<s>[INST] What does the tilde ("$\sim$") mean in $\tan^{-1}\frac{M_1\sim M_2}{M_1+M_2}$? 
$$  \theta_{} \leq \pi/2  $$
$~ H_{x} ~$ can be positive or negative but $~ H_{y} ~$  can be assumed which takes only upward vertical value.
$$  \tan \left( \theta_{}  \right) = \frac{  H_{x}  }{  H_{y}  } = \frac{ \left( M_{1}-M_{2} \right) }{  \left( M_{1}+M_{2} \right)   }   $$
The problem has been provoked from the below equation.
$$  \therefore ~~  \left| \theta_{}  \right| = \tan^{-1} \left( \frac{  M_{1}  \sim  M_{2}  }{   \left( M_{1}+M_{2} \right)  }  \right)  $$
What it this "$~ \sim ~$"? misprint? Or a new conception for me? How should I interpret it ?
Moreover, I've been confused of bars of $~ \theta_{}  ~$ . How do I interpret it?
p.s
I have to go to bed in 10 minutes.
 [/INST] $$  \tan \left( -\theta_{}  \right) =-\tan \left( \theta_{}  \right)     $$
$$  \tan \left( \theta_{}  \right) =\frac{  M_{1}-M_{2}  }{ M_{1}+ M_{ 2 }    }  $$
$$ \therefore ~~ \theta_{} = \tan^{-1} \left( \frac{  M_{ 1 } -M_{ 2 }   }{  M_{ 1 } +M_{ 2 }   }  \right)  $$
$$
    \theta_{} = 
\begin{cases}
    \tan^{-1} \left( \frac{  M_{ 1 } -M_{ 2 }   }{  M_{ 1 } +M_{ 2 }   }  \right) &~  \left( M_{ 1 } \geq M_{ 2 } \right)     \\\\
    \tan^{-1} \left( \frac{  -\left| M_{ 1 } -M_{ 2 }  \right|   }{  M_{ 1 }+ M_{ 2 }    }  \right)  &~   \left( M_{ 1 } \leq M_{ 2 } \right) 
\end{cases}
$$
$$
    = 
\begin{cases}
    \tan^{-1} \left( \frac{ \left|  M_{ 1 } -M_{ 2 } \right|    }{  M_{ 1 } +M_{ 2 }   }  \right) &~  \left( M_{ 1 } \geq M_{ 2 } \right)     \\\\
    \tan^{-1} \left( \frac{  -\left| M_{ 1 } -M_{ 2 }  \right|   }{  M_{ 1 }+ M_{ 2 }    }  \right)  &~   \left( M_{ 1 } \leq M_{ 2 } \right) 
\end{cases}
$$
$$
    = 
\begin{cases}
    \tan^{-1} \left( \frac{ \left|  M_{ 1 } -M_{ 2 } \right|    }{  M_{ 1 } +M_{ 2 }   }  \right) &~  \left( M_{ 1 } \geq M_{ 2 } \right)     \\\\
    -\tan^{-1} \left( \frac{  \left| M_{ 1 } -M_{ 2 }  \right|   }{  M_{ 1 }+ M_{ 2 }    }  \right)  &~   \left( M_{ 1 } \leq M_{ 2 } \right) 
\end{cases}
$$
$$ = \pm \tan^{-1} \left( \frac{  \left| M_{ 1 } -M_{ 2 }  \right|   }{  M_{ 1 } + M_{ 2 }   }  \right)  $$
$$   \therefore ~~ \left| \theta_{}  \right| =\tan^{-1} \left( \frac{  \left| M_{ 1 } -M_{ 2 }  \right|   }{  M_{ 1 } + M_{ 2 }   }  \right)   $$
</s>