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Resolver desigualdades/ sqrt(2)*(sin(x)-cos(x))<=sqrt(3)

sqrt(2)*(sin(x)-cos(x))<=sqrt(3) desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
  ___                        ___
\/ 2 *(sin(x) - cos(x)) <= \/ 3
2(sin⁡(x)−cos⁡(x))≤3\sqrt{2} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \leq \sqrt{3} 
sqrt(2)*(sin(x) - cos(x)) <= sqrt(3)
Solución detallada
Se da la desigualdad:
2(sin⁡(x)−cos⁡(x))≤3\sqrt{2} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \leq \sqrt{3}
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
2(sin⁡(x)−cos⁡(x))=3\sqrt{2} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) = \sqrt{3}
Resolvemos:
x1=2atan⁡(1−2−3+2)x_{1} = 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
x2=−2atan⁡(1+2−3+2)x_{2} = - 2 \operatorname{atan}{\left(\frac{1 + \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
x1=2atan⁡(1−2−3+2)x_{1} = 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
x2=−2atan⁡(1+2−3+2)x_{2} = - 2 \operatorname{atan}{\left(\frac{1 + \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
Las raíces dadas
x1=2atan⁡(1−2−3+2)x_{1} = 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
x2=−2atan⁡(1+2−3+2)x_{2} = - 2 \operatorname{atan}{\left(\frac{1 + \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x1x_{0} \leq x_{1}
Consideremos, por ejemplo, el punto
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−110+2atan⁡(1−2−3+2)- \frac{1}{10} + 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
=
−110+2atan⁡(1−2−3+2)- \frac{1}{10} + 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
lo sustituimos en la expresión
2(sin⁡(x)−cos⁡(x))≤3\sqrt{2} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \leq \sqrt{3}
2(−cos⁡(−110+2atan⁡(1−2−3+2))+sin⁡(−110+2atan⁡(1−2−3+2)))≤3\sqrt{2} \left(- \cos{\left(- \frac{1}{10} + 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)} \right)} + \sin{\left(- \frac{1}{10} + 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)} \right)}\right) \leq \sqrt{3}
      /     /           /        ___  \\      /           /        ___  \\\         
  ___ |     |1          |  1 - \/ 2   ||      |1          |  1 - \/ 2   |||      ___
\/ 2 *|- cos|-- - 2*atan|-------------|| - sin|-- - 2*atan|-------------||| <= \/ 3 
      |     |10         |  ___     ___||      |10         |  ___     ___|||    
      \     \           \\/ 2  - \/ 3 //      \           \\/ 2  - \/ 3 ///

significa que una de las soluciones de nuestra ecuación será con:
x≤2atan⁡(1−2−3+2)x \leq 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
 _____           _____          
      \         /
-------•-------•-------
       x1      x2

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≤2atan⁡(1−2−3+2)x \leq 2 \operatorname{atan}{\left(\frac{1 - \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
x≥−2atan⁡(1+2−3+2)x \geq - 2 \operatorname{atan}{\left(\frac{1 + \sqrt{2}}{- \sqrt{3} + \sqrt{2}} \right)}
Solución de la desigualdad en el gráfico
0-70-60-50-40-30-20-10102030405060705-5
Respuesta rápida [src] 
  /   /                      /  ___     ___\\     /                    /  ___     ___\     \\
  |   |                      |\/ 2  + \/ 6 ||     |                    |\/ 2  - \/ 6 |     ||
Or|And|0 <= x, x <= pi + atan|-------------||, And|x <= 2*pi, pi + atan|-------------| <= x||
  |   |                      |  ___     ___||     |                    |  ___     ___|     ||
  \   \                      \\/ 2  - \/ 6 //     \                    \\/ 2  + \/ 6 /     //
(0≤x∧x≤atan⁡(2+6−6+2)+π)∨(x≤2π∧atan⁡(−6+22+6)+π≤x)\left(0 \leq x \wedge x \leq \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{6} + \sqrt{2}} \right)} + \pi\right) \vee \left(x \leq 2 \pi \wedge \operatorname{atan}{\left(\frac{- \sqrt{6} + \sqrt{2}}{\sqrt{2} + \sqrt{6}} \right)} + \pi \leq x\right) 
((0 <= x)∧(x <= pi + atan((sqrt(2) + sqrt(6))/(sqrt(2) - sqrt(6)))))∨((x <= 2*pi)∧(pi + atan((sqrt(2) - sqrt(6))/(sqrt(2) + sqrt(6))) <= x))
Respuesta rápida 2 [src] 
             /  ___     ___\              /  ___     ___\       
             |\/ 2  + \/ 6 |              |\/ 2  - \/ 6 |       
[0, pi + atan|-------------|] U [pi + atan|-------------|, 2*pi]
             |  ___     ___|              |  ___     ___|       
             \\/ 2  - \/ 6 /              \\/ 2  + \/ 6 /
x in [0,atan⁡(2+6−6+2)+π]∪[atan⁡(−6+22+6)+π,2π]x\ in\ \left[0, \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{6} + \sqrt{2}} \right)} + \pi\right] \cup \left[\operatorname{atan}{\left(\frac{- \sqrt{6} + \sqrt{2}}{\sqrt{2} + \sqrt{6}} \right)} + \pi, 2 \pi\right] 
x in Union(Interval(0, atan((sqrt(2) + sqrt(6))/(-sqrt(6) + sqrt(2))) + pi), Interval(atan((-sqrt(6) + sqrt(2))/(sqrt(2) + sqrt(6))) + pi, 2*pi))
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