Let $S=\{(\theta \sin \frac{\pi \theta }{1+\theta },\frac{1}{\theta }\cos \frac{\pi \theta }{1+\theta });\theta \in R,\theta >0\}$  and $T=\{(x,y);x\in R,y\in R,xy=\frac{1}{2}\}$  Then the number of elements in $S\cap T$  is ___

Assume that   $(x,y)\in S\cap T,$then for some real   $\theta >0,$$x=\theta \sin \frac{\pi \theta }{1+\theta },y=\frac{1}{\theta }\cos \frac{\pi \theta }{1+\theta }$
And it follows that  $\frac{1}{2}=xy=\sin \frac{\pi \theta }{1+\theta }\cos \frac{\pi \theta }{1+\theta }$
The latter equality is true if and only if  $1=2\sin \frac{\pi \theta }{1+\theta }\cos \frac{\pi \theta }{1+\theta }=\sin \frac{2\pi \theta }{1+\theta }$
                                                                              $\frac{2\pi \theta }{1+\theta }=\frac{\pi }{2}+2\pi k$
For some integer k, equivalently $k=-\frac{1}{4}+\frac{\theta }{1+\theta }$
Since  $0<\frac{\theta }{1+\theta }<1,$ we get  $-\frac{1}{4}<k<\frac{3}{4}.$
The only such integer is k = 0. Hence,  $\frac{\theta }{1+\theta }=\frac{1}{4},\theta =\frac{1}{3}$
There is only one element in $S\cap T,$ namely $(\frac{1}{3}\sin \frac{\pi }{4},3\cos \frac{\pi }{4}).$