Mister Exam

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sin(2*x)<-√(3)/2 inequation

You have entered [src]

              ___ 
           -\/ 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|- + -- - 2*pi*n| < -------
    \5   3          /      2

one of the solutions of our inequality is:

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

 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:

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

x > \pi n + \frac{2 \pi}{3}

Solving inequality on a graph

Rapid solution [src]

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

\frac{2 \pi}{3} < x \wedge x < \frac{5 \pi}{6}

(2*pi/3 < x)∧(x < 5*pi/6)

Rapid solution 2 [src]

 2*pi  5*pi 
(----, ----)
  3     6

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

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