Exploring the Value of cos(π/7)cos(2π/7)cos(4π/7)

Understanding the value of cos(π/7)cos(2π/7)cos(4π/7) involves a deep dive into trigonometric identities and mathematical logic. This problem is not just a simple calculation, but a beautiful example of how trigonometric properties interplay to simplify complex expressions.

Trigonometric Identity Introduction

The expression cos(π/7)cos(2π/7)cos(4π/7) can be evaluated using fundamental trigonometric identities. These identities are the backbone of trigonometry and are instrumental in simplifying complex expressions into manageable forms. Some of the key identities used in solving this expression include:

Product-to-Sum Identities Tau and Sine Relations

Step-by-Step Solution

Let's consider the expression again:

cos(π/7)cos(2π/7)cos(4π/7)

First, we employ the product-to-sum identity, which states that:

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

However, in this context, we will use a more direct approach leveraging the known product of cosines identity:

cos(A)cos(2A)cos(4A) (1/8)sin(8A)/sin(A)

Here, A π/7. Let's see how this simplification is performed:

Substitute A for π/7 in the identity: Step 1: cos(π/7)cos(2π/7)cos(4π/7) (1/8)sin(8π/7)/sin(π/7) Step 2: Simplify using the fact that sin(8π/7) -sin(π/7) (since sin(θ π) -sin(θ)): Step 3: cos(π/7)cos(2π/7)cos(4π/7) (1/8)(-sin(π/7))/sin(π/7) -1/8

Verification and Proof

To ensure the correctness of the solution, we can verify it through a few steps:

Start with the basic trigonometric product identity. Use the substitution and simplification techniques. Apply the known trigonometric properties to verify the final result.

By carefully applying these principles, we can confidently state that the value of cos(π/7)cos(2π/7)cos(4π/7) is indeed -1/8.

Conclusion

The evaluation of cos(π/7)cos(2π/7)cos(4π/7) is a crucial example in trigonometry that demonstrates the power of trigonometric identities and logical reasoning. By leveraging these tools, even seemingly complex expressions can be broken down into simpler forms, making them easier to understand and solve.

Frequently Asked Questions (FAQs)

How is the value -1/8 derived from cos(π/7)cos(2π/7)cos(4π/7)?

The value -1/8 is derived through the application of trigonometric identities, particularly the identity for the product of cosines, and simplifying using known properties of sine and cosine.

What does the process of evaluating cos(π/7)cos(2π/7)cos(4π/7) teach us about trigonometric functions?

This process illustrates the intricate relationships between trigonometric functions and how they can be manipulated to simplify expressions. It emphasizes the importance of memorizing and understanding key trigonometric identities.