Mister Exam

Other calculators

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)} > \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)} > \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} < 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)} > \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)} > \frac{\left(-1\right) \sqrt{3}}{2}
                            ___ 
    /1    pi         \   -\/ 3  
-sin|-- + -- - 2*pi*n| > -------
    \10   3          /      2

Then
x<2πn−π3x < 2 \pi n - \frac{\pi}{3}
no execute
one of the solutions of our inequality is:
x>2πn−π3∧x<2πn+4π3x > 2 \pi n - \frac{\pi}{3} \wedge x < 2 \pi n + \frac{4 \pi}{3}
         _____  
        /     \  
-------ο-------ο-------
       x1      x2
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 <= 2*pi, ---- < x||
  \   \             3  /     \            3      //
(0≤x∧x<4π3)∨(x≤2π∧5π3<x)\left(0 \leq x \wedge x < \frac{4 \pi}{3}\right) \vee \left(x \leq 2 \pi \wedge \frac{5 \pi}{3} < x\right) 
((0 <= x)∧(x < 4*pi/3))∨((x <= 2*pi)∧(5*pi/3 < x))
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.Ropen(0, 4*pi/3), Interval.Lopen(5*pi/3, 2*pi))
    © Mister Exam – Calculator