Complex Numbers
Argument and Modulus
Grade None
Question:
<p>If <i>z</i> is a complex number of unit modulus and argument <i>θ</i>, then <i>arg</i> <i>(1 + z)/(1 + z̄)</i> equals to</p>
<p>(a) <i>π</i>/2 - <i>θ</i></p>
<p>(b) <i>θ</i></p>
<p>(c) <i>π</i> - <i>θ</i></p>
<p>(d) -<i>θ</i></p>
Step-by-Step Solution
Key Concept: Use the exponential form of complex numbers with unit modulus to simplify the ratio and find its argument.
<p><strong>Solution:</strong> Since <i>z</i> has unit modulus and argument <i>θ</i>, we have <i>z</i> = <i>e</i><sup><i>iθ</i></sup>. Therefore <i>z̄</i> = <i>e</i><sup>-<i>iθ</i></sup>.</p><p>1 + <i>z</i> = 1 + <i>e</i><sup><i>iθ</i></sup> = <i>e</i><sup><i>iθ</i>/2</sup>(2cos(<i>θ</i>/2))</p><p>1 + <i>z̄</i> = 1 + <i>e</i><sup>-<i>iθ</i></sup> = <i>e</i><sup>-<i>iθ</i>/2</sup>(2cos(<i>θ</i>/2))</p><p><i>(1 + z)/(1 + z̄)</i> = <i>e</i><sup><i>iθ</i></sup></p><p>∴ <i>arg</i> ((1 + <i>z</i>)/(1 + <i>z̄</i>)) = <i>θ</i></p>
Correct Answer: A