Complex Numbers
Argument of complex numbers
Grade 11
Question:
<p><strong>For Problems 1–4</strong><br>Consider the complex numbers \(z = (1 - i\sin\theta)/(1 + i\cos\theta)\).</p><p><strong>Problem 4.</strong> If argument of \(z\) is \(\pi/4\), then</p>
<p>(1) \(\theta = n\pi,\, n \in I\) only</p>
<p>(2) \(\theta = (2n+1)\dfrac{\pi}{2},\, n \in I\) only</p>
<p>(3) both \(\theta = n\pi\) and \(\theta = (2n+1)\dfrac{\pi}{2},\, n \in I\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: To find θ when arg(z) = π/4, multiply numerator and denominator by the conjugate of the denominator, then use the condition that arg(z) = arg(numerator) - arg(denominator) = π/4 by comparing imaginary and real parts of the simplified form.
<p><strong>Step 1:</strong> Rationalize by multiplying by conjugate (1 - i cos θ)/(1 - i cos θ):</p><p>z = [(1 - i sin θ)(1 - i cos θ)] / [(1 + i cos θ)(1 - i cos θ)]</p><p><strong>Step 2:</strong> Expand numerator: (1 - i sin θ)(1 - i cos θ) = 1 - i cos θ - i sin θ + i² sin θ cos θ = (1 - sin θ cos θ) - i(sin θ + cos θ)</p><p><strong>Step 3:</strong> Expand denominator: (1 + i cos θ)(1 - i cos θ) = 1 + cos² θ</p><p><strong>Step 4:</strong> Therefore: z = [(1 - sin θ cos θ) - i(sin θ + cos θ)] / (1 + cos² θ)</p><p><strong>Step 5:</strong> For arg(z) = π/4, we need Re(z) = Im(z), which means:<br>(1 - sin θ cos θ) = (sin θ + cos θ)</p><p><strong>Step 6:</strong> Let u = sin θ + cos θ. Then sin θ cos θ = (u² - 1)/2<br>Substituting: 1 - (u² - 1)/2 = u<br>⟹ (3 - u²)/2 = u<br>⟹ u² + 2u - 3 = 0<br>⟹ (u + 3)(u - 1) = 0</p><p><strong>Step 7:</strong> Since sin θ + cos θ ≤ √2, we have u = 1, so sin θ + cos θ = 1<br>This gives θ = 0, π/2, or solve from √2 sin(θ + π/4) = 1</p><p>∴ Answer: C</p>
Correct Answer: C