Trigonometry & Inverse Trigonometry
Product of Trigonometric Functions
Grade 11
Question:
<p>Find the value of \(\cos\dfrac{\pi}{2^2} \cdot \cos\dfrac{\pi}{2^3} \cdots \cos\dfrac{\pi}{2^{10}} \cdot \sin\dfrac{\pi}{2^{10}}\).</p>
<p>\(\dfrac{1}{256}\)</p>
<p>\(\dfrac{1}{512}\)</p>
<p>\(\dfrac{1}{1024}\)</p>
<p>\(\dfrac{1}{128}\)</p>
Step-by-Step Solution
Key Concept: Use the telescoping product formula sin(2θ) = 2sin(θ)cos(θ) repeatedly to collapse the product. Working backwards from sin(π/2^10), multiply by cos(π/2^10) to get sin(π/2^9)/2, then continue collapsing upward.
<p><strong>Step 1:</strong> Denote the product as P = cos(π/2²)·cos(π/2³)·...·cos(π/2¹⁰)·sin(π/2¹⁰)</p><p><strong>Step 2:</strong> Start from the rightmost terms and use sin(2θ) = 2sin(θ)cos(θ), so sin(θ)cos(θ) = sin(2θ)/2</p><p><strong>Step 3:</strong> sin(π/2¹⁰)·cos(π/2¹⁰) = sin(π/2⁹)/2</p><p><strong>Step 4:</strong> Now multiply by cos(π/2⁹): [sin(π/2⁹)/2]·cos(π/2⁹) = sin(π/2⁸)/(2²)</p><p><strong>Step 5:</strong> Continue this telescoping process. Each step multiplies by the next cosine term going left and divides by 2</p><p><strong>Step 6:</strong> After applying this 8 times (from 2¹⁰ up to 2²), we get: P = sin(π/2²)/(2⁸) = sin(π/4)/(2⁸)</p><p><strong>Step 7:</strong> sin(π/4) = 1/√2 = √2/2</p><p><strong>Step 8:</strong> P = (√2/2)/(2⁸) = √2/(2⁹) = √2/512 = <strong>1/(256√2) = √2/512</strong></p><p>∴ Answer: <strong>√2/512</strong> or equivalently <strong>1/(256√2)</strong></p>
Correct Answer: B