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Resolver desigualdades/ 4sinxcosx>sqrt2

4sinxcosx>sqrt2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
                    ___
4*sin(x)*cos(x) > \/ 2
4sin⁡(x)cos⁡(x)>24 \sin{\left(x \right)} \cos{\left(x \right)} > \sqrt{2} 
(4*sin(x))*cos(x) > sqrt(2)
Solución detallada
Se da la desigualdad:
4sin⁡(x)cos⁡(x)>24 \sin{\left(x \right)} \cos{\left(x \right)} > \sqrt{2}
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
4sin⁡(x)cos⁡(x)=24 \sin{\left(x \right)} \cos{\left(x \right)} = \sqrt{2}
Resolvemos:
x1=−2atan⁡(−1+22−2+2)x_{1} = - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=−2atan⁡(1+2+22+2)x_{2} = - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x3=2atan⁡(−2+1+22−2)x_{3} = 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x4=−2atan⁡(−22+2+1+2)x_{4} = - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
x1=−2atan⁡(−1+22−2+2)x_{1} = - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=−2atan⁡(1+2+22+2)x_{2} = - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x3=2atan⁡(−2+1+22−2)x_{3} = 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x4=−2atan⁡(−22+2+1+2)x_{4} = - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
Las raíces dadas
x2=−2atan⁡(1+2+22+2)x_{2} = - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x1=−2atan⁡(−1+22−2+2)x_{1} = - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x4=−2atan⁡(−22+2+1+2)x_{4} = - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
x3=2atan⁡(−2+1+22−2)x_{3} = 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \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<x2x_{0} < x_{2}
Consideremos, por ejemplo, el punto
x0=x2−110x_{0} = x_{2} - \frac{1}{10}
=
−2atan⁡(1+2+22+2)−110- 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)} - \frac{1}{10}
=
−2atan⁡(1+2+22+2)−110- 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)} - \frac{1}{10}
lo sustituimos en la expresión
4sin⁡(x)cos⁡(x)>24 \sin{\left(x \right)} \cos{\left(x \right)} > \sqrt{2}
4sin⁡(−2atan⁡(1+2+22+2)−110)cos⁡(−2atan⁡(1+2+22+2)−110)>24 \sin{\left(- 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)} - \frac{1}{10} \right)} \cos{\left(- 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)} - \frac{1}{10} \right)} > \sqrt{2}
      /           /                     ___________\\    /           /                     ___________\\        
      |1          |      ___     ___   /       ___ ||    |1          |      ___     ___   /       ___ ||     ___
-4*cos|-- + 2*atan\1 + \/ 2  + \/ 2 *\/  2 + \/ 2  /|*sin|-- + 2*atan\1 + \/ 2  + \/ 2 *\/  2 + \/ 2  /| > \/ 2 
      \10                                           /    \10                                           /

Entonces
x<−2atan⁡(1+2+22+2)x < - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x>−2atan⁡(1+2+22+2)∧x<−2atan⁡(−1+22−2+2)x > - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)} \wedge x < - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
         _____           _____  
        /     \         /     \  
-------ο-------ο-------ο-------ο-------
       x2      x1      x4      x3

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x>−2atan⁡(1+2+22+2)∧x<−2atan⁡(−1+22−2+2)x > - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)} \wedge x < - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x>−2atan⁡(−22+2+1+2)∧x<2atan⁡(−2+1+22−2)x > - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)} \wedge x < 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
Solución de la desigualdad en el gráfico
0-70-60-50-40-30-20-10102030405060705-5
Respuesta rápida 2 [src] 
     /   ___________\      /   ___________\ 
     |  /       ___ |      |  /       ___ | 
     |\/  2 - \/ 2  |      |\/  2 + \/ 2  | 
(atan|--------------|, atan|--------------|)
     |   ___________|      |   ___________| 
     |  /       ___ |      |  /       ___ | 
     \\/  2 + \/ 2  /      \\/  2 - \/ 2  /
x in (atan⁡(2−22+2),atan⁡(2+22−2))x\ in\ \left(\operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}, \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)}\right) 
x in Interval.open(atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2)), atan(sqrt(sqrt(2) + 2)/sqrt(2 - sqrt(2))))
Respuesta rápida [src] 
   /        /   ___________\      /   ___________\    \
   |        |  /       ___ |      |  /       ___ |    |
   |        |\/  2 + \/ 2  |      |\/  2 - \/ 2  |    |
And|x < atan|--------------|, atan|--------------| < x|
   |        |   ___________|      |   ___________|    |
   |        |  /       ___ |      |  /       ___ |    |
   \        \\/  2 - \/ 2  /      \\/  2 + \/ 2  /    /
x<atan⁡(2+22−2)∧atan⁡(2−22+2)<xx < \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)} \wedge \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)} < x 
(x < atan(sqrt(2 + sqrt(2))/sqrt(2 - sqrt(2))))∧(atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))) < x)
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