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tg(x)>-(√3)/3

tg(x)>-(√3)/3 inequation

A inequation with variable

The solution

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

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