<p>Find the principal argument of <strong>(a)</strong> \(-1 - i\sqrt{3}\)</p>
Step-by-Step Solution
Key Concept: The principal argument lies in (-π, π]. For -1 - i√3 in the third quadrant, use arg(z) = -π + arctan(√3/1) = -π + π/3 = -2π/3, not +4π/3.
<p><strong>Step 1:</strong> Identify the complex number: z = -1 - i√3</p><p><strong>Step 2:</strong> Determine the quadrant. Since Re(z) = -1 < 0 and Im(z) = -√3 < 0, the point lies in the <strong>third quadrant</strong>.</p><p><strong>Step 3:</strong> Calculate the reference angle: tan(θ) = |Im(z)|/|Re(z)| = √3/1 = √3, so the reference angle is π/3.</p><p><strong>Step 4:</strong> Apply quadrant adjustment. For the third quadrant, the principal argument is: arg(z) = -π + (π/3) = <strong>-2π/3</strong></p><p><strong>Verification:</strong> z = r(cos(-2π/3) + i·sin(-2π/3)) = 2(-1/2 - i√3/2) = -1 - i√3 ✓</p><p>∴ Answer: <strong>-2π/3</strong></p>
Correct Answer: -2π/3