<p>The value of \(\sin 10°\sin 30°\sin 50°\sin 70°\) is __________ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Recognize that sin(90°-θ) = cos(θ), so sin 50° = cos 40° and sin 70° = cos 20°. Then use the product-to-sum formula strategically: sin A cos A = ½sin(2A), applied with complementary angle pairing to reduce the expression to a simple form.
<p><strong>Step 1:</strong> Use complementary angles: sin 70° = cos 20° and sin 50° = cos 40°</p><p>sin 10° sin 30° sin 50° sin 70° = sin 10° · (1/2) · cos 40° · cos 20°</p><p><strong>Step 2:</strong> Rearrange: = (1/2) · sin 10° cos 40° cos 20°</p><p><strong>Step 3:</strong> Use sin 10° cos 40° = (1/2)[sin(50°) + sin(-30°)] = (1/2)[sin 50° - 1/2]</p><p>Alternatively, use the identity: sin 10° sin 30° sin 50° sin 70° = sin 10° sin 50° · sin 30° sin 70°</p><p><strong>Step 4:</strong> Apply sin A sin(60°-A) sin(60°+A) = (1/4)sin(3A). With A = 10°:</p><p>sin 10° sin 50° sin 70° = sin 10° sin(60°-10°) sin(60°+10°) = (1/4)sin 30° = (1/4)(1/2) = 1/8</p><p><strong>Step 5:</strong> Therefore: sin 10° sin 30° sin 50° sin 70° = (1/8) · sin 30° = (1/8) · (1/2) = 1/16</p><p>∴ Answer: <strong>0.0625</strong></p>
Correct Answer: 0.0625