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Solving inequalities/ cot(x)>-1/sqrt3

cot(x)>-1/sqrt3 inequation

A inequation with variable

The solution

You have entered [src] 
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cot(x) > -----
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cot⁡(x)>−13\cot{\left(x \right)} > - \frac{1}{\sqrt{3}} 
cot(x) > -1/sqrt(3)
Detail solution
Given the inequality:
cot⁡(x)>−13\cot{\left(x \right)} > - \frac{1}{\sqrt{3}}
To solve this inequality, we must first solve the corresponding equation:
cot⁡(x)=−13\cot{\left(x \right)} = - \frac{1}{\sqrt{3}}
Solve:
Given the equation
cot⁡(x)=−13\cot{\left(x \right)} = - \frac{1}{\sqrt{3}}
transform
cot⁡(x)−1+33=0\cot{\left(x \right)} - 1 + \frac{\sqrt{3}}{3} = 0
cot⁡(x)−1+13=0\cot{\left(x \right)} - 1 + \frac{1}{\sqrt{3}} = 0
Do replacement
w=cot⁡(x)w = \cot{\left(x \right)}
Expand brackets in the left part
-1 + w + 1/sqrt+1/3) = 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} < 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)>−13\cot{\left(x \right)} > - \frac{1}{\sqrt{3}}
cot⁡(−π3−110)>−13\cot{\left(- \frac{\pi}{3} - \frac{1}{10} \right)} > - \frac{1}{\sqrt{3}}
                   ___ 
    /1    pi\   -\/ 3  
-cot|-- + --| > -------
    \10   3 /      3

the solution of our inequality is:
x<−π3x < - \frac{\pi}{3}
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       x1
Rapid solution 2 [src] 
    2*pi 
(0, ----)
     3
x in (0,2π3)x\ in\ \left(0, \frac{2 \pi}{3}\right) 
x in Interval.open(0, 2*pi/3)
Rapid solution [src] 
   /           2*pi\
And|0 < x, x < ----|
   \            3  /
0<x∧x<2π30 < x \wedge x < \frac{2 \pi}{3} 
(0 < x)∧(x < 2*pi/3)
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