Mister Exam

Other calculators

sin(7*x)>-sqrt3/2
Solving inequalities/ sin(7*x)>-sqrt3/2

sin(7*x)>-sqrt3/2 inequation

A inequation with variable

The solution

You have entered [src] 
              ___ 
           -\/ 3  
sin(7*x) > -------
              2
sin⁡(7x)>(−1)32\sin{\left(7 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2} 
sin(7*x) > -sqrt(3)/2
Detail solution
Given the inequality:
sin⁡(7x)>(−1)32\sin{\left(7 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(7x)=(−1)32\sin{\left(7 x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
Solve:
Given the equation
sin⁡(7x)=(−1)32\sin{\left(7 x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}
- this is the simplest trigonometric equation
This equation is transformed to
7x=2πn+asin⁡(−32)7 x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}
7x=2πn−asin⁡(−32)+π7 x = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi
Or
7x=2πn−π37 x = 2 \pi n - \frac{\pi}{3}
7x=2πn+4π37 x = 2 \pi n + \frac{4 \pi}{3}
, where n - is a integer
Divide both parts of the equation by
77
x1=2πn7−π21x_{1} = \frac{2 \pi n}{7} - \frac{\pi}{21}
x2=2πn7+4π21x_{2} = \frac{2 \pi n}{7} + \frac{4 \pi}{21}
x1=2πn7−π21x_{1} = \frac{2 \pi n}{7} - \frac{\pi}{21}
x2=2πn7+4π21x_{2} = \frac{2 \pi n}{7} + \frac{4 \pi}{21}
This roots
x1=2πn7−π21x_{1} = \frac{2 \pi n}{7} - \frac{\pi}{21}
x2=2πn7+4π21x_{2} = \frac{2 \pi n}{7} + \frac{4 \pi}{21}
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}
=
(2πn7−π21)−110\left(\frac{2 \pi n}{7} - \frac{\pi}{21}\right) - \frac{1}{10}
=
2πn7−π21−110\frac{2 \pi n}{7} - \frac{\pi}{21} - \frac{1}{10}
substitute to the expression
sin⁡(7x)>(−1)32\sin{\left(7 x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
sin⁡(7⋅(2πn7−π21−110))>(−1)32\sin{\left(7 \cdot \left(\frac{2 \pi n}{7} - \frac{\pi}{21} - \frac{1}{10}\right) \right)} > \frac{\left(-1\right) \sqrt{3}}{2}
                   ___ 
    /7    pi\   -\/ 3  
-sin|-- + --| > -------
    \10   3 /      2

Then
x<2πn7−π21x < \frac{2 \pi n}{7} - \frac{\pi}{21}
no execute
one of the solutions of our inequality is:
x>2πn7−π21∧x<2πn7+4π21x > \frac{2 \pi n}{7} - \frac{\pi}{21} \wedge x < \frac{2 \pi n}{7} + \frac{4 \pi}{21}
         _____  
        /     \  
-------ο-------ο-------
       x_1      x_2
Solving inequality on a graph
0-80-60-40-20204060802-2
Rapid solution 2 [src] 
    4*pi     5*pi  2*pi 
[0, ----) U (----, ----)
     21       21    7
x in [0,4π21)∪(5π21,2π7)x\ in\ \left[0, \frac{4 \pi}{21}\right) \cup \left(\frac{5 \pi}{21}, \frac{2 \pi}{7}\right) 
x in Union(Interval.Ropen(0, 4*pi/21), Interval.open(5*pi/21, 2*pi/7))
Rapid solution [src] 
  /   /            4*pi\     /5*pi          2*pi\\
Or|And|0 <= x, x < ----|, And|---- < x, x < ----||
  \   \             21 /     \ 21            7  //
(0≤x∧x<4π21)∨(5π21<x∧x<2π7)\left(0 \leq x \wedge x < \frac{4 \pi}{21}\right) \vee \left(\frac{5 \pi}{21} < x \wedge x < \frac{2 \pi}{7}\right) 
((0 <= x)∧(x < 4*pi/21))∨((5*pi/21 < x)∧(x < 2*pi/7))
The graph
sin(7*x)>-sqrt3/2 inequation
    © Mister Exam – Calculator