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Solving inequalities/ sin(x/7)>-1/5

sin(x/7)>-1/5 inequation

A inequation with variable

The solution

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   /x\       
sin|-| > -1/5
   \7/
sin⁡(x7)>−15\sin{\left(\frac{x}{7} \right)} > - \frac{1}{5} 
sin(x/7) > -1/5
Detail solution
Given the inequality:
sin⁡(x7)>−15\sin{\left(\frac{x}{7} \right)} > - \frac{1}{5}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(x7)=−15\sin{\left(\frac{x}{7} \right)} = - \frac{1}{5}
Solve:
Given the equation
sin⁡(x7)=−15\sin{\left(\frac{x}{7} \right)} = - \frac{1}{5}
- this is the simplest trigonometric equation
This equation is transformed to
x7=2πn+asin⁡(−15)\frac{x}{7} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{5} \right)}
x7=2πn−asin⁡(−15)+π\frac{x}{7} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{5} \right)} + \pi
Or
x7=2πn−asin⁡(15)\frac{x}{7} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)}
x7=2πn+asin⁡(15)+π\frac{x}{7} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi
, where n - is a integer
Divide both parts of the equation by
17\frac{1}{7}
x1=14πn−7asin⁡(15)x_{1} = 14 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)}
x2=14πn+7asin⁡(15)+7πx_{2} = 14 \pi n + 7 \operatorname{asin}{\left(\frac{1}{5} \right)} + 7 \pi
x1=14πn−7asin⁡(15)x_{1} = 14 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)}
x2=14πn+7asin⁡(15)+7πx_{2} = 14 \pi n + 7 \operatorname{asin}{\left(\frac{1}{5} \right)} + 7 \pi
This roots
x1=14πn−7asin⁡(15)x_{1} = 14 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)}
x2=14πn+7asin⁡(15)+7πx_{2} = 14 \pi n + 7 \operatorname{asin}{\left(\frac{1}{5} \right)} + 7 \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}
=
(14πn−7asin⁡(15))+−110\left(14 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)}\right) + - \frac{1}{10}
=
14πn−7asin⁡(15)−11014 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)} - \frac{1}{10}
substitute to the expression
sin⁡(x7)>−15\sin{\left(\frac{x}{7} \right)} > - \frac{1}{5}
sin⁡(14πn−7asin⁡(15)−1107)>−15\sin{\left(\frac{14 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)} - \frac{1}{10}}{7} \right)} > - \frac{1}{5}
-sin(1/70 - 2*pi*n + asin(1/5)) > -1/5

Then
x<14πn−7asin⁡(15)x < 14 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)}
no execute
one of the solutions of our inequality is:
x>14πn−7asin⁡(15)∧x<14πn+7asin⁡(15)+7πx > 14 \pi n - 7 \operatorname{asin}{\left(\frac{1}{5} \right)} \wedge x < 14 \pi n + 7 \operatorname{asin}{\left(\frac{1}{5} \right)} + 7 \pi
         _____  
        /     \  
-------ο-------ο-------
       x1      x2
Solving inequality on a graph
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.01-1
Rapid solution [src] 
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Or\And\0 <= x, x < - 14*atan\5 + 2*\/ 6 / + 14*pi/, And\x <= 14*pi, - 14*atan\5 - 2*\/ 6 / + 14*pi < x//
(0≤x∧x<−14atan⁡(26+5)+14π)∨(x≤14π∧−14atan⁡(5−26)+14π<x)\left(0 \leq x \wedge x < - 14 \operatorname{atan}{\left(2 \sqrt{6} + 5 \right)} + 14 \pi\right) \vee \left(x \leq 14 \pi \wedge - 14 \operatorname{atan}{\left(5 - 2 \sqrt{6} \right)} + 14 \pi < x\right) 
((0 <= x)∧(x < -14*atan(5 + 2*sqrt(6)) + 14*pi))∨((x <= 14*pi)∧(-14*atan(5 - 2*sqrt(6)) + 14*pi < x))
Rapid solution 2 [src] 
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[0, - 14*atan\5 + 2*\/ 6 / + 14*pi) U (- 14*atan\5 - 2*\/ 6 / + 14*pi, 14*pi]
x in [0,−14atan⁡(26+5)+14π)∪(−14atan⁡(5−26)+14π,14π]x\ in\ \left[0, - 14 \operatorname{atan}{\left(2 \sqrt{6} + 5 \right)} + 14 \pi\right) \cup \left(- 14 \operatorname{atan}{\left(5 - 2 \sqrt{6} \right)} + 14 \pi, 14 \pi\right] 
x in Union(Interval.Ropen(0, -14*atan(2*sqrt(6) + 5) + 14*pi), Interval.Lopen(-14*atan(5 - 2*sqrt(6)) + 14*pi, 14*pi))
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