Complex Numbers
Properties of Conjugate
Grade 11
Question:
<p>If <i>a</i> = cos θ + <i>i</i> sin θ, then <span>\(\frac{1+a}{1-a}\)</span> is equal to</p>
<p>(a) <i>i</i> cot <span>\(\frac{\theta}{2}\)</span></p>
<p>(b) <i>i</i> tan <span>\(\frac{\theta}{2}\)</span></p>
<p>(c) <i>i</i> cos <span>\(\frac{\theta}{2}\)</span></p>
<p>(d) <i>i</i> cosec <span>\(\frac{\theta}{2}\)</span></p>
Step-by-Step Solution
Key Concept: Convert the complex number to trigonometric form using half-angle formulas and simplify using the conjugate multiplication technique.
<p><strong>Solution:</strong> Given, <i>a</i> = cos θ + <i>i</i> sin θ</p><p><span>$\frac{1+a}{1-a} = \frac{1 + \cos\theta + i\sin\theta}{1 - \cos\theta - i\sin\theta}$</span></p><p>Multiplying numerator and denominator by the conjugate of denominator:</p><p><span>$= \frac{(1 + \cos\theta + i\sin\theta)(1 - \cos\theta + i\sin\theta)}{(1 - \cos\theta - i\sin\theta)(1 - \cos\theta + i\sin\theta)}$</span></p><p><span>$= \frac{(1 + \cos\theta)^2 + 2i\sin\theta - \sin^2\theta}{(1 - \cos\theta)^2 + \sin^2\theta}$</span></p><p><span>$= \frac{4\sin\frac{\theta}{2}\cos\frac{\theta}{2}}{4\sin^2\frac{\theta}{2}}$</span></p><p><span>$= i\cot\frac{\theta}{2}$</span></p><p>∴ Answer is (a).</p>
Correct Answer: A