Minimum value of this particular trigonomterical function : really interesting

First take a look at the problem :

Find the minimum value of the trigonometrical function : $\displaystyle y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2}$

If we proceed by perfect-square method , we get ,

$\displaystyle {(2\sin x - 3/ \ sin x)}^{ 2} \geqslant 0 $

=> $\displaystyle {(2\sin x - 3/ \ sin x)}^{ 2} + 12 \geqslant 12 $

=> $\displaystyle {(2\sin x - 3/ \ sin x)}^{ 2} + 2 \cdot 2\sin x \cdot 3 (1/ \sin x) \geqslant 12 $

Therefore , $\displaystyle y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2} \geqslant 12 $

But this minimum value will be achieved if and only if ,

$\displaystyle {(2\sin x - 3/ \ sin x)}^{ 2} = 0 $

which when solved gives $\displaystyle \sin x = 1.2247 $ ( approximately ) , which is certainly not possible since maximum value of $\displaystyle \sin x $ is 1

Does this mean that the function $\displaystyle y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2}$ does not have any minimum value .

But if we plot the graph of $\displaystyle y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2}$ , it looks like it has a minimum value of 13 .

Please analyze and explain !