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Solving inequalities/ sin(3*x)>-sqrt(3)/2

sin(3*x)>-sqrt(3)/2 inequation

A inequation with variable

The solution

You have entered [src] 
              ___ 
           -\/ 3  
sin(3*x) > -------
              2
sin⁡(3x)>(−1)32\sin{\left(3 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2} 
sin(3*x) > (-sqrt(3))/2
Detail solution
Given the inequality:
sin⁡(3x)>(−1)32\sin{\left(3 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(3x)=(−1)32\sin{\left(3 x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
Solve:
Given the equation
sin⁡(3x)=(−1)32\sin{\left(3 x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
- this is the simplest trigonometric equation
This equation is transformed to
3x=2πn+asin⁡(−32)3 x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}
3x=2πn−asin⁡(−32)+π3 x = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi
Or
3x=2πn−π33 x = 2 \pi n - \frac{\pi}{3}
3x=2πn+4π33 x = 2 \pi n + \frac{4 \pi}{3}
, where n - is a integer
Divide both parts of the equation by
33
x1=2πn3−π9x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{9}
x2=2πn3+4π9x_{2} = \frac{2 \pi n}{3} + \frac{4 \pi}{9}
x1=2πn3−π9x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{9}
x2=2πn3+4π9x_{2} = \frac{2 \pi n}{3} + \frac{4 \pi}{9}
This roots
x1=2πn3−π9x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{9}
x2=2πn3+4π9x_{2} = \frac{2 \pi n}{3} + \frac{4 \pi}{9}
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}
=
(2πn3−π9)+−110\left(\frac{2 \pi n}{3} - \frac{\pi}{9}\right) + - \frac{1}{10}
=
2πn3−π9−110\frac{2 \pi n}{3} - \frac{\pi}{9} - \frac{1}{10}
substitute to the expression
sin⁡(3x)>(−1)32\sin{\left(3 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
sin⁡(3(2πn3−π9−110))>(−1)32\sin{\left(3 \left(\frac{2 \pi n}{3} - \frac{\pi}{9} - \frac{1}{10}\right) \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
                            ___ 
    /3    pi         \   -\/ 3  
-sin|-- + -- - 2*pi*n| > -------
    \10   3          /      2

Then
x<2πn3−π9x < \frac{2 \pi n}{3} - \frac{\pi}{9}
no execute
one of the solutions of our inequality is:
x>2πn3−π9∧x<2πn3+4π9x > \frac{2 \pi n}{3} - \frac{\pi}{9} \wedge x < \frac{2 \pi n}{3} + \frac{4 \pi}{9}
         _____  
        /     \  
-------ο-------ο-------
       x1      x2
Solving inequality on a graph
0-80-60-40-20204060802-2
Rapid solution [src] 
  /   /            4*pi\     /     2*pi  5*pi    \\
Or|And|0 <= x, x < ----|, And|x <= ----, ---- < x||
  \   \             9  /     \      3     9      //
(0≤x∧x<4π9)∨(x≤2π3∧5π9<x)\left(0 \leq x \wedge x < \frac{4 \pi}{9}\right) \vee \left(x \leq \frac{2 \pi}{3} \wedge \frac{5 \pi}{9} < x\right) 
((0 <= x)∧(x < 4*pi/9))∨((x <= 2*pi/3)∧(5*pi/9 < x))
Rapid solution 2 [src] 
    4*pi     5*pi  2*pi 
[0, ----) U (----, ----]
     9        9     3
x in [0,4π9)∪(5π9,2π3]x\ in\ \left[0, \frac{4 \pi}{9}\right) \cup \left(\frac{5 \pi}{9}, \frac{2 \pi}{3}\right] 
x in Union(Interval.Ropen(0, 4*pi/9), Interval.Lopen(5*pi/9, 2*pi/3))
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