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Solving inequalities/ sin(4x/3)>-√3/2

sin(4x/3)>-√3/2 inequation

A inequation with variable

The solution

You have entered [src] 
              ___ 
   /4*x\   -\/ 3  
sin|---| > -------
   \ 3 /      2
sin⁡(4x3)>(−1)32\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2} 
sin((4*x)/3) > (-sqrt(3))/2
Detail solution
Given the inequality:
sin⁡(4x3)>(−1)32\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(4x3)=(−1)32\sin{\left(\frac{4 x}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
Solve:
Given the equation
sin⁡(4x3)=(−1)32\sin{\left(\frac{4 x}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
- this is the simplest trigonometric equation
This equation is transformed to
4x3=2πn+asin⁡(−32)\frac{4 x}{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}
4x3=2πn−asin⁡(−32)+π\frac{4 x}{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi
Or
4x3=2πn−π3\frac{4 x}{3} = 2 \pi n - \frac{\pi}{3}
4x3=2πn+4π3\frac{4 x}{3} = 2 \pi n + \frac{4 \pi}{3}
, where n - is a integer
Divide both parts of the equation by
43\frac{4}{3}
x1=3πn2−π4x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}
x2=3πn2+πx_{2} = \frac{3 \pi n}{2} + \pi
x1=3πn2−π4x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}
x2=3πn2+πx_{2} = \frac{3 \pi n}{2} + \pi
This roots
x1=3πn2−π4x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}
x2=3πn2+πx_{2} = \frac{3 \pi n}{2} + \pi
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}
=
(3πn2−π4)+−110\left(\frac{3 \pi n}{2} - \frac{\pi}{4}\right) + - \frac{1}{10}
=
3πn2−π4−110\frac{3 \pi n}{2} - \frac{\pi}{4} - \frac{1}{10}
substitute to the expression
sin⁡(4x3)>(−1)32\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
sin⁡(4(3πn2−π4−110)3)>(−1)32\sin{\left(\frac{4 \left(\frac{3 \pi n}{2} - \frac{\pi}{4} - \frac{1}{10}\right)}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
                            ___ 
    /2    pi         \   -\/ 3  
-sin|-- + -- - 2*pi*n| > -------
    \15   3          /      2

Then
x<3πn2−π4x < \frac{3 \pi n}{2} - \frac{\pi}{4}
no execute
one of the solutions of our inequality is:
x>3πn2−π4∧x<3πn2+πx > \frac{3 \pi n}{2} - \frac{\pi}{4} \wedge x < \frac{3 \pi n}{2} + \pi
         _____  
        /     \  
-------ο-------ο-------
       x1      x2
Solving inequality on a graph
0-50-40-30-20-1010203040502-2
Rapid solution 2 [src] 
           5*pi  3*pi 
[0, pi) U (----, ----]
            4     2
x in [0,π)∪(5π4,3π2]x\ in\ \left[0, \pi\right) \cup \left(\frac{5 \pi}{4}, \frac{3 \pi}{2}\right] 
x in Union(Interval.Ropen(0, pi), Interval.Lopen(5*pi/4, 3*pi/2))
Rapid solution [src] 
  /                        /     3*pi  5*pi    \\
Or|And(0 <= x, x < pi), And|x <= ----, ---- < x||
  \                        \      2     4      //
(0≤x∧x<π)∨(x≤3π2∧5π4<x)\left(0 \leq x \wedge x < \pi\right) \vee \left(x \leq \frac{3 \pi}{2} \wedge \frac{5 \pi}{4} < x\right) 
((0 <= x)∧(x < pi))∨((x <= 3*pi/2)∧(5*pi/4 < x))
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