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(sqrt3*tgx+1)/(2*sinx-1)=0 equation

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Numerical solution:

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The solution

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  ___               
\/ 3 *tan(x) + 1    
---------------- = 0
  2*sin(x) - 1      
$$\frac{\sqrt{3} \tan{\left(x \right)} + 1}{2 \sin{\left(x \right)} - 1} = 0$$
The graph
Sum and product of roots [src]
sum
-pi 
----
 6  
$$- \frac{\pi}{6}$$
=
-pi 
----
 6  
$$- \frac{\pi}{6}$$
product
-pi 
----
 6  
$$- \frac{\pi}{6}$$
=
-pi 
----
 6  
$$- \frac{\pi}{6}$$
-pi/6
Rapid solution [src]
     -pi 
x1 = ----
      6  
$$x_{1} = - \frac{\pi}{6}$$
x1 = -pi/6
Numerical answer [src]
x1 = -75.9218224617533
x2 = 445.582558034152
x3 = 30.8923277602996
x4 = -1684.41726109973
x5 = -38.2227106186758
x6 = 7099.47579833733
x7 = 1959.83021706443
x8 = -57.0722665402146
x9 = 18.3259571459405
x10 = -0.523598775598299
x11 = -44.5058959258554
x12 = -113.620934304831
x13 = -94.7713783832921
x14 = -82.2050077689329
x15 = -6.80678408277789
x16 = -157.603231455088
x17 = -25.6563400043166
x18 = 74.8746249105567
x19 = 37.1755130674792
x20 = 12.0427718387609
x21 = 24.60914245312
x22 = -88.4881930761125
x23 = -50.789081233035
x24 = 49.7418836818384
x25 = 43.4586983746588
x26 = 62.3082542961976
x27 = -748.222650329969
x28 = 68.5914396033772
x29 = -13.0899693899575
x30 = 100.007366139275
x31 = -19.3731546971371
x32 = -69.6386371545737
x33 = 93.7241808320955
x34 = 81.1578102177363
x35 = -63.3554518473942
x36 = -729.37309440843
x37 = 87.4409955249159
x38 = -31.9395253114962
x39 = 56.025068989018
x40 = 5.75958653158129
x41 = -207.868713912525
x41 = -207.868713912525