Correct Answer - Option 3 : 240°
Concept:
The argument of a complex number z = x + iy
arg(z) = tan-1\(\rm \left(y\over x\right)\)
The angle is according to the sign of the y and x
- Both positive then angle ∈ [0°, 90°]
- Negative x and positive y then angle ∈ [90°, 180°]
- Negative x and negative y then angle ∈ [180°, 270°]
- Positive x and negative y then angle ∈ [270°, 360°]
Calculation:
Given that:
\(\rm \frac{1 - i√3}{1 + i√3}\)
Multiply numerator and denominator both by (1 - i√3), we get
\(\rm (\frac{1 - i√3}{1 + i√3}) \times (\frac {1 - i√3}{1 - i √3})\) = \(\rm \frac{(1 - i√3)^2}{(1)^2 - (i√3)^2}\) = \(\rm \frac{[1 + (i√3)^2 - 2i√3]}{1 + 3}\) = \(\rm \frac{(- 2 - 2i√3]}{4}\) = \(\rm \frac{-1}{2} - \frac{√3}{2}i\)
For a complex number, Z = x + yi
Argument (θ) = tan-1(y/x)
Here, x = \(\rm -1\over2\) and y = \(\rm -{√3\over2}i\)
Argument (θ) = tan-1(y/x) = tan-1\(\rm \frac {(\frac{√3}{2})}{(\frac{1}{2})}\) = tan-1\(\rm (√ 3)\)
Here θ = 60° But x and y is negative so the angle lies in 3rd quardrant
argument (θ) = 180° + 60° = 240°