Mister Exam

Other calculators

Solving inequalities/ 2sinx<=1

2sinx<=1 inequation

A inequation with variable

The solution

You have entered [src] 
2*sin(x) <= 1
2sin⁡(x)≤12 \sin{\left(x \right)} \leq 1 
2*sin(x) <= 1
Detail solution
Given the inequality:
2sin⁡(x)≤12 \sin{\left(x \right)} \leq 1
To solve this inequality, we must first solve the corresponding equation:
2sin⁡(x)=12 \sin{\left(x \right)} = 1
Solve:
Given the equation
2sin⁡(x)=12 \sin{\left(x \right)} = 1
- this is the simplest trigonometric equation
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+5π6x = 2 \pi n + \frac{5 \pi}{6}
, where n - is a integer
x1=2πn+π6x_{1} = 2 \pi n + \frac{\pi}{6}
x2=2πn+5π6x_{2} = 2 \pi n + \frac{5 \pi}{6}
x1=2πn+π6x_{1} = 2 \pi n + \frac{\pi}{6}
x2=2πn+5π6x_{2} = 2 \pi n + \frac{5 \pi}{6}
This roots
x1=2πn+π6x_{1} = 2 \pi n + \frac{\pi}{6}
x2=2πn+5π6x_{2} = 2 \pi n + \frac{5 \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−110+π62 \pi n - \frac{1}{10} + \frac{\pi}{6}
substitute to the expression
2sin⁡(x)≤12 \sin{\left(x \right)} \leq 1
2sin⁡(2πn−110+π6)≤12 \sin{\left(2 \pi n - \frac{1}{10} + \frac{\pi}{6} \right)} \leq 1
     /  1    pi         \     
2*sin|- -- + -- + 2*pi*n| <= 1
     \  10   6          /

one of the solutions of our inequality is:
x≤2πn+π6x \leq 2 \pi n + \frac{\pi}{6}
 _____           _____          
      \         /
-------•-------•-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
x≤2πn+π6x \leq 2 \pi n + \frac{\pi}{6}
x≥2πn+5π6x \geq 2 \pi n + \frac{5 \pi}{6}
Solving inequality on a graph
0-60-50-40-30-20-101020304050605-5
Rapid solution 2 [src] 
    pi     5*pi       
[0, --] U [----, 2*pi]
    6       6
x in [0,π6]∪[5π6,2π]x\ in\ \left[0, \frac{\pi}{6}\right] \cup \left[\frac{5 \pi}{6}, 2 \pi\right] 
x in Union(Interval(0, pi/6), Interval(5*pi/6, 2*pi))
Rapid solution [src] 
  /   /             pi\     /5*pi                \\
Or|And|0 <= x, x <= --|, And|---- <= x, x <= 2*pi||
  \   \             6 /     \ 6                  //
(0≤x∧x≤π6)∨(5π6≤x∧x≤2π)\left(0 \leq x \wedge x \leq \frac{\pi}{6}\right) \vee \left(\frac{5 \pi}{6} \leq x \wedge x \leq 2 \pi\right) 
((0 <= x)∧(x <= pi/6))∨((5*pi/6 <= x)∧(x <= 2*pi))
    © Mister Exam – Calculator