Proof Of Sin(20°)sin(40°)sin(60°)sin(80°) = 3/16 Trigonometric Identity

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Introduction

The fascinating world of trigonometry often presents us with elegant identities that require clever manipulation and a deep understanding of trigonometric relationships. One such identity is the proof that the product of sines of certain angles equals a specific fraction. In this article, we will embark on a journey to prove the trigonometric identity: sin(20°)sin(40°)sin(60°)sin(80°) = 3/16. This identity showcases the beautiful interplay between different trigonometric functions and angles. We will explore various trigonometric identities and formulas to arrive at the final solution. This exploration will not only solidify your understanding of trigonometry but also enhance your problem-solving skills. Our main keywords in this article revolve around the trigonometric identity, the sines of the specified angles, and the step-by-step proof process.

Initial Observations and Strategy

Before diving into the intricate details, let's make some initial observations. We are dealing with the product of sines of angles that are multiples of 20 degrees. This suggests that we might be able to use trigonometric identities that relate sines of multiple angles. The presence of sin(60°), which has a known value of √3/2, simplifies the problem to some extent. Our primary strategy will revolve around employing trigonometric identities to simplify the expression sin(20°)sin(40°)sin(80°) and then multiply the result by sin(60°). We will specifically focus on using product-to-sum identities and double-angle formulas to achieve this simplification. The goal is to transform the product of sines into a form where we can easily evaluate the expression and ultimately prove that it equals 3/16. This strategy involves careful selection and application of trigonometric identities, a skill crucial for solving complex trigonometric problems. Remember, the beauty of trigonometry lies in its ability to transform seemingly complicated expressions into simpler, more manageable forms.

Utilizing the Product-to-Sum Identity

To begin our journey of simplifying the expression, we will strategically apply the product-to-sum trigonometric identity. This identity is a powerful tool that allows us to convert the product of trigonometric functions into a sum or difference. Specifically, we'll focus on the following identity:

  • 2sin(A)sin(B) = cos(A - B) - cos(A + B)

We will first apply this identity to the product sin(20°)sin(40°). Let A = 40° and B = 20°. Substituting these values into the identity, we get:

  • 2sin(40°)sin(20°) = cos(40° - 20°) - cos(40° + 20°)
  • 2sin(40°)sin(20°) = cos(20°) - cos(60°)

Since we know that cos(60°) = 1/2, we can further simplify this to:

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

Now, dividing both sides by 2, we obtain:

  • sin(40°)sin(20°) = (1/2)cos(20°) - 1/4

This transformation is a crucial step in our proof. We have successfully converted the product sin(20°)sin(40°) into an expression involving cos(20°). This form is more amenable to further simplification when we consider the remaining term, sin(80°). By strategically applying the product-to-sum identity, we have paved the way for the next stage of our proof, which will involve incorporating sin(80°) and further simplifying the expression. This highlights the importance of recognizing and utilizing appropriate trigonometric identities to solve complex problems.

Incorporating sin(80°) and Further Simplification

Now that we have simplified sin(20°)sin(40°), our next step involves incorporating the sin(80°) term and continuing the simplification process. We recall that:

  • sin(40°)sin(20°) = (1/2)cos(20°) - 1/4

We need to multiply this expression by sin(80°) to incorporate the final sine term from the original identity. Multiplying both sides by sin(80°), we get:

  • sin(20°)sin(40°)sin(80°) = [(1/2)cos(20°) - 1/4]sin(80°)
  • sin(20°)sin(40°)sin(80°) = (1/2)cos(20°)sin(80°) - (1/4)sin(80°)

At this point, we encounter another product of trigonometric functions: cos(20°)sin(80°). To simplify this, we will again employ the product-to-sum identity, but this time we'll use the following form:

  • 2sin(A)cos(B) = sin(A + B) + sin(A - B)

Let A = 80° and B = 20°. Substituting these values, we have:

  • 2sin(80°)cos(20°) = sin(80° + 20°) + sin(80° - 20°)
  • 2sin(80°)cos(20°) = sin(100°) + sin(60°)

Dividing both sides by 2, we get:

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

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

  • sin(80°)cos(20°) = (1/2)sin(80°) + (1/2)(√3/2)
  • sin(80°)cos(20°) = (1/2)sin(80°) + √3/4

Now, we can substitute this back into our equation for sin(20°)sin(40°)sin(80°):

  • sin(20°)sin(40°)sin(80°) = (1/2)[(1/2)sin(80°) + √3/4] - (1/4)sin(80°)
  • sin(20°)sin(40°)sin(80°) = (1/4)sin(80°) + √3/8 - (1/4)sin(80°)

Notice that the (1/4)sin(80°) terms cancel out, leaving us with:

  • sin(20°)sin(40°)sin(80°) = √3/8

This simplification is a significant achievement. We have reduced the product of three sine terms to a single, simple fraction involving the square root of 3. This form is now ready to be multiplied by the remaining term, sin(60°), to complete the proof. This step demonstrates the power of strategically applying trigonometric identities to simplify complex expressions and isolate the desired result.

Final Step: Multiplying by sin(60°) and Concluding the Proof

We have successfully simplified the product sin(20°)sin(40°)sin(80°) to √3/8. Now, we need to incorporate the final term, sin(60°), to complete the proof of the identity.

Recall that our original identity is:

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

We have already established that:

  • sin(20°)sin(40°)sin(80°) = √3/8

We also know that:

  • sin(60°) = √3/2

Therefore, we can multiply the simplified expression by sin(60°):

  • sin(20°)sin(40°)sin(60°)sin(80°) = (√3/8) * (√3/2)
  • sin(20°)sin(40°)sin(60°)sin(80°) = (√3 * √3) / (8 * 2)
  • sin(20°)sin(40°)sin(60°)sin(80°) = 3 / 16

And there we have it! We have successfully proven the trigonometric identity:

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

This final step beautifully ties together all the previous simplifications and calculations. By multiplying the simplified expression by sin(60°), we arrived at the desired result, thus completing the proof. This demonstrates the elegance and precision of trigonometric identities and their ability to reveal hidden relationships between seemingly disparate trigonometric functions and angles. The journey to prove this identity has been a testament to the power of strategic application of trigonometric identities and formulas.

Conclusion

In conclusion, we have successfully proven the trigonometric identity sin(20°)sin(40°)sin(60°)sin(80°) = 3/16 through a series of strategic applications of trigonometric identities. We began by making initial observations and outlining our strategy, which involved using product-to-sum identities and double-angle formulas. We then meticulously applied the product-to-sum identity to simplify the expression sin(20°)sin(40°). Next, we incorporated the sin(80°) term and further simplified the expression using another application of the product-to-sum identity. Finally, we multiplied the simplified expression by sin(60°) and arrived at the desired result, thus completing the proof. This exercise highlights the importance of a strong understanding of trigonometric identities and the ability to apply them strategically to solve complex problems. The elegance of trigonometry lies in its ability to transform seemingly complicated expressions into simpler, more manageable forms, and this proof serves as a beautiful example of this principle. By mastering these techniques, you can unlock the power of trigonometry and apply it to a wide range of mathematical and scientific problems. The journey through this proof has not only demonstrated a specific trigonometric identity but also reinforced the fundamental principles of trigonometric manipulation and problem-solving.