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cos(3x+1)<=-(√2/2)
Resolver desigualdades/ cos(3x+1)<=-(√2/2)

cos(3x+1)<=-(√2/2) desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
                   ___ 
                -\/ 2  
cos(3*x + 1) <= -------
                   2
cos⁡(3x+1)≤−22\cos{\left(3 x + 1 \right)} \leq - \frac{\sqrt{2}}{2} 
cos(3*x + 1) <= -sqrt(2)/2
Solución detallada
Se da la desigualdad:
cos⁡(3x+1)≤−22\cos{\left(3 x + 1 \right)} \leq - \frac{\sqrt{2}}{2}
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
cos⁡(3x+1)=−22\cos{\left(3 x + 1 \right)} = - \frac{\sqrt{2}}{2}
Resolvemos:
Tenemos la ecuación
cos⁡(3x+1)=−22\cos{\left(3 x + 1 \right)} = - \frac{\sqrt{2}}{2}
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
3x+1=πn+acos⁡(−22)3 x + 1 = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}
3x+1=πn−π+acos⁡(−22)3 x + 1 = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}
O
3x+1=πn+3π43 x + 1 = \pi n + \frac{3 \pi}{4}
3x+1=πn−π43 x + 1 = \pi n - \frac{\pi}{4}
, donde n es cualquier número entero
Transportemos
11
al miembro derecho de la ecuación
con el signo opuesto, en total:
3x=πn−1+3π43 x = \pi n - 1 + \frac{3 \pi}{4}
3x=πn−1−π43 x = \pi n - 1 - \frac{\pi}{4}
Dividamos ambos miembros de la ecuación obtenida en
33
x1=πn3−13+π4x_{1} = \frac{\pi n}{3} - \frac{1}{3} + \frac{\pi}{4}
x2=πn3−13−π12x_{2} = \frac{\pi n}{3} - \frac{1}{3} - \frac{\pi}{12}
x1=πn3−13+π4x_{1} = \frac{\pi n}{3} - \frac{1}{3} + \frac{\pi}{4}
x2=πn3−13−π12x_{2} = \frac{\pi n}{3} - \frac{1}{3} - \frac{\pi}{12}
Las raíces dadas
x1=πn3−13+π4x_{1} = \frac{\pi n}{3} - \frac{1}{3} + \frac{\pi}{4}
x2=πn3−13−π12x_{2} = \frac{\pi n}{3} - \frac{1}{3} - \frac{\pi}{12}
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}
=
(πn3−13+π4)+−110\left(\frac{\pi n}{3} - \frac{1}{3} + \frac{\pi}{4}\right) + - \frac{1}{10}
=
πn3−1330+π4\frac{\pi n}{3} - \frac{13}{30} + \frac{\pi}{4}
lo sustituimos en la expresión
cos⁡(3x+1)≤−22\cos{\left(3 x + 1 \right)} \leq - \frac{\sqrt{2}}{2}
cos⁡(3(πn3−1330+π4)+1)≤−22\cos{\left(3 \left(\frac{\pi n}{3} - \frac{13}{30} + \frac{\pi}{4}\right) + 1 \right)} \leq - \frac{\sqrt{2}}{2}
                             ___ 
    /  3    pi       \    -\/ 2  
-sin|- -- + -- + pi*n| <= -------
    \  10   4        /       2

pero
                             ___ 
    /  3    pi       \    -\/ 2  
-sin|- -- + -- + pi*n| >= -------
    \  10   4        /       2

