The Value of sin12°sin48°sin54° and Its Trigonometric Insights

Introduction

Understanding the value of sin 12°sin 48°sin 54° involves a deep dive into trigonometric identities and some basic calculations. This article delves into the process, explaining the use of product-to-sum identities and other trigonometric principles to find the exact or approximate value of this expression.

Step-by-Step Solution Using Trigonometric Identities

To find the value of sin 12°sin 48°sin 54°, we start by leveraging some known trigonometric identities. Firstly, we recognize that sin 54° cos 36°, derived from the identity sin(90° - x) cos x.

Step 1: Simplify the Expression

Using the simplified form, our original expression can be rewritten as:

sin 12°sin 48°cos 36°

Step 2: Apply the Product-to-Sum Identity

Next, we use the product-to-sum identity for sine, which states:

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

Applying this to sin 12°sin 48°:

sin 12°sin 48° 1/2 [cos(12° - 48°) - cos(12° 48°)]

Simplifying the angles:

cos(12° - 48°) cos(-36°) cos 36°

cos(12° 48°) cos 60° 1/2

Substituting these back into the equation:

sin 12°sin 48° 1/2 [cos 36° - 1/2]

Step 3: Integrate the Cosine Term

Now, we reintegrate this expression into our original problem:

sin 12°sin 48°cos 36° 1/2 [cos 36° - 1/2] cos 36°

Distributing cos 36°:

1/2 [cos^2 36° - 1/2 cos 36°]

Step 4: Use the Half-Angle Identity

Using the half-angle identity for cosine:

cos^2 x 1/2 (1 cos 2x)

Here, we substitute for x 36°:

cos^2 36° 1/2 (1 cos 72°)

And, knowing that sin 18° cos 72°, we have:

cos^2 36° 1/2 (1 sin 18°)

Substituting this back into our expression, let c cos 36°:

1/2 [1/2 (1 sin 18°) - 1/2 c]

Numerical Approximation

While finding an exact value is complex, a numerical approximation can provide a clearer picture. Using a calculator, we approximate the following:

sin 12° ≈ 0.2079

sin 48° ≈ 0.7431

sin 54° ≈ 0.8090

Multiplying these values together:

sin 12°sin 48°sin 54° ≈ 0.2079 × 0.7431 × 0.8090 ≈ 0.1230

Conclusion

The value of sin 12°sin 48°sin 54° is approximately 0.1230. This numerical approximation confirms the complexity of exact value calculations but provides a practical answer in many real-world applications.