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Solving inequalities/ cos(x)>=0.5

cos(x)>=0.5 inequation

A inequation with variable

The solution

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cos(x) >= 1/2
cos⁡(x)≥12\cos{\left(x \right)} \geq \frac{1}{2} 
cos(x) >= 1/2
Detail solution
Given the inequality:
cos⁡(x)≥12\cos{\left(x \right)} \geq \frac{1}{2}
To solve this inequality, we must first solve the corresponding equation:
cos⁡(x)=12\cos{\left(x \right)} = \frac{1}{2}
Solve:
Given the equation
cos⁡(x)=12\cos{\left(x \right)} = \frac{1}{2}
- this is the simplest trigonometric equation
This equation is transformed to
x=πn+acos⁡(12)x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}
x=πn−π+acos⁡(12)x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}
Or
x=πn+π3x = \pi n + \frac{\pi}{3}
x=πn−2π3x = \pi n - \frac{2 \pi}{3}
, where n - is a integer
x1=πn+π3x_{1} = \pi n + \frac{\pi}{3}
x2=πn−2π3x_{2} = \pi n - \frac{2 \pi}{3}
x1=πn+π3x_{1} = \pi n + \frac{\pi}{3}
x2=πn−2π3x_{2} = \pi n - \frac{2 \pi}{3}
This roots
x1=πn+π3x_{1} = \pi n + \frac{\pi}{3}
x2=πn−2π3x_{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:
x0≤x1x_{0} \leq x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
(πn+π3)+−110\left(\pi n + \frac{\pi}{3}\right) + - \frac{1}{10}
=
πn−110+π3\pi n - \frac{1}{10} + \frac{\pi}{3}
substitute to the expression
cos⁡(x)≥12\cos{\left(x \right)} \geq \frac{1}{2}
cos⁡(πn−110+π3)≥12\cos{\left(\pi n - \frac{1}{10} + \frac{\pi}{3} \right)} \geq \frac{1}{2}
   /  1    pi       \       
cos|- -- + -- + pi*n| >= 1/2
   \  10   3        /

but
   /  1    pi       \      
cos|- -- + -- + pi*n| < 1/2
   \  10   3        /

Then
x≤πn+π3x \leq \pi n + \frac{\pi}{3}
no execute
one of the solutions of our inequality is:
x≥πn+π3∧x≤πn−2π3x \geq \pi n + \frac{\pi}{3} \wedge x \leq \pi n - \frac{2 \pi}{3}
         _____  
        /     \  
-------•-------•-------
       x1      x2
Solving inequality on a graph
05-20-15-10-51015202-2
Rapid solution 2 [src] 
    pi     5*pi       
[0, --] U [----, 2*pi]
    3       3
x in [0,π3]∪[5π3,2π]x\ in\ \left[0, \frac{\pi}{3}\right] \cup \left[\frac{5 \pi}{3}, 2 \pi\right] 
x in Union(Interval(0, pi/3), Interval(5*pi/3, 2*pi))
Rapid solution [src] 
  /   /             pi\     /5*pi                \\
Or|And|0 <= x, x <= --|, And|---- <= x, x <= 2*pi||
  \   \             3 /     \ 3                  //
(0≤x∧x≤π3)∨(5π3≤x∧x≤2π)\left(0 \leq x \wedge x \leq \frac{\pi}{3}\right) \vee \left(\frac{5 \pi}{3} \leq x \wedge x \leq 2 \pi\right) 
((0 <= x)∧(x <= pi/3))∨((5*pi/3 <= x)∧(x <= 2*pi))
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