A) \[{{\sin }^{-1}}\left( \tan \,i \right)\]
B) \[ta{{n}^{-1}}\left( sin\,i \right)\]
C) \[si{{n}^{-1}}\left( \cot \,i \right)\]
D) \[{{\cos }^{-1}}\left( \tan \,i \right)\]
Correct Answer: C
Solution :
From law of reflection, \[\angle i=\angle r\] ...(i) and \[\frac{\sin r}{\sin i}=\frac{{{\mu }_{d}}}{{{\mu }_{r}}}\] ...(ii) From the figure \[r+r+{{90}^{o}}={{180}^{o}}\] \[\Rightarrow \] \[r+r={{90}^{o}}\] or \[i+r={{90}^{o}}\] \[r=({{90}^{o}}-i)\] ?(iii) From Eq. (ii)\[\frac{\sin ({{90}^{o}}-i)}{\sin i}=\frac{{{\mu }_{d}}}{{{\mu }_{r}}}\] Or \[\frac{\cos i}{\sin i}=\frac{{{\mu }_{d}}}{{{\mu }_{r}}}\Rightarrow \cot i=\frac{{{\mu }_{d}}}{{{\mu }_{r}}}\] But \[\frac{{{\mu }_{d}}}{{{\mu }_{r}}}=\sin C\] (where C is critical angle) \[\therefore \]\[\cot i=\sin C\Rightarrow C={{\sin }^{-1}}(coti)\]You need to login to perform this action.
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