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Solving inequalities/ sin(x/4-3)<1/2

sin(x/4-3)<1/2 inequation

A inequation with variable

The solution

You have entered [src] 
   /x    \      
sin|- - 3| < 1/2
   \4    /
sin⁡(x4−3)<12\sin{\left(\frac{x}{4} - 3 \right)} < \frac{1}{2} 
sin(x/4 - 3) < 1/2
Detail solution
Given the inequality:
sin⁡(x4−3)<12\sin{\left(\frac{x}{4} - 3 \right)} < \frac{1}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(x4−3)=12\sin{\left(\frac{x}{4} - 3 \right)} = \frac{1}{2}
Solve:
Given the equation
sin⁡(x4−3)=12\sin{\left(\frac{x}{4} - 3 \right)} = \frac{1}{2}
- this is the simplest trigonometric equation
This equation is transformed to
x4−3=2πn+asin⁡(12)\frac{x}{4} - 3 = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}
x4−3=2πn−asin⁡(12)+π\frac{x}{4} - 3 = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi
Or
x4−3=2πn+π6\frac{x}{4} - 3 = 2 \pi n + \frac{\pi}{6}
x4−3=2πn+5π6\frac{x}{4} - 3 = 2 \pi n + \frac{5 \pi}{6}
, where n - is a integer
Move
−3-3
to right part of the equation
with the opposite sign, in total:
x4=2πn+π6+3\frac{x}{4} = 2 \pi n + \frac{\pi}{6} + 3
x4=2πn+5π6+3\frac{x}{4} = 2 \pi n + \frac{5 \pi}{6} + 3
Divide both parts of the equation by
14\frac{1}{4}
x1=8πn+2π3+12x_{1} = 8 \pi n + \frac{2 \pi}{3} + 12
x2=8πn+10π3+12x_{2} = 8 \pi n + \frac{10 \pi}{3} + 12
x1=8πn+2π3+12x_{1} = 8 \pi n + \frac{2 \pi}{3} + 12
x2=8πn+10π3+12x_{2} = 8 \pi n + \frac{10 \pi}{3} + 12
This roots
x1=8πn+2π3+12x_{1} = 8 \pi n + \frac{2 \pi}{3} + 12
x2=8πn+10π3+12x_{2} = 8 \pi n + \frac{10 \pi}{3} + 12
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+2π3+12)+−110\left(8 \pi n + \frac{2 \pi}{3} + 12\right) + - \frac{1}{10}
=
8πn+2π3+119108 \pi n + \frac{2 \pi}{3} + \frac{119}{10}
substitute to the expression
sin⁡(x4−3)<12\sin{\left(\frac{x}{4} - 3 \right)} < \frac{1}{2}
sin⁡(8πn+2π3+119104−3)<12\sin{\left(\frac{8 \pi n + \frac{2 \pi}{3} + \frac{119}{10}}{4} - 3 \right)} < \frac{1}{2}
   /  1    pi         \      
sin|- -- + -- + 2*pi*n| < 1/2
   \  40   6          /

