Proving the Trigonometric Identity: 16 sin20° × sin40° × sin60° × sin80° 3

Proving the Trigonometric Identity: 16 sin20° × sin40° × sin60° × sin80° 3

In this article, we will explore the steps to prove the trigonometric identity 16 sin20° × sin40° × sin60° × sin80° 3. We will use properties and identities of sine functions to simplify the expression and arrive at the desired result. Let's break down the proof into a series of logical steps.

Step 1: Using the Identity for Sine of Complementary Angles

Recall that sin80° cos10°. This allows us to rewrite the product:

16 sin20° × sin40° × sin60° × cos10°.

Step 2: Expressing sin60°

We know that sin60° frac{sqrt{3}}{2}. Substituting this in gives:

16 sin20° × sin40° × frac{sqrt{3}}{2} × cos10°.

This simplifies to:

8sqrt{3} sin20° × sin40° × cos10°.

Step 3: Utilizing Product-to-Sum Formulas

We can use the product-to-sum identity:

sinA sinB frac{1}{2} [cos(A - B) - cos(A B)]

Applying this identity to sin20° × sin40°:

sin20° × sin40° frac{1}{2} [cos(20° - 40°) - cos(20° 40°)] frac{1}{2} [cos(-20°) - cos60°] frac{1}{2} [cos20° - frac{1}{2}].

Thus, we can rewrite the expression:

8sqrt{3} × frac{1}{2} [cos20° - frac{1}{2}] × cos10°.

This simplifies to:

4sqrt{3} (cos20° - frac{1}{2}) × cos10°.

Step 4: Expansion and Simplification

Now, let's expand this expression:

4sqrt{3} cos20° cos10° - 2sqrt{3} cos10°.

Using the product-to-sum formulas again for cos20° × cos10°:

cos20° cos10° frac{1}{2} [cos(20° - 10°) cos(20° 10°)] frac{1}{2} [cos10° cos30°] frac{1}{2} [cos10° frac{sqrt{3}}{2}].

Substituting this in gives:

4sqrt{3} × frac{1}{2} (cos10° frac{sqrt{3}}{2}) - 2sqrt{3} cos10° 2sqrt{3} cos10° 3 - 2sqrt{3} cos10°.

This simplifies to:

3.

Step 5: Final Calculation

Now, we can use numerical values to verify the expression:

sin20° ≈ 0.3420 sin40° ≈ 0.6428 sin60° frac{sqrt{3}}{2} ≈ 0.8660 sin80° ≈ 0.9848

Calculating the product:

16 × 0.3420 × 0.6428 × 0.8660 × 0.9848 ≈ 3.

Through both algebraic manipulation and numerical approximation, we can conclude:

boxed{3}.

Keywords: trigonometric identity, algebraic manipulation, product-to-sum formulas