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Solving inequalities/ 2sinx+1>_0

2sinx+1>_0 inequation

A inequation with variable

The solution

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2*sin(x) + 1 >= 0
2sin⁡(x)+1≥02 \sin{\left(x \right)} + 1 \geq 0 
2*sin(x) + 1 >= 0
Detail solution
Given the inequality:
2sin⁡(x)+1≥02 \sin{\left(x \right)} + 1 \geq 0
To solve this inequality, we must first solve the corresponding equation:
2sin⁡(x)+1=02 \sin{\left(x \right)} + 1 = 0
Solve:
Given the equation
2sin⁡(x)+1=02 \sin{\left(x \right)} + 1 = 0
- this is the simplest trigonometric equation
Move 1 to right part of the equation

with the change of sign in 1

We get:
2sin⁡(x)=−12 \sin{\left(x \right)} = -1
Divide both parts of the equation by 2

The equation is transformed to
sin⁡(x)=−12\sin{\left(x \right)} = - \frac{1}{2}
This equation is transformed to
x=2πn+asin⁡(−12)x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}
x=2πn−asin⁡(−12)+πx = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi
Or
x=2πn−π6x = 2 \pi n - \frac{\pi}{6}
x=2πn+7π6x = 2 \pi n + \frac{7 \pi}{6}
, where n - is a integer
x1=2πn−π6x_{1} = 2 \pi n - \frac{\pi}{6}
x2=2πn+7π6x_{2} = 2 \pi n + \frac{7 \pi}{6}
x1=2πn−π6x_{1} = 2 \pi n - \frac{\pi}{6}
x2=2πn+7π6x_{2} = 2 \pi n + \frac{7 \pi}{6}
This roots
x1=2πn−π6x_{1} = 2 \pi n - \frac{\pi}{6}
x2=2πn+7π6x_{2} = 2 \pi n + \frac{7 \pi}{6}
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}
=
(2πn−π6)+−110\left(2 \pi n - \frac{\pi}{6}\right) + - \frac{1}{10}
=
2πn−π6−1102 \pi n - \frac{\pi}{6} - \frac{1}{10}
substitute to the expression
2sin⁡(x)+1≥02 \sin{\left(x \right)} + 1 \geq 0
2sin⁡(2πn−π6−110)+1≥02 \sin{\left(2 \pi n - \frac{\pi}{6} - \frac{1}{10} \right)} + 1 \geq 0
         /1    pi         \     
1 - 2*sin|-- + -- - 2*pi*n| >= 0
         \10   6          /

but
         /1    pi         \    
1 - 2*sin|-- + -- - 2*pi*n| < 0
         \10   6          /

Then
x≤2πn−π6x \leq 2 \pi n - \frac{\pi}{6}
no execute
one of the solutions of our inequality is:
x≥2πn−π6∧x≤2πn+7π6x \geq 2 \pi n - \frac{\pi}{6} \wedge x \leq 2 \pi n + \frac{7 \pi}{6}
         _____  
        /     \  
-------•-------•-------
       x1      x2
Solving inequality on a graph
0-60-50-40-30-20-101020304050605-5
Rapid solution [src] 
  /   /             7*pi\     /11*pi                \\
Or|And|0 <= x, x <= ----|, And|----- <= x, x <= 2*pi||
  \   \              6  /     \  6                  //
(0≤x∧x≤7π6)∨(11π6≤x∧x≤2π)\left(0 \leq x \wedge x \leq \frac{7 \pi}{6}\right) \vee \left(\frac{11 \pi}{6} \leq x \wedge x \leq 2 \pi\right) 
((0 <= x)∧(x <= 7*pi/6))∨((11*pi/6 <= x)∧(x <= 2*pi))
Rapid solution 2 [src] 
    7*pi     11*pi       
[0, ----] U [-----, 2*pi]
     6         6
x in [0,7π6]∪[11π6,2π]x\ in\ \left[0, \frac{7 \pi}{6}\right] \cup \left[\frac{11 \pi}{6}, 2 \pi\right] 
x in Union(Interval(0, 7*pi/6), Interval(11*pi/6, 2*pi))
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