Elucidate or Evaluate

$\begin{aligned} \Theta & = \arctan 1 + \arctan 2 + \arctan 3 \\ & = \arctan \left(\frac {1+2}{1-2} \right) + \arctan 3 \\ & = \arctan \left(-3\right) + \arctan 3 \\ & = \arctan \left(\frac {-3+3}{1+9} \right) \\ & = \arctan 0 & \small {\color{#3D99F6}\text{Since } 0 < \Theta < \frac {3\pi}2} \\ & = \boxed{\pi} \end{aligned}$

Let
${ z }_{ 1 }$ =1+i
${ z }_{ 2 }$ =1+2i
${ z }_{ 3 }$ =1+3i

Therefore

${ z }_{ 1 }$ = ${ r }_{ 1 }$ ${ e }^{ \arctan { 1 } }$

${ z }_{ 2 }$ = ${ r }_{ 2 }$ ${ e }^{ \arctan { 2 } }$

${ z }_{ 3 }$ = ${ r }_{ 3 }$ ${ e }^{ \arctan { 3 } }$

(1+i)(1+2i)(1+3i)=1+6i-11-6i

$\Rightarrow$ ${ z }_{ 1 }$ $\times$ ${ z }_{ 2 }$ $\times$ ${ z }_{ 3 }$ = -10 $\Rightarrow$ argument of[ ${ z }_{ 1 }$ $\times$ ${ z }_{ 2 }$ $\times$ ${ z }_{ 3 }$ ]=argument of (-10)

$\Rightarrow$ $\arctan { 1 }$ + $\arctan { 2 }$ + $\arctan { 3 }$ = $\pi$