<p>Find the modulus, argument and the principal argument of the complex number \((\tan 1 - i)^2\).</p>
Step-by-Step Solution
Key Concept: Expand (tan 1 - i)² using (a - b)² = a² - 2ab + b², then convert to modulus-argument form using |z|² = (Real)² + (Imaginary)² and arg(z²) = 2·arg(z) carefully accounting for the principal argument range.
<p><strong>Step 1:</strong> Expand (tan 1 - i)²</p><p>(tan 1 - i)² = tan²1 - 2i·tan 1 + i²</p><p>= tan²1 - 2i·tan 1 - 1</p><p>= (tan²1 - 1) - 2i·tan 1</p><p><strong>Step 2:</strong> Find the modulus</p><p>|z|² = (tan²1 - 1)² + (-2tan 1)²</p><p>= (tan²1 - 1)² + 4tan²1</p><p>= tan⁴1 - 2tan²1 + 1 + 4tan²1</p><p>= tan⁴1 + 2tan²1 + 1</p><p>= (tan²1 + 1)²</p><p>= sec⁴1</p><p>∴ Modulus = sec²1</p><p><strong>Step 3:</strong> Find the argument</p><p>For z = (tan²1 - 1) - 2i·tan 1, let z₀ = tan 1 - i</p><p>arg(z₀) = arctan(-1/tan 1) = arctan(-cot 1)</p><p>Since tan 1 > 0 and Im(z₀) < 0, z₀ is in the fourth quadrant</p><p>arg(z₀) = -arctan(cot 1) = -(π/2 - 1) = 1 - π/2</p><p>Therefore, arg(z) = 2·arg(z₀) = 2(1 - π/2) = 2 - π</p><p><strong>Step 4:</strong> Verify principal argument</p><p>Since 2 - π ≈ -1.14 ∈ (-π, π], this is already the principal argument</p><p>∴ <strong>Modulus = sec²1, Argument = 2 - π, Principal Argument = 2 - π</strong></p>
Correct Answer: Modulus = sec²1, Argument = 2 - π