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tan(x/2)=-√3/3 equation

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Numerical solution:

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The solution

You have entered [src]
            ___ 
   /x\   -\/ 3  
tan|-| = -------
   \2/      3   
$$\tan{\left(\frac{x}{2} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Detail solution
Given the equation
$$\tan{\left(\frac{x}{2} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{atan}{\left(- \frac{\sqrt{3}}{3} \right)}$$
Or
$$\frac{x}{2} = \pi n - \frac{\pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n - \frac{\pi}{3}$$
The graph
Rapid solution [src]
     -pi 
x1 = ----
      3  
$$x_{1} = - \frac{\pi}{3}$$
x1 = -pi/3
Sum and product of roots [src]
sum
-pi 
----
 3  
$$- \frac{\pi}{3}$$
=
-pi 
----
 3  
$$- \frac{\pi}{3}$$
product
-pi 
----
 3  
$$- \frac{\pi}{3}$$
=
-pi 
----
 3  
$$- \frac{\pi}{3}$$
-pi/3
Numerical answer [src]
x1 = 30.3687289847013
x2 = 36.6519142918809
x3 = 55.5014702134197
x4 = 74.3510261349584
x5 = -51.3126800086333
x6 = -45.0294947014537
x7 = 99.4837673636768
x8 = -19.8967534727354
x9 = -13.6135681655558
x10 = 61.7846555205993
x11 = 11.5191730631626
x12 = 93.2005820564972
x13 = 80.634211442138
x14 = -38.7463093942741
x15 = -76.4454212373516
x16 = -95.2949771588904
x17 = -57.5958653158129
x18 = 24.0855436775217
x19 = 5.23598775598299
x20 = -32.4631240870945
x21 = -63.8790506229925
x22 = -26.1799387799149
x23 = -70.162235930172
x24 = -101.57816246607
x25 = 17.8023583703422
x26 = -7.33038285837618
x27 = 49.2182849062401
x28 = -89.0117918517108
x29 = -1.0471975511966
x30 = 68.0678408277789
x31 = 86.9173967493176
x32 = -82.7286065445312
x33 = 42.9350995990605
x33 = 42.9350995990605