If [x] denotes the greatest integer le | vedclass

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Question

$\left[ {\sin 	heta } ight]x + \left[ { - \cos 	heta } ight]y = 0\,\,\,\,\,\,\,.......\left( 1 ight)$

$\left[ {\cot 	heta } ight]x + y =

Vedclass · Accepted Answer

have infinitely many solutions if $	heta  \in \left( {\frac{\pi }{2},\frac{{2\pi }}{3}} ight)$ and and has a unique solution if $	heta  \in \left( {\pi ,\frac{{7\pi }}{6}} ight)$

Vedclass · Answer

$\left[ {\sin 	heta } ight]x + \left[ { - \cos 	heta } ight]y = 0\,\,\,\,\,\,\,.......\left( 1 ight)$

$\left[ {\cot 	heta } ight]x + y = 0\,\,\,\,\,\,\,......\left( 2 ight)$

Case $I$

Whene $	heta  \in \left( {\frac{\pi }{2},\frac{{2\pi }}{3}} ight)$

$\sin 	heta  \in \left( {\frac{{\sqrt 3 }}{2},1} ight)$

$\cos 	heta  \in \left( { - \frac{1}{2},0} ight) - \cos 	heta  \in \left( {0,\frac{1}{2}} ight)$

$\cot 	heta  \in \left( { - \frac{1}{{\sqrt 3 }},0} ight)$

$\left[ {\sin 	heta } ight] = 0\,\,\,\,\,\left[ { - \cos 	heta } ight] = 0\,\,\,\,\,\left[ {\cot 	heta } ight] =  - 1$

Equation $(1)$ and $(2)$ will

$\left. \begin{array}{l}
0x + 0y = 0\
 - x + y = 0
\end{array} ight]$ ystem will have infinitely many solution 

Case $II$

When $	heta  \in \left( {\pi ,\frac{{7\pi }}{6}} ight)\,\,\sin 	heta  \in \left( { - \frac{1}{2},0} ight)$

$\cos 	heta  \in \left( { - 1,\frac{{\sqrt 3 }}{2}} ight)$

$\cot 	heta  \in \left( {\sqrt 3 ,\infty } ight)$

$\left[ {\sin 	heta } ight] =  - 1,\left[ {\cos 	heta } ight] =  - 1$

$\left[ {\cot 	heta } ight] = \left\{ {1,2,3,......} ight\}$

$-x-y=0$

$Ix+y=0$              I={1,2,.....}

It will have unique solution in all cases $x=0,y=0$

If $[x]$ denotes the greatest integer $ \leq x$, then the system of linear equations
$[sin \,\theta ] x + [-cos\,\theta ] y = 0$

$[cot \,\theta ] x + y = 0$

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