Mister Exam

Other calculators

Solving inequalities/ sin(2x+pi/4)≥sin3pi/4

sin(2x+pi/4)≥sin3pi/4 inequation

A inequation with variable

The solution

You have entered [src] 
   /      pi\    sin(3*pi)
sin|2*x + --| >= ---------
   \      4 /        4
sin⁡(2x+π4)≥sin⁡(3π)4\sin{\left(2 x + \frac{\pi}{4} \right)} \geq \frac{\sin{\left(3 \pi \right)}}{4} 
sin(2*x + pi/4) >= sin(3*pi)/4
Detail solution
Given the inequality:
sin⁡(2x+π4)≥sin⁡(3π)4\sin{\left(2 x + \frac{\pi}{4} \right)} \geq \frac{\sin{\left(3 \pi \right)}}{4}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(2x+π4)=sin⁡(3π)4\sin{\left(2 x + \frac{\pi}{4} \right)} = \frac{\sin{\left(3 \pi \right)}}{4}
Solve:
Given the equation
sin⁡(2x+π4)=sin⁡(3π)4\sin{\left(2 x + \frac{\pi}{4} \right)} = \frac{\sin{\left(3 \pi \right)}}{4}
- this is the simplest trigonometric equation
This equation is transformed to
2x+π4=2πn+asin⁡(0)2 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}
2x+π4=2πn−asin⁡(0)+π2 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi
Or
2x+π4=2πn2 x + \frac{\pi}{4} = 2 \pi n
2x+π4=2πn+π2 x + \frac{\pi}{4} = 2 \pi n + \pi
, where n - is a integer
Move
π4\frac{\pi}{4}
to right part of the equation
with the opposite sign, in total:
2x=2πn−π42 x = 2 \pi n - \frac{\pi}{4}
2x=2πn+3π42 x = 2 \pi n + \frac{3 \pi}{4}
Divide both parts of the equation by
22
x1=πn−π8x_{1} = \pi n - \frac{\pi}{8}
x2=πn+3π8x_{2} = \pi n + \frac{3 \pi}{8}
x1=πn−π8x_{1} = \pi n - \frac{\pi}{8}
x2=πn+3π8x_{2} = \pi n + \frac{3 \pi}{8}
This roots
x1=πn−π8x_{1} = \pi n - \frac{\pi}{8}
x2=πn+3π8x_{2} = \pi n + \frac{3 \pi}{8}
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}
=
(πn−π8)+−110\left(\pi n - \frac{\pi}{8}\right) + - \frac{1}{10}
=
πn−π8−110\pi n - \frac{\pi}{8} - \frac{1}{10}
substitute to the expression
sin⁡(2x+π4)≥sin⁡(3π)4\sin{\left(2 x + \frac{\pi}{4} \right)} \geq \frac{\sin{\left(3 \pi \right)}}{4}
sin⁡(2(πn−π8−110)+π4)≥sin⁡(3π)4\sin{\left(2 \left(\pi n - \frac{\pi}{8} - \frac{1}{10}\right) + \frac{\pi}{4} \right)} \geq \frac{\sin{\left(3 \pi \right)}}{4}
sin(-1/5 + 2*pi*n) >= 0

but
sin(-1/5 + 2*pi*n) < 0

Then
x≤πn−π8x \leq \pi n - \frac{\pi}{8}
no execute
one of the solutions of our inequality is:
x≥πn−π8∧x≤πn+3π8x \geq \pi n - \frac{\pi}{8} \wedge x \leq \pi n + \frac{3 \pi}{8}
         _____  
        /     \  
-------•-------•-------
       x1      x2
Solving inequality on a graph
0-80-60-40-20204060802-2
Rapid solution [src] 
  /   /                 /   ___________\\     /                  /   ___________\     \\
  |   |                 |  /       ___ ||     |                  |  /       ___ |     ||
  |   |                 |\/  2 + \/ 2  ||     |                  |\/  2 - \/ 2  |     ||
Or|And|0 <= x, x <= atan|--------------||, And|x <= pi, pi - atan|--------------| <= x||
  |   |                 |   ___________||     |                  |   ___________|     ||
  |   |                 |  /       ___ ||     |                  |  /       ___ |     ||
  \   \                 \\/  2 - \/ 2  //     \                  \\/  2 + \/ 2  /     //
(0≤x∧x≤atan⁡(2+22−2))∨(x≤π∧π−atan⁡(2−22+2)≤x)\left(0 \leq x \wedge x \leq \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)}\right) \vee \left(x \leq \pi \wedge \pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)} \leq x\right) 
((0 <= x)∧(x <= atan(sqrt(2 + sqrt(2))/sqrt(2 - sqrt(2)))))∨((x <= pi)∧(pi - atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))) <= x))
Rapid solution 2 [src] 
        /   ___________\              /   ___________\     
        |  /       ___ |              |  /       ___ |     
        |\/  2 + \/ 2  |              |\/  2 - \/ 2  |     
[0, atan|--------------|] U [pi - atan|--------------|, pi]
        |   ___________|              |   ___________|     
        |  /       ___ |              |  /       ___ |     
        \\/  2 - \/ 2  /              \\/  2 + \/ 2  /
x in [0,atan⁡(2+22−2)]∪[π−atan⁡(2−22+2),π]x\ in\ \left[0, \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)}\right] \cup \left[\pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}, \pi\right] 
x in Union(Interval(0, atan(sqrt(sqrt(2) + 2)/sqrt(2 - sqrt(2)))), Interval(pi - atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2)), pi))
    © Mister Exam – Calculator