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sin(x)≥-sqrt(3)/2
Solving inequalities/ sin(x)≥-sqrt(3)/2

sin(x)≥-sqrt(3)/2 inequation

A inequation with variable

The solution

You have entered [src] 
             ___ 
          -\/ 3  
sin(x) >= -------
             2
sin⁡(x)≥(−1)32\sin{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{2} 
sin(x) >= -sqrt(3)/2
Detail solution
Given the inequality:
sin⁡(x)≥(−1)32\sin{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(x)=(−1)32\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
Solve:
Given the equation
sin⁡(x)=(−1)32\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
- this is the simplest trigonometric equation
This equation is transformed to
x=2πn+asin⁡(−32)x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}
x=2πn−asin⁡(−32)+πx = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi
Or
x=2πn−π3x = 2 \pi n - \frac{\pi}{3}
x=2πn+4π3x = 2 \pi n + \frac{4 \pi}{3}
, where n - is a integer
x1=2πn−π3x_{1} = 2 \pi n - \frac{\pi}{3}
x2=2πn+4π3x_{2} = 2 \pi n + \frac{4 \pi}{3}
x1=2πn−π3x_{1} = 2 \pi n - \frac{\pi}{3}
x2=2πn+4π3x_{2} = 2 \pi n + \frac{4 \pi}{3}
This roots
x1=2πn−π3x_{1} = 2 \pi n - \frac{\pi}{3}
x2=2πn+4π3x_{2} = 2 \pi n + \frac{4 \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}
=
(2πn−π3)−110\left(2 \pi n - \frac{\pi}{3}\right) - \frac{1}{10}
=
2πn−π3−1102 \pi n - \frac{\pi}{3} - \frac{1}{10}
substitute to the expression
sin⁡(x)≥(−1)32\sin{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{2}
sin⁡(2πn−π3−110)≥(−1)32\sin{\left(2 \pi n - \frac{\pi}{3} - \frac{1}{10} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{2}
                    ___ 
    /1    pi\    -\/ 3  
-sin|-- + --| >= -------
    \10   3 /       2

but
                   ___ 
    /1    pi\   -\/ 3  
-sin|-- + --| < -------
    \10   3 /      2

Then
x≤2πn−π3x \leq 2 \pi n - \frac{\pi}{3}
no execute
one of the solutions of our inequality is:
x≥2πn−π3∧x≤2πn+4π3x \geq 2 \pi n - \frac{\pi}{3} \wedge x \leq 2 \pi n + \frac{4 \pi}{3}
         _____  
        /     \  
-------•-------•-------
       x_1      x_2
Solving inequality on a graph
0-60-50-40-30-20-101020304050602-2
Rapid solution [src] 
  /   /             4*pi\     /5*pi               \\
Or|And|0 <= x, x <= ----|, And|---- <= x, x < 2*pi||
  \   \              3  /     \ 3                 //
(0≤x∧x≤4π3)∨(5π3≤x∧x<2π)\left(0 \leq x \wedge x \leq \frac{4 \pi}{3}\right) \vee \left(\frac{5 \pi}{3} \leq x \wedge x < 2 \pi\right) 
((0 <= x)∧(x <= 4*pi/3))∨((5*pi/3 <= x)∧(x < 2*pi))
Rapid solution 2 [src] 
    4*pi     5*pi       
[0, ----] U [----, 2*pi)
     3        3
x in [0,4π3]∪[5π3,2π)x\ in\ \left[0, \frac{4 \pi}{3}\right] \cup \left[\frac{5 \pi}{3}, 2 \pi\right) 
x in Union(Interval(0, 4*pi/3), Interval.Ropen(5*pi/3, 2*pi))
The graph
sin(x)≥-sqrt(3)/2 inequation
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