The value of \(arg (\frac{(1\ + \ \sqrt 3 i)(\sqrt 3\ + \ i)}{1\ +\ i})\) will be:

CONCEPT:

Complex Numbers: For a complex number z = a + ib, the following are defined:

arg(z) = θ = \(\rm\tan^{-1}\left(b\over a\right)\).

Properties of argument:

Let z, z1 and z2 be three complex numbers. Then

1. \(\arg \left( {\bar z\;} \right) = \; - \arg z\)

2. arg (z1 ⋅ z2) = arg (z1) + arg (z2)

3. \(\arg \left( {{z_1} \cdot \overline {{z_2}} } \right) = \arg {z_1} - \arg {z_2}\)

4. arg (zn) = n ⋅ arg (z)

5. arg (z1 / z2) = arg (z1) – arg(z2)

6. The argument of zero is not defined

7. If arg (z) = 0 ⇒ z is real

Calculation:

Let,

\(arg (\frac{(1\ + \ √ 3 i)(√ 3\ + \ i)}{1\ +\ i})\ =\ θ \)    ----(1)

∵ arg (z1 / z2) = arg (z1) – arg(z2)

⇒ θ = arg(1 + √3i)(√3 + i) - arg(1 + i) 

∵ arg (z1 ⋅ z2) = arg (z1) + arg (z2)

⇒ θ = arg(1 + √3i) + arg(√3 + i) - arg(1 + i)

\(\Rightarrow\ \theta \ = \ tan^{-1}(\frac{\sqrt 3}{1})\ +\ tan^{-1}(\frac{1}{\sqrt 3})\ -\ tan^{-1}(1)\)

\(\Rightarrow\ \theta \ = \ (\frac{\pi}{3})\ +\ (\frac{\pi}{6})\ -\ (\frac{\pi}{4})\)

\(\Rightarrow\ \theta \ = \ (\frac{\pi}{4})\)

Hence, option 3  is correct.