one of the solutions of our inequality is:
x<8πn+2π3+12x < 8 \pi n + \frac{2 \pi}{3} + 12
 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
x<8πn+2π3+12x < 8 \pi n + \frac{2 \pi}{3} + 12
x>8πn+10π3+12x > 8 \pi n + \frac{10 \pi}{3} + 12
Solving inequality on a graph
012345-10-9-8-7-6-5-4-3-2-12-2
Rapid solution [src] 
  /   /                         /      ___ /       2     \                                     \\     /                        /    ___ /       2     \                                     \    \\
  |   |                         |    \/ 3 *\1 + tan (3/2)/      2*(1 - tan(3/2))*(1 + tan(3/2))||     |                        |  \/ 3 *\1 + tan (3/2)/      2*(1 - tan(3/2))*(1 + tan(3/2))|    ||
Or|And|0 <= x, x < 8*pi + 8*atan|- -------------------------- + -------------------------------||, And|x <= 8*pi, 8*pi + 8*atan|-------------------------- + -------------------------------| < x||
  |   |                         |         2                               2                    ||     |                        |       2                               2                    |    ||
  \   \                         \  1 + tan (3/2) - 4*tan(3/2)      1 + tan (3/2) - 4*tan(3/2)  //     \                        \1 + tan (3/2) - 4*tan(3/2)      1 + tan (3/2) - 4*tan(3/2)  /    //
(0≤x∧x<8atan⁡(2(1−tan⁡(32))(1+tan⁡(32))−4tan⁡(32)+1+tan⁡2(32)−3(1+tan⁡2(32))−4tan⁡(32)+1+tan⁡2(32))+8π)∨(x≤8π∧8atan⁡(2(1−tan⁡(32))(1+tan⁡(32))−4tan⁡(32)+1+tan⁡2(32)+3(1+tan⁡2(32))−4tan⁡(32)+1+tan⁡2(32))+8π<x)\left(0 \leq x \wedge x < 8 \operatorname{atan}{\left(\frac{2 \left(1 - \tan{\left(\frac{3}{2} \right)}\right) \left(1 + \tan{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} - \frac{\sqrt{3} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi\right) \vee \left(x \leq 8 \pi \wedge 8 \operatorname{atan}{\left(\frac{2 \left(1 - \tan{\left(\frac{3}{2} \right)}\right) \left(1 + \tan{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} + \frac{\sqrt{3} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi < x\right) 
((0 <= x)∧(x < 8*pi + 8*atan(-sqrt(3)*(1 + tan(3/2)^2)/(1 + tan(3/2)^2 - 4*tan(3/2)) + 2*(1 - tan(3/2))*(1 + tan(3/2))/(1 + tan(3/2)^2 - 4*tan(3/2)))))∨((x <= 8*pi)∧(8*pi + 8*atan(sqrt(3)*(1 + tan(3/2)^2)/(1 + tan(3/2)^2 - 4*tan(3/2)) + 2*(1 - tan(3/2))*(1 + tan(3/2))/(1 + tan(3/2)^2 - 4*tan(3/2))) < x))
Rapid solution 2 [src] 
                 /      ___ /       2     \                                     \                  /    ___ /       2     \                                     \       
                 |    \/ 3 *\1 + tan (3/2)/      2*(1 - tan(3/2))*(1 + tan(3/2))|                  |  \/ 3 *\1 + tan (3/2)/      2*(1 - tan(3/2))*(1 + tan(3/2))|       
[0, 8*pi + 8*atan|- -------------------------- + -------------------------------|) U (8*pi + 8*atan|-------------------------- + -------------------------------|, 8*pi]
                 |         2                               2                    |                  |       2                               2                    |       
                 \  1 + tan (3/2) - 4*tan(3/2)      1 + tan (3/2) - 4*tan(3/2)  /                  \1 + tan (3/2) - 4*tan(3/2)      1 + tan (3/2) - 4*tan(3/2)  /
x in [0,8atan⁡(2(1−tan⁡(32))(1+tan⁡(32))−4tan⁡(32)+1+tan⁡2(32)−3(1+tan⁡2(32))−4tan⁡(32)+1+tan⁡2(32))+8π)∪(8atan⁡(2(1−tan⁡(32))(1+tan⁡(32))−4tan⁡(32)+1+tan⁡2(32)+3(1+tan⁡2(32))−4tan⁡(32)+1+tan⁡2(32))+8π,8π]x\ in\ \left[0, 8 \operatorname{atan}{\left(\frac{2 \left(1 - \tan{\left(\frac{3}{2} \right)}\right) \left(1 + \tan{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} - \frac{\sqrt{3} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi\right) \cup \left(8 \operatorname{atan}{\left(\frac{2 \left(1 - \tan{\left(\frac{3}{2} \right)}\right) \left(1 + \tan{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} + \frac{\sqrt{3} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + 1 + \tan^{2}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi, 8 \pi\right] 
x in Union(Interval.Ropen(0, 8*atan(2*(1 - tan(3/2))*(1 + tan(3/2))/(-4*tan(3/2) + 1 + tan(3/2)^2) - sqrt(3)*(1 + tan(3/2)^2)/(-4*tan(3/2) + 1 + tan(3/2)^2)) + 8*pi), Interval.Lopen(8*atan(2*(1 - tan(3/2))*(1 + tan(3/2))/(-4*tan(3/2) + 1 + tan(3/2)^2) + sqrt(3)*(1 + tan(3/2)^2)/(-4*tan(3/2) + 1 + tan(3/2)^2)) + 8*pi, 8*pi))
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