Proving The Identity Sin(20°)sin(40°)sin(60°)sin(80°) = 3/16 A Step-by-Step Guide

by StackCamp Team 82 views

Introduction

In the fascinating realm of trigonometry, certain identities stand out for their elegance and the ingenuity required to prove them. One such identity is the seemingly simple yet intriguing equation: sin(20°)sin(40°)sin(60°)sin(80°) = 3/16. This article delves into a comprehensive exploration of this trigonometric gem, providing a step-by-step proof and shedding light on the underlying trigonometric principles that make this identity hold true. We will embark on a journey through trigonometric transformations, product-to-sum formulas, and algebraic manipulations to unravel the beauty of this equation. This exploration is not just an exercise in mathematical rigor; it is a testament to the interconnectedness of trigonometric functions and the power of mathematical reasoning.

Understanding the Problem

Before we dive into the proof, let’s clearly state the problem. Our mission is to demonstrate that the product of the sines of angles 20°, 40°, 60°, and 80° equals 3/16. In mathematical notation, this is expressed as:

sin(20°)sin(40°)sin(60°)sin(80°) = 3/16

This identity is not immediately obvious, and a direct calculation using a calculator might only provide numerical evidence. To truly prove it, we need to employ trigonometric identities and transformations. We will leverage the well-known value of sin(60°), which is √3/2, and then focus on simplifying the remaining product of sines. The challenge lies in manipulating the product of sines into a form where we can clearly see the 3/16 emerge. This requires a strategic application of trigonometric identities, particularly the product-to-sum formulas, which will help us transform the product of sines into a more manageable expression. The journey towards proving this identity is a beautiful illustration of how seemingly complex trigonometric expressions can be simplified using fundamental principles.

Strategy for the Proof

Our strategy for proving this identity involves a series of steps, each building upon the previous one. We'll begin by substituting the known value of sin(60°) into the equation. This will simplify the expression and allow us to focus on the product of the remaining sine terms. The core of our strategy lies in using the product-to-sum trigonometric identities. These identities are crucial for transforming products of trigonometric functions into sums or differences, which are often easier to manipulate. Specifically, we will use the identity:

sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]

This identity will be applied to pairs of sine terms in our expression. By strategically pairing the terms and applying this identity, we aim to create terms that can be further simplified or combined. After applying the product-to-sum identities, we will simplify the resulting expression by evaluating cosine values of specific angles and using other trigonometric identities, such as the double-angle formula for sine and cosine. The ultimate goal is to reduce the expression to a numerical value, which we expect to be 3/16. This step-by-step approach ensures that we maintain clarity and rigor throughout the proof, making it easier to follow and understand the underlying logic.

Step-by-Step Proof

Let's embark on the proof step by step:

  1. Start with the given equation:

    sin(20°)sin(40°)sin(60°)sin(80°) = 3/16

  2. Substitute the value of sin(60°):

    We know that sin(60°) = √3/2. Substituting this value into the equation, we get:

    sin(20°)sin(40°)(√3/2)sin(80°) = 3/16

  3. Rearrange the terms:

    To make the next steps clearer, let's rearrange the terms:

    (√3/2)sin(20°)sin(40°)sin(80°) = 3/16

  4. Apply the product-to-sum identity:

    We'll apply the identity sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)] to sin(20°)sin(40°):

    sin(20°)sin(40°) = 1/2[cos(40°-20°) - cos(40°+20°)]

    sin(20°)sin(40°) = 1/2[cos(20°) - cos(60°)]

  5. Substitute the value of cos(60°):

    We know that cos(60°) = 1/2. Substituting this value, we get:

    sin(20°)sin(40°) = 1/2[cos(20°) - 1/2]

  6. Substitute back into the equation:

    Now, substitute this back into our main equation:

    (√3/2) [1/2[cos(20°) - 1/2]] sin(80°) = 3/16

  7. Simplify the equation:

    Let's simplify the equation by multiplying the constants:

    (√3/4) [cos(20°) - 1/2] sin(80°) = 3/16

  8. Multiply through by sin(80°):

    Multiply the terms inside the brackets by sin(80°):

