<p>Find the modulus, argument, and the principal argument of the complex number \[\dfrac{i-1}{i\left(1 - \cos\dfrac{2\pi}{5}\right) + \sin\dfrac{2\pi}{5}}\]</p>
Step-by-Step Solution
Key Concept: Recognize the denominator as a complex number in the form (sinθ + i(1-cosθ)) which can be rewritten using the identity 1-cosθ = 2sin²(θ/2) and sinθ = 2sin(θ/2)cos(θ/2), then simplify to exponential form e^(iθ/2).
<p><strong>Step 1:</strong> Rewrite the numerator: i - 1 = -1 + i = √2 · e^(i3π/4)</p><p><strong>Step 2:</strong> Simplify the denominator using identities:<br/>• 1 - cos(2π/5) = 2sin²(π/5)<br/>• sin(2π/5) = 2sin(π/5)cos(π/5)<br/>Denominator = i·2sin²(π/5) + 2sin(π/5)cos(π/5) = 2sin(π/5)[cos(π/5) + i·sin(π/5)] = 2sin(π/5)·e^(iπ/5)</p><p><strong>Step 3:</strong> Divide the complex numbers:<br/>z = (√2·e^(i3π/4))/(2sin(π/5)·e^(iπ/5))</p><p><strong>Step 4:</strong> Calculate modulus:<br/>|z| = √2/(2sin(π/5)) = (1/√2)·cosec(π/5)</p><p><strong>Step 5:</strong> Calculate argument:<br/>arg(z) = 3π/4 - π/5 = 15π/20 - 4π/20 = 11π/20</p><p>∴ <strong>Modulus:</strong> (1/√2)cosec(π/5), <strong>Argument:</strong> 11π/20</p>
Correct Answer: Modulus = (1/√2)cosec(π/5), argument = 11π/20