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Solving inequalities/ cot(x)<=-√3/3

cot(x)<=-√3/3 inequation

A inequation with variable

The solution

You have entered [src] 
             ___ 
          -\/ 3  
cot(x) <= -------
             3
cot⁡(x)≤(−1)33\cot{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3} 
cot(x) <= (-sqrt(3))/3
Detail solution
Given the inequality:
cot⁡(x)≤(−1)33\cot{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}
To solve this inequality, we must first solve the corresponding equation:
cot⁡(x)=(−1)33\cot{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}
Solve:
Given the equation
cot⁡(x)=(−1)33\cot{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}
transform
cot⁡(x)−1+33=0\cot{\left(x \right)} - 1 + \frac{\sqrt{3}}{3} = 0
cot⁡(x)−1−(−1)33=0\cot{\left(x \right)} - 1 - \frac{\left(-1\right) \sqrt{3}}{3} = 0
Do replacement
w=cot⁡(x)w = \cot{\left(x \right)}
Expand brackets in the left part
-1 + w - -sqrt+3)/3 = 0

Move free summands (without w)
from left part to right part, we given:
w+33=1w + \frac{\sqrt{3}}{3} = 1
Divide both parts of the equation by (w + sqrt(3)/3)/w
w = 1 / ((w + sqrt(3)/3)/w)

We get the answer: w = 1 - sqrt(3)/3
do backward replacement
cot⁡(x)=w\cot{\left(x \right)} = w
substitute w:
x1=−π3x_{1} = - \frac{\pi}{3}
x1=−π3x_{1} = - \frac{\pi}{3}
This roots
x1=−π3x_{1} = - \frac{\pi}{3}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0≤x1x_{0} \leq x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−π3−110- \frac{\pi}{3} - \frac{1}{10}
=
−π3−110- \frac{\pi}{3} - \frac{1}{10}
substitute to the expression
cot⁡(x)≤(−1)33\cot{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}
cot⁡(−π3−110)≤(−1)33\cot{\left(- \frac{\pi}{3} - \frac{1}{10} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}
                    ___ 
    /1    pi\    -\/ 3  
-cot|-- + --| <= -------
    \10   3 /       3

but
                    ___ 
    /1    pi\    -\/ 3  
-cot|-- + --| >= -------
    \10   3 /       3

Then
x≤−π3x \leq - \frac{\pi}{3}
no execute
the solution of our inequality is:
x≥−π3x \geq - \frac{\pi}{3}
         _____  
        /
-------•-------
       x1
Rapid solution [src] 
   /2*pi             \
And|---- <= x, x < pi|
   \ 3               /
2π3≤x∧x<π\frac{2 \pi}{3} \leq x \wedge x < \pi 
(x < pi)∧(2*pi/3 <= x)
Rapid solution 2 [src] 
 2*pi     
[----, pi)
  3
x in [2π3,π)x\ in\ \left[\frac{2 \pi}{3}, \pi\right) 
x in Interval.Ropen(2*pi/3, pi)
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