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sin2x>-sqrt3/2 inequation

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              ___ 
           -\/ 3  
sin(2*x) > -------
              2

\sin{\left(2 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}

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

Detail solution

Given the inequality:

\sin{\left(2 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}

To solve this inequality, we must first solve the corresponding equation:

\sin{\left(2 x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}

Solve:
Given the equation

\sin{\left(2 x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}

- this is the simplest trigonometric equation
This equation is transformed to

2 x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}

2 x = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi

2 x = 2 \pi n - \frac{\pi}{3}

2 x = 2 \pi n + \frac{4 \pi}{3}

, where n - is a integer
Divide both parts of the equation by

2

x_{1} = \pi n - \frac{\pi}{6}

x_{2} = \pi n + \frac{2 \pi}{3}

x_{1} = \pi n - \frac{\pi}{6}

x_{2} = \pi n + \frac{2 \pi}{3}

This roots

x_{1} = \pi n - \frac{\pi}{6}

x_{2} = \pi n + \frac{2 \pi}{3}

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

\sin{\left(2 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}

\sin{\left(2 \left(\pi n - \frac{\pi}{6} - \frac{1}{10}\right) \right)} > \frac{\left(-1\right) \sqrt{3}}{2}

                  ___ 
    /1   pi\   -\/ 3  
-sin|- + --| > -------
    \5   3 /      2

Then

x < \pi n - \frac{\pi}{6}

no execute
one of the solutions of our inequality is:

x > \pi n - \frac{\pi}{6} \wedge x < \pi n + \frac{2 \pi}{3}

         _____  
        /     \  
-------ο-------ο-------
       x_1      x_2

Solving inequality on a graph

Rapid solution [src]

  /   /            2*pi\     /5*pi            \\
Or|And|0 <= x, x < ----|, And|---- < x, x < pi||
  \   \             3  /     \ 6              //

\left(0 \leq x \wedge x < \frac{2 \pi}{3}\right) \vee \left(\frac{5 \pi}{6} < x \wedge x < \pi\right)

((0 <= x)∧(x < 2*pi/3))∨((x < pi)∧(5*pi/6 < x))

Rapid solution 2 [src]

    2*pi     5*pi     
[0, ----) U (----, pi)
     3        6

x\ in\ \left[0, \frac{2 \pi}{3}\right) \cup \left(\frac{5 \pi}{6}, \pi\right)

x in Union(Interval.Ropen(0, 2*pi/3), Interval.open(5*pi/6, pi))

The graph