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tan(x/3+pi/6)=-1 equation

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Numerical solution:

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The solution

You have entered [src]
   /x   pi\     
tan|- + --| = -1
   \3   6 /     
$$\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = -1$$
Detail solution
Given the equation
$$\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = -1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(-1 \right)}$$
Or
$$\frac{x}{3} + \frac{\pi}{6} = \pi n - \frac{\pi}{4}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = \pi n - \frac{5 \pi}{12}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 3 \pi n - \frac{5 \pi}{4}$$
The graph
Rapid solution [src]
     -5*pi
x1 = -----
       4  
$$x_{1} = - \frac{5 \pi}{4}$$
x1 = -5*pi/4
Sum and product of roots [src]
sum
-5*pi
-----
  4  
$$- \frac{5 \pi}{4}$$
=
-5*pi
-----
  4  
$$- \frac{5 \pi}{4}$$
product
-5*pi
-----
  4  
$$- \frac{5 \pi}{4}$$
=
-5*pi
-----
  4  
$$- \frac{5 \pi}{4}$$
-5*pi/4
Numerical answer [src]
x1 = -60.4756585816035
x2 = 52.621676947629
x3 = 90.3207887907066
x4 = -79.3252145031423
x5 = 43.1968989868597
x6 = -88.7499924639117
x7 = -98.174770424681
x8 = -41.6261026600648
x9 = 14.9225651045515
x10 = -3.92699081698724
x11 = -51.0508806208341
x12 = -69.9004365423729
x13 = -13.3517687777566
x14 = -32.2013246992954
x15 = 24.3473430653209
x16 = 80.8960108299372
x17 = -22.776546738526
x18 = 71.4712328691678
x19 = 62.0464549083984
x20 = 99.7455667514759
x21 = 5.49778714378214
x22 = 33.7721210260903
x22 = 33.7721210260903