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Solving inequalities/ sinx<-1/4

sinx<-1/4 inequation

A inequation with variable

The solution

You have entered [src] 
sin(x) < -1/4
sin⁡(x)<−14\sin{\left(x \right)} < - \frac{1}{4} 
sin(x) < -1/4
Detail solution
Given the inequality:
sin⁡(x)<−14\sin{\left(x \right)} < - \frac{1}{4}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(x)=−14\sin{\left(x \right)} = - \frac{1}{4}
Solve:
Given the equation
sin⁡(x)=−14\sin{\left(x \right)} = - \frac{1}{4}
- this is the simplest trigonometric equation
This equation is transformed to
x=2πn+asin⁡(−14)x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{4} \right)}
x=2πn−asin⁡(−14)+πx = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{4} \right)} + \pi
Or
x=2πn−asin⁡(14)x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}
x=2πn+asin⁡(14)+πx = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi
, where n - is a integer
x1=2πn−asin⁡(14)x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}
x2=2πn+asin⁡(14)+πx_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi
x1=2πn−asin⁡(14)x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}
x2=2πn+asin⁡(14)+πx_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi
This roots
x1=2πn−asin⁡(14)x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}
x2=2πn+asin⁡(14)+πx_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \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}
=
(2πn−asin⁡(14))+−110\left(2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}\right) + - \frac{1}{10}
=
2πn−asin⁡(14)−1102 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)} - \frac{1}{10}
substitute to the expression
sin⁡(x)<−14\sin{\left(x \right)} < - \frac{1}{4}
sin⁡(2πn−asin⁡(14)−110)<−14\sin{\left(2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)} - \frac{1}{10} \right)} < - \frac{1}{4}
-sin(1/10 - 2*pi*n + asin(1/4)) < -1/4

one of the solutions of our inequality is:
x<2πn−asin⁡(14)x < 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}
 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
x<2πn−asin⁡(14)x < 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}
x>2πn+asin⁡(14)+πx > 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi
Solving inequality on a graph
0-60-50-40-30-20-101020304050602-2
Rapid solution [src] 
   /          /  ____\                  /  ____\    \
   |          |\/ 15 |                  |\/ 15 |    |
And|x < - atan|------| + 2*pi, pi + atan|------| < x|
   \          \  15  /                  \  15  /    /
x<−atan⁡(1515)+2π∧atan⁡(1515)+π<xx < - \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + 2 \pi \wedge \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + \pi < x 
(pi + atan(sqrt(15)/15) < x)∧(x < -atan(sqrt(15)/15) + 2*pi)
Rapid solution 2 [src] 
          /  ____\        /  ____\        
          |\/ 15 |        |\/ 15 |        
(pi + atan|------|, - atan|------| + 2*pi)
          \  15  /        \  15  /
x in (atan⁡(1515)+π,−atan⁡(1515)+2π)x\ in\ \left(\operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + \pi, - \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + 2 \pi\right) 
x in Interval.open(atan(sqrt(15)/15) + pi, -atan(sqrt(15)/15) + 2*pi)
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