    (√3/4) [cos(20°)sin(80°) - 1/2 sin(80°)] = 3/16

  9. Apply the product-to-sum identity again:

    We'll apply the identity cos(A)sin(B) = 1/2[sin(A+B) - sin(A-B)] to cos(20°)sin(80°):

    cos(20°)sin(80°) = 1/2[sin(80°+20°) - sin(80°-20°)]

    cos(20°)sin(80°) = 1/2[sin(100°) - sin(60°)]

  10. Substitute the value of sin(60°) and use sin(100°) = sin(80°):

    We know that sin(60°) = √3/2 and sin(100°) = sin(180°-80°) = sin(80°). Substituting these values, we get:

    cos(20°)sin(80°) = 1/2[sin(80°) - √3/2]

  11. Substitute back into the equation:

    Now, substitute this back into our equation:

    (√3/4) [1/2[sin(80°) - √3/2] - 1/2 sin(80°)] = 3/16

  12. Simplify the equation:

    Let's simplify the equation:

    (√3/4) [1/2 sin(80°) - √3/4 - 1/2 sin(80°)] = 3/16

    (√3/4) [- √3/4] = 3/16

    -3/16 = 3/16

  13. Final simplification and verification:

    (√3/4) [1/2 sin(80°) - √3/4 - 1/2 sin(80°)] = (√3/4) [-√3/4] = -3/16

    However, there seems to be a sign error in the calculation. Let's re-evaluate from step 8.

From step 8:

(√3/4) [cos(20°)sin(80°) - 1/2 sin(80°)] = 3/16

Using the product-to-sum identity for cos(20°)sin(80°):

cos(20°)sin(80°) = 1/2 [sin(100°) + sin(60°)]

Since sin(100°) = sin(180°-80°) = sin(80°):

cos(20°)sin(80°) = 1/2 [sin(80°) + sin(60°)] = 1/2 [sin(80°) + √3/2]

Substituting back into the equation from step 8:

(√3/4) [1/2 [sin(80°) + √3/2] - 1/2 sin(80°)] = 3/16

(√3/4) [1/2 sin(80°) + √3/4 - 1/2 sin(80°)] = 3/16

(√3/4) [√3/4] = 3/16

3/16 = 3/16

  1. Conclusion:

    Thus, we have successfully proven that sin(20°)sin(40°)sin(60°)sin(80°) = 3/16.

Alternative Approaches

While we have presented a detailed proof using product-to-sum identities, there are alternative approaches to tackle this trigonometric identity. One such approach involves using the triple angle formula for sine, which states:

sin(3x) = 3sin(x) - 4sin³(x)

By strategically applying this formula and making appropriate substitutions, we can arrive at the same result. Another approach involves using complex numbers and De Moivre's theorem, which provides a powerful tool for dealing with trigonometric identities. These alternative methods not only offer different perspectives on the problem but also highlight the interconnectedness of various mathematical concepts. Exploring these approaches can deepen our understanding of trigonometry and enhance our problem-solving skills.

Significance of the Identity

The identity sin(20°)sin(40°)sin(60°)sin(80°) = 3/16 is more than just a mathematical curiosity; it exemplifies the elegance and inherent structure within trigonometry. This identity showcases how seemingly disparate trigonometric functions can be related through specific and concise equations. Its proof requires a blend of trigonometric identities, algebraic manipulation, and strategic problem-solving, making it a valuable exercise for students and enthusiasts alike. Furthermore, this identity serves as a reminder of the beauty and order that underlies the mathematical world, where seemingly complex relationships can be distilled into simple and elegant forms. The significance of this identity lies not only in its result but also in the journey of its proof, which enhances our appreciation for the power and versatility of trigonometric principles.

Conclusion

In conclusion, we have successfully demonstrated that sin(20°)sin(40°)sin(60°)sin(80°) = 3/16 through a rigorous step-by-step proof. This journey through trigonometric transformations and identities has not only validated the identity but also deepened our understanding of trigonometric principles. The strategic use of product-to-sum formulas, along with careful algebraic manipulation, allowed us to unravel the intricate relationships between these trigonometric functions. This exploration serves as a testament to the beauty and interconnectedness of mathematics, where seemingly complex problems can be elegantly solved with the right tools and techniques. The significance of this identity extends beyond its numerical result; it embodies the power of mathematical reasoning and the inherent elegance of trigonometric relationships.