Question

The polar form of the complex number \(z = \frac{1}{1+i}\), (where \(i = \sqrt{-1}\)) is

• A. \(\frac{1}{\sqrt{2}}(cos\frac{7π}{4}+isin\frac{7π}{4})\)
• B. \(\frac{1}{\sqrt{2}}(cos\frac{π}{4}+isin\frac{π}{4})\)
• C. \(\frac{1}{2}(cos\frac{π}{4}+isin\frac{π}{4})\)
• D. \(\frac{1}{2}(cos\frac{7π}{4}+isin\frac{7π}{4})\)

Hint:

Rationalise to get the real and imaginary parts, then find modulus and argument, remembering the quadrant.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The polar form of \(z = x + iy\) is \(r(\cos\theta + i\sin\theta)\), where \(r = |z|\) and \(\theta\) is the argument, chosen according to the quadrant of \((x, y)\).

Step 2: Step to a + ib form.
Multiply top and bottom by the conjugate of the denominator.
\[ z = \frac{1}{1+i} \times \frac{1-i}{1-i} = \frac{1-i}{2} = \frac{1}{2} - \frac{1}{2}i \]

Step 3: Find the modulus.
\[ r = \sqrt{\left(\tfrac{1}{2}\right)^2 + \left(\tfrac{1}{2}\right)^2} = \sqrt{\tfrac{1}{2}} = \frac{1}{\sqrt{2}} \]

Step 4: Find the argument.
The real part is positive and the imaginary part is negative, so \(z\) lies in the fourth quadrant. The reference angle is \(\tan^{-1}(1) = \pi/4\), so \(\theta = 2\pi - \pi/4 = 7\pi/4\) (or \(-\pi/4\)).

Step 5: Check the options.
Options (B) and (C) use \(\pi/4\), which is the first quadrant. Options (C) and (D) use the modulus \(1/2\), which is wrong. Only (A) has both \(1/\sqrt{2}\) and \(7\pi/4\).

Final Answer:
In polar form \(z = \dfrac{1}{\sqrt{2}}\left(\cos\dfrac{7\pi}{4} + i\sin\dfrac{7\pi}{4}\right)\), option (A). \[ \boxed{\frac{1}{\sqrt{2}}\left(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}\right)} \]