<p>Find the principal argument of <strong>(b)</strong> \(\dfrac{1+\sqrt{3}i}{\sqrt{3}+i}\)</p>
Step-by-Step Solution
Key Concept: Convert both complex numbers to polar form using r = √(a² + b²) and θ = tan⁻¹(b/a), then use the property that arg(z₁/z₂) = arg(z₁) - arg(z₂).
<p><strong>Step 1:</strong> Find argument of numerator (1 + √3i)</p><p>Here a = 1, b = √3, so tan(θ) = √3/1 = √3</p><p>Since both real and imaginary parts are positive (first quadrant), arg(1 + √3i) = π/3</p><p><strong>Step 2:</strong> Find argument of denominator (√3 + i)</p><p>Here a = √3, b = 1, so tan(θ) = 1/√3</p><p>Since both parts are positive (first quadrant), arg(√3 + i) = π/6</p><p><strong>Step 3:</strong> Apply quotient rule for arguments</p><p>arg(z₁/z₂) = arg(z₁) - arg(z₂)</p><p>arg[(1 + √3i)/(√3 + i)] = π/3 - π/6 = 2π/6 - π/6 = π/6</p><p>∴ <strong>Principal Argument = π/6</strong></p>
Correct Answer: π/6