<p>Find the principal argument of <strong>(d)</strong> \((1+i\sqrt{3})^2\)</p>
Step-by-Step Solution
Key Concept: First simplify the complex number by expanding the square, then identify its position in the complex plane (real and imaginary parts) to find the argument using tan⁻¹(imaginary/real).
<p><strong>Step 1:</strong> Expand (1+i√3)²</p><p>(1+i√3)² = 1 + 2i√3 + (i√3)² = 1 + 2i√3 + i²·3 = 1 + 2i√3 - 3 = -2 + 2i√3</p><p><strong>Step 2:</strong> Identify the complex number in the form a + ib where a = -2, b = 2√3</p><p>Since a < 0 and b > 0, the complex number lies in the second quadrant.</p><p><strong>Step 3:</strong> Apply the argument formula for second quadrant</p><p>tan(θ) = b/a = (2√3)/(-2) = -√3</p><p>For second quadrant: arg(z) = π - tan⁻¹(|b/a|) = π - tan⁻¹(√3) = π - π/3 = 2π/3</p><p><strong>Verification:</strong> The modulus is |z| = √(4 + 12) = 4, and 4(cos(2π/3) + i·sin(2π/3)) = 4(-1/2 + i√3/2) = -2 + 2i√3 ✓</p><p>∴ Principal argument = <strong>2π/3</strong></p>
Correct Answer: 2π/3