6 - π/3)]
                 = 1/2[cos(π/2) + cos(-π/6)]
                 = 1/2[0 + cos(π/6)]
                 = 1/2[cos(π/6)]

Now, we multiply this result by cos(2π/3):

1/2[cos(π/6)] * cos(2π/3)

Again, we apply the identity:

= 1/2[1/2(cos(π/6 + 2π/3) + cos(π/6 - 2π/3))]
= 1/2[1/2(cos(5π/6) + cos(-π/2))]
= 1/2[1/2(-√3/2 + 0)]
= -√3/8

Therefore, the product of cosines for the given angles in GP with r = 2 is -√3/8.

Table of Differences and Important Points

Common Ratio (r)	Sequence of Angles	Product of Cosines Example	Simplification Strategy
2	α, 2α, 4α, ...	cos(α) * cos(2α) * cos(4α)	Use trigonometric identities iteratively to simplify the product.
1/2	α, α/2, α/4, ...	cos(α) * cos(α/2) * cos(α/4)	Same as above, but note that the angles are getting smaller.

Conclusion

The product of cosines where angles are in geometric progression with a common ratio of 2 or 1/2 can be simplified using trigonometric identities. Understanding the sequence of angles and how to apply these identities is crucial for solving such problems in trigonometry. With practice, you can master these techniques and apply them to various trigonometric expressions and equations.