Mister Exam

2sinx+1>_0 inequation

A inequation with variable

The solution

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

with the change of sign in 1

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

The equation is transformed to
$$\sin{\left(x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$x = 2 \pi n - \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
This roots
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$2 \sin{\left(x \right)} + 1 \geq 0$$
$$2 \sin{\left(2 \pi n - \frac{\pi}{6} - \frac{1}{10} \right)} + 1 \geq 0$$
         /1    pi         \     
1 - 2*sin|-- + -- - 2*pi*n| >= 0
         \10   6          /     

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

Then
$$x \leq 2 \pi n - \frac{\pi}{6}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n - \frac{\pi}{6} \wedge x \leq 2 \pi n + \frac{7 \pi}{6}$$
         _____  
        /     \  
-------•-------•-------
       x1      x2
Solving inequality on a graph
Rapid solution [src]
  /   /             7*pi\     /11*pi                \\
Or|And|0 <= x, x <= ----|, And|----- <= x, x <= 2*pi||
  \   \              6  /     \  6                  //
$$\left(0 \leq x \wedge x \leq \frac{7 \pi}{6}\right) \vee \left(\frac{11 \pi}{6} \leq x \wedge x \leq 2 \pi\right)$$
((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         
$$x\ in\ \left[0, \frac{7 \pi}{6}\right] \cup \left[\frac{11 \pi}{6}, 2 \pi\right]$$
x in Union(Interval(0, 7*pi/6), Interval(11*pi/6, 2*pi))