<p>Find the value of \(\sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ\).</p>
Step-by-Step Solution
Key Concept: Recognize that sin 10° sin 70° and sin 30° sin 50° are complementary angle pairs. Use the product-to-sum formula and the fact that sin 70° = cos 20° and sin 50° = cos 40° to simplify. The crucial insight is that sin 30° = 1/2, reducing the problem to finding sin 10° sin 50° sin 70°.
<p><strong>Step 1:</strong> Rewrite using complementary angles. Note that sin 70° = cos 20° and sin 50° = cos 40°.</p><p>sin 10° sin 30° sin 50° sin 70° = sin 10° · (1/2) · sin 50° · cos 20°</p><p><strong>Step 2:</strong> Use the product formula sin A cos B = (1/2)[sin(A+B) + sin(A-B)]. First, pair sin 50° cos 20°:</p><p>sin 50° cos 20° = (1/2)[sin 70° + sin 30°] = (1/2)[sin 70° + 1/2]</p><p><strong>Step 3:</strong> So the expression becomes: sin 10° · (1/2) · (1/2)[sin 70° + 1/2]</p><p>= (1/4) sin 10° [sin 70° + 1/2]</p><p>= (1/4) sin 10° sin 70° + (1/8) sin 10°</p><p><strong>Step 4:</strong> For sin 10° sin 70° = sin 10° cos 20°, use product formula:</p><p>sin 10° cos 20° = (1/2)[sin 30° + sin(-10°)] = (1/2)[1/2 - sin 10°]</p><p><strong>Step 5:</strong> Solving the system: Let x = sin 10° sin 70°. From the identity sin 10° sin 70° = (1/2)[1/2 - sin 10°], and combining all terms systematically yields:</p><p>sin 10° sin 30° sin 50° sin 70° = (1/16)</p><p>∴ Answer: 0.0625 = 1/16</p>
Correct Answer: 0.0625