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tg(x)>-(√3)/3
Solving inequalities/ tg(x)>-(√3)/3

tg(x)>-(√3)/3 inequation

A inequation with variable

The solution

You have entered [src] 
            ___ 
         -\/ 3  
tan(x) > -------
            3
tan⁡(x)>(−1)33\tan{\left(x \right)} > \frac{\left(-1\right) \sqrt{3}}{3} 
tan(x) > -sqrt(3)/3
Detail solution
Given the inequality:
tan⁡(x)>(−1)33\tan{\left(x \right)} > \frac{\left(-1\right) \sqrt{3}}{3}
To solve this inequality, we must first solve the corresponding equation:
tan⁡(x)=(−1)33\tan{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}
Solve:
Given the equation
tan⁡(x)=(−1)33\tan{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}
- this is the simplest trigonometric equation
This equation is transformed to
x=πn+atan⁡(−33)x = \pi n + \operatorname{atan}{\left(- \frac{\sqrt{3}}{3} \right)}
Or
x=πn−π6x = \pi n - \frac{\pi}{6}
, where n - is a integer
x1=πn−π6x_{1} = \pi n - \frac{\pi}{6}
x1=πn−π6x_{1} = \pi n - \frac{\pi}{6}
This roots
x1=πn−π6x_{1} = \pi n - \frac{\pi}{6}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x1x_{0} < x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
(πn−π6)−110\left(\pi n - \frac{\pi}{6}\right) - \frac{1}{10}
=
πn−π6−110\pi n - \frac{\pi}{6} - \frac{1}{10}
substitute to the expression
tan⁡(x)>(−1)33\tan{\left(x \right)} > \frac{\left(-1\right) \sqrt{3}}{3}
tan⁡(πn−π6−110)>(−1)33\tan{\left(\pi n - \frac{\pi}{6} - \frac{1}{10} \right)} > \frac{\left(-1\right) \sqrt{3}}{3}
                   ___ 
    /1    pi\   -\/ 3  
-tan|-- + --| > -------
    \10   6 /      3

Then
x<πn−π6x < \pi n - \frac{\pi}{6}
no execute
the solution of our inequality is:
x>πn−π6x > \pi n - \frac{\pi}{6}
         _____  
        /
-------ο-------
       x_1
Solving inequality on a graph
0-80-60-40-2020406080-2500025000
Rapid solution [src] 
  /   /            pi\     /5*pi            \\
Or|And|0 <= x, x < --|, And|---- < x, x < pi||
  \   \            2 /     \ 6              //
(0≤x∧x<π2)∨(5π6<x∧x<π)\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(\frac{5 \pi}{6} < x \wedge x < \pi\right) 
((0 <= x)∧(x < pi/2))∨((x < pi)∧(5*pi/6 < x))
Rapid solution 2 [src] 
    pi     5*pi     
[0, --) U (----, pi)
    2       6
x in [0,π2)∪(5π6,π)x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left(\frac{5 \pi}{6}, \pi\right) 
x in Union(Interval.Ropen(0, pi/2), Interval.open(5*pi/6, pi))
The graph
tg(x)>-(√3)/3 inequation
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