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Solving inequalities/ sin(x/4)<-sqrt3/2

sin(x/4)<-sqrt3/2 inequation

A inequation with variable

The solution

You have entered [src] 
            ___ 
   /x\   -\/ 3  
sin|-| < -------
   \4/      2
sin⁡(x4)<(−1)32\sin{\left(\frac{x}{4} \right)} < \frac{\left(-1\right) \sqrt{3}}{2} 
sin(x/4) < (-sqrt(3))/2
Detail solution
Given the inequality:
sin⁡(x4)<(−1)32\sin{\left(\frac{x}{4} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(x4)=(−1)32\sin{\left(\frac{x}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
Solve:
Given the equation
sin⁡(x4)=(−1)32\sin{\left(\frac{x}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
- this is the simplest trigonometric equation
This equation is transformed to
x4=2πn+asin⁡(−32)\frac{x}{4} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}
x4=2πn−asin⁡(−32)+π\frac{x}{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi
Or
x4=2πn−π3\frac{x}{4} = 2 \pi n - \frac{\pi}{3}
x4=2πn+4π3\frac{x}{4} = 2 \pi n + \frac{4 \pi}{3}
, where n - is a integer
Divide both parts of the equation by
14\frac{1}{4}
x1=8πn−4π3x_{1} = 8 \pi n - \frac{4 \pi}{3}
x2=8πn+16π3x_{2} = 8 \pi n + \frac{16 \pi}{3}
x1=8πn−4π3x_{1} = 8 \pi n - \frac{4 \pi}{3}
x2=8πn+16π3x_{2} = 8 \pi n + \frac{16 \pi}{3}
This roots
x1=8πn−4π3x_{1} = 8 \pi n - \frac{4 \pi}{3}
x2=8πn+16π3x_{2} = 8 \pi n + \frac{16 \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}
=
(8πn−4π3)+−110\left(8 \pi n - \frac{4 \pi}{3}\right) + - \frac{1}{10}
=
8πn−4π3−1108 \pi n - \frac{4 \pi}{3} - \frac{1}{10}
substitute to the expression
sin⁡(x4)<(−1)32\sin{\left(\frac{x}{4} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}
sin⁡(8πn−4π3−1104)<(−1)32\sin{\left(\frac{8 \pi n - \frac{4 \pi}{3} - \frac{1}{10}}{4} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}
                            ___ 
    /1    pi         \   -\/ 3  
-sin|-- + -- - 2*pi*n| < -------
    \40   3          /      2

one of the solutions of our inequality is:
x<8πn−4π3x < 8 \pi n - \frac{4 \pi}{3}
 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
x<8πn−4π3x < 8 \pi n - \frac{4 \pi}{3}
x>8πn+16π3x > 8 \pi n + \frac{16 \pi}{3}
Solving inequality on a graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.52-2
Rapid solution [src] 
   /16*pi          20*pi\
And|----- < x, x < -----|
   \  3              3  /
16π3<x∧x<20π3\frac{16 \pi}{3} < x \wedge x < \frac{20 \pi}{3} 
(16*pi/3 < x)∧(x < 20*pi/3)
Rapid solution 2 [src] 
 16*pi  20*pi 
(-----, -----)
   3      3
x in (16π3,20π3)x\ in\ \left(\frac{16 \pi}{3}, \frac{20 \pi}{3}\right) 
x in Interval.open(16*pi/3, 20*pi/3)
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