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Resolver desigualdades/ cos(x)*cos(pi/4)-sin(x)*sin(pi/4)>=1/2

cos(x)*cos(pi/4)-sin(x)*sin(pi/4)>=1/2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
          /pi\             /pi\       
cos(x)*cos|--| - sin(x)*sin|--| >= 1/2
          \4 /             \4 /
−sin⁡(x)sin⁡(π4)+cos⁡(x)cos⁡(π4)≥12- \sin{\left(x \right)} \sin{\left(\frac{\pi}{4} \right)} + \cos{\left(x \right)} \cos{\left(\frac{\pi}{4} \right)} \geq \frac{1}{2} 
-sin(x)*sin(pi/4) + cos(x)*cos(pi/4) >= 1/2
Solución detallada
Se da la desigualdad:
−sin⁡(x)sin⁡(π4)+cos⁡(x)cos⁡(π4)≥12- \sin{\left(x \right)} \sin{\left(\frac{\pi}{4} \right)} + \cos{\left(x \right)} \cos{\left(\frac{\pi}{4} \right)} \geq \frac{1}{2}
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
−sin⁡(x)sin⁡(π4)+cos⁡(x)cos⁡(π4)=12- \sin{\left(x \right)} \sin{\left(\frac{\pi}{4} \right)} + \cos{\left(x \right)} \cos{\left(\frac{\pi}{4} \right)} = \frac{1}{2}
Resolvemos:
x1=2atan⁡(−6−2+2+3)x_{1} = 2 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)}
x2=−2atan⁡(−6−2+3+2)x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} + \sqrt{3} + 2 \right)}
x1=2atan⁡(−6−2+2+3)x_{1} = 2 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)}
x2=−2atan⁡(−6−2+3+2)x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} + \sqrt{3} + 2 \right)}
Las raíces dadas
x1=2atan⁡(−6−2+2+3)x_{1} = 2 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)}
x2=−2atan⁡(−6−2+3+2)x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} + \sqrt{3} + 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}
=
2atan⁡(−6−2+2+3)−1102 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)} - \frac{1}{10}
=
2atan⁡(−6−2+2+3)−1102 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)} - \frac{1}{10}
lo sustituimos en la expresión
−sin⁡(x)sin⁡(π4)+cos⁡(x)cos⁡(π4)≥12- \sin{\left(x \right)} \sin{\left(\frac{\pi}{4} \right)} + \cos{\left(x \right)} \cos{\left(\frac{\pi}{4} \right)} \geq \frac{1}{2}
cos⁡(π4)cos⁡(2atan⁡(−6−2+2+3)−110)−sin⁡(π4)sin⁡(2atan⁡(−6−2+2+3)−110)≥12\cos{\left(\frac{\pi}{4} \right)} \cos{\left(2 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)} - \frac{1}{10} \right)} - \sin{\left(\frac{\pi}{4} \right)} \sin{\left(2 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)} - \frac{1}{10} \right)} \geq \frac{1}{2}
  ___    /  1          /       ___     ___     ___\\     ___    /  1          /       ___     ___     ___\\       
\/ 2 *cos|- -- + 2*atan\-2 + \/ 2  + \/ 3  - \/ 6 /|   \/ 2 *sin|- -- + 2*atan\-2 + \/ 2  + \/ 3  - \/ 6 /|       
         \  10                                     /            \  10                                     / >= 1/2
---------------------------------------------------- - ----------------------------------------------------       
                         2                                                      2

pero
  ___    /  1          /       ___     ___     ___\\     ___    /  1          /       ___     ___     ___\\      
\/ 2 *cos|- -- + 2*atan\-2 + \/ 2  + \/ 3  - \/ 6 /|   \/ 2 *sin|- -- + 2*atan\-2 + \/ 2  + \/ 3  - \/ 6 /|      
         \  10                                     /            \  10                                     / < 1/2
---------------------------------------------------- - ----------------------------------------------------      
                         2                                                      2

Entonces
x≤2atan⁡(−6−2+2+3)x \leq 2 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)}
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥2atan⁡(−6−2+2+3)∧x≤−2atan⁡(−6−2+3+2)x \geq 2 \operatorname{atan}{\left(- \sqrt{6} - 2 + \sqrt{2} + \sqrt{3} \right)} \wedge x \leq - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} + \sqrt{3} + 2 \right)}
         _____  
        /     \  
-------•-------•-------
       x1      x2
Solución de la desigualdad en el gráfico
0-60-50-40-30-20-101020304050602-2
Respuesta rápida [src] 
  /   /                 /  ___     ___\\     /                    /  ___     ___\     \\
  |   |                 |\/ 6  - \/ 2 ||     |                    |\/ 2  + \/ 6 |     ||
Or|And|0 <= x, x <= atan|-------------||, And|x <= 2*pi, pi - atan|-------------| <= x||
  |   |                 |  ___     ___||     |                    |  ___     ___|     ||
  \   \                 \\/ 2  + \/ 6 //     \                    \\/ 2  - \/ 6 /     //
(0≤x∧x≤atan⁡(−2+62+6))∨(x≤2π∧π−atan⁡(2+6−6+2)≤x)\left(0 \leq x \wedge x \leq \operatorname{atan}{\left(\frac{- \sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}} \right)}\right) \vee \left(x \leq 2 \pi \wedge \pi - \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{6} + \sqrt{2}} \right)} \leq x\right) 
((0 <= x)∧(x <= atan((sqrt(6) - sqrt(2))/(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, -atan|-------------|] U [pi - atan|-------------|, 2*pi]
         |  ___     ___|              |  ___     ___|       
         \\/ 2  + \/ 6 /              \\/ 2  - \/ 6 /
x in [0,−atan⁡(−6+22+6)]∪[π−atan⁡(2+6−6+2),2π]x\ in\ \left[0, - \operatorname{atan}{\left(\frac{- \sqrt{6} + \sqrt{2}}{\sqrt{2} + \sqrt{6}} \right)}\right] \cup \left[\pi - \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{6} + \sqrt{2}} \right)}, 2 \pi\right] 
x in Union(Interval(0, -atan((-sqrt(6) + sqrt(2))/(sqrt(2) + sqrt(6)))), Interval(pi - atan((sqrt(2) + sqrt(6))/(-sqrt(6) + sqrt(2))), 2*pi))
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