Entonces
x≤πn3−13+π4x \leq \frac{\pi n}{3} - \frac{1}{3} + \frac{\pi}{4}
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥πn3−13+π4∧x≤πn3−13−π12x \geq \frac{\pi n}{3} - \frac{1}{3} + \frac{\pi}{4} \wedge x \leq \frac{\pi n}{3} - \frac{1}{3} - \frac{\pi}{12}
         _____  
        /     \  
-------•-------•-------
       x1      x2
Solución de la desigualdad en el gráfico
0-80-60-40-20204060802-2
Respuesta rápida 2 [src] 
       /                                                         ___ /       2     \           \        /                                                         ___ /       2     \           \ 
       |                4*tan(1/2)                             \/ 2 *\1 + tan (1/2)/           |        |                4*tan(1/2)                             \/ 2 *\1 + tan (1/2)/           | 
 2*atan|------------------------------------------ - ------------------------------------------|  2*atan|------------------------------------------ + ------------------------------------------| 
       |       ___        2          ___    2               ___        2          ___    2     |        |       ___        2          ___    2               ___        2          ___    2     | 
       \-2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)   -2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)/        \-2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)   -2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)/ 
[-----------------------------------------------------------------------------------------------, -----------------------------------------------------------------------------------------------]
                                                3                                                                                                3
x in [2atan⁡(−2(tan⁡2(12)+1)−2+2tan⁡2(12)+2tan⁡2(12)+2+4tan⁡(12)−2+2tan⁡2(12)+2tan⁡2(12)+2)3,2atan⁡(2(tan⁡2(12)+1)−2+2tan⁡2(12)+2tan⁡2(12)+2+4tan⁡(12)−2+2tan⁡2(12)+2tan⁡2(12)+2)3]x\ in\ \left[\frac{2 \operatorname{atan}{\left(- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{1}{2} \right)} + 1\right)}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} \right)}}{3}, \frac{2 \operatorname{atan}{\left(\frac{\sqrt{2} \left(\tan^{2}{\left(\frac{1}{2} \right)} + 1\right)}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} \right)}}{3}\right] 
x in Interval(2*atan(-sqrt(2)*(tan(1/2)^2 + 1)/(-2 + sqrt(2)*tan(1/2)^2 + 2*tan(1/2)^2 + sqrt(2)) + 4*tan(1/2)/(-2 + sqrt(2)*tan(1/2)^2 + 2*tan(1/2)^2 + sqrt(2)))/3, 2*atan(sqrt(2)*(tan(1/2)^2 + 1)/(-2 + sqrt(2)*tan(1/2)^2 + 2*tan(1/2)^2 + sqrt(2)) + 4*tan(1/2)/(-2 + sqrt(2)*tan(1/2)^2 + 2*tan(1/2)^2 + sqrt(2)))/3)
Respuesta rápida [src] 
   /           /                                                         ___ /       2     \           \        /                                                         ___ /       2     \           \     \
   |           |                4*tan(1/2)                             \/ 2 *\1 + tan (1/2)/           |        |                4*tan(1/2)                             \/ 2 *\1 + tan (1/2)/           |     |
   |     2*atan|------------------------------------------ + ------------------------------------------|  2*atan|------------------------------------------ - ------------------------------------------|     |
   |           |       ___        2          ___    2               ___        2          ___    2     |        |       ___        2          ___    2               ___        2          ___    2     |     |
   |           \-2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)   -2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)/        \-2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)   -2 + \/ 2  + 2*tan (1/2) + \/ 2 *tan (1/2)/     |
And|x <= -----------------------------------------------------------------------------------------------, ----------------------------------------------------------------------------------------------- <= x|
   \                                                    3                                                                                                3                                                    /
x≤2atan⁡(2(tan⁡2(12)+1)−2+2tan⁡2(12)+2tan⁡2(12)+2+4tan⁡(12)−2+2tan⁡2(12)+2tan⁡2(12)+2)3∧2atan⁡(−2(tan⁡2(12)+1)−2+2tan⁡2(12)+2tan⁡2(12)+2+4tan⁡(12)−2+2tan⁡2(12)+2tan⁡2(12)+2)3≤xx \leq \frac{2 \operatorname{atan}{\left(\frac{\sqrt{2} \left(\tan^{2}{\left(\frac{1}{2} \right)} + 1\right)}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} \right)}}{3} \wedge \frac{2 \operatorname{atan}{\left(- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{1}{2} \right)} + 1\right)}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{-2 + \sqrt{2} \tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + \sqrt{2}} \right)}}{3} \leq x 
(x <= 2*atan(4*tan(1/2)/(-2 + sqrt(2) + 2*tan(1/2)^2 + sqrt(2)*tan(1/2)^2) + sqrt(2)*(1 + tan(1/2)^2)/(-2 + sqrt(2) + 2*tan(1/2)^2 + sqrt(2)*tan(1/2)^2))/3)∧(2*atan(4*tan(1/2)/(-2 + sqrt(2) + 2*tan(1/2)^2 + sqrt(2)*tan(1/2)^2) - sqrt(2)*(1 + tan(1/2)^2)/(-2 + sqrt(2) + 2*tan(1/2)^2 + sqrt(2)*tan(1/2)^2))/3 <= x)
Gráfico
cos(3x+1)<=-(√2/2) desigualdades
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