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sin(p*x/4)=-sqrt(2)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
              ___ 
   /p*x\   -\/ 2  
sin|---| = -------
   \ 4 /      2
sin⁡(px4)=(−1)22\sin{\left(\frac{p x}{4} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}
Simplificar
Solución detallada
Tenemos la ecuación
sin⁡(px4)=(−1)22\sin{\left(\frac{p x}{4} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
px4=2πn+asin⁡(−22)\frac{p x}{4} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}
px4=2πn−asin⁡(−22)+π\frac{p x}{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi
O
px4=2πn−π4\frac{p x}{4} = 2 \pi n - \frac{\pi}{4}
px4=2πn+5π4\frac{p x}{4} = 2 \pi n + \frac{5 \pi}{4}
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
p4\frac{p}{4}
obtenemos la respuesta:
x1=4(2πn−π4)px_{1} = \frac{4 \left(2 \pi n - \frac{\pi}{4}\right)}{p}
x2=4(2πn+5π4)px_{2} = \frac{4 \left(2 \pi n + \frac{5 \pi}{4}\right)}{p}
Gráfica
Respuesta rápida [src] 
           pi*re(p)         pi*I*im(p)  
x1 = - --------------- + ---------------
         2        2        2        2   
       im (p) + re (p)   im (p) + re (p)
x1=−πre⁡(p)(re⁡(p))2+(im⁡(p))2+iπim⁡(p)(re⁡(p))2+(im⁡(p))2x_{1} = - \frac{\pi \operatorname{re}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} + \frac{i \pi \operatorname{im}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} 
        5*pi*re(p)       5*pi*I*im(p) 
x2 = --------------- - ---------------
       2        2        2        2   
     im (p) + re (p)   im (p) + re (p)
x2=5πre⁡(p)(re⁡(p))2+(im⁡(p))2−5iπim⁡(p)(re⁡(p))2+(im⁡(p))2x_{2} = \frac{5 \pi \operatorname{re}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} - \frac{5 i \pi \operatorname{im}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} 
x2 = 5*pi*re(p)/(re(p)^2 + im(p)^2) - 5*i*pi*im(p)/(re(p)^2 + im(p)^2)
Suma y producto de raíces [src] 
suma
      pi*re(p)         pi*I*im(p)        5*pi*re(p)       5*pi*I*im(p) 
- --------------- + --------------- + --------------- - ---------------
    2        2        2        2        2        2        2        2   
  im (p) + re (p)   im (p) + re (p)   im (p) + re (p)   im (p) + re (p)
(−πre⁡(p)(re⁡(p))2+(im⁡(p))2+iπim⁡(p)(re⁡(p))2+(im⁡(p))2)+(5πre⁡(p)(re⁡(p))2+(im⁡(p))2−5iπim⁡(p)(re⁡(p))2+(im⁡(p))2)\left(- \frac{\pi \operatorname{re}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} + \frac{i \pi \operatorname{im}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}}\right) + \left(\frac{5 \pi \operatorname{re}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} - \frac{5 i \pi \operatorname{im}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}}\right) 
=
   4*pi*re(p)       4*pi*I*im(p) 
--------------- - ---------------
  2        2        2        2   
im (p) + re (p)   im (p) + re (p)
4πre⁡(p)(re⁡(p))2+(im⁡(p))2−4iπim⁡(p)(re⁡(p))2+(im⁡(p))2\frac{4 \pi \operatorname{re}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} - \frac{4 i \pi \operatorname{im}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} 
producto
/      pi*re(p)         pi*I*im(p)  \ /   5*pi*re(p)       5*pi*I*im(p) \
|- --------------- + ---------------|*|--------------- - ---------------|
|    2        2        2        2   | |  2        2        2        2   |
\  im (p) + re (p)   im (p) + re (p)/ \im (p) + re (p)   im (p) + re (p)/
(−πre⁡(p)(re⁡(p))2+(im⁡(p))2+iπim⁡(p)(re⁡(p))2+(im⁡(p))2)(5πre⁡(p)(re⁡(p))2+(im⁡(p))2−5iπim⁡(p)(re⁡(p))2+(im⁡(p))2)\left(- \frac{\pi \operatorname{re}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} + \frac{i \pi \operatorname{im}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}}\right) \left(\frac{5 \pi \operatorname{re}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}} - \frac{5 i \pi \operatorname{im}{\left(p\right)}}{\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}}\right) 
=
     2                   2
-5*pi *(-I*im(p) + re(p)) 
--------------------------
                     2    
    /  2        2   \     
    \im (p) + re (p)/
−5π2(re⁡(p)−iim⁡(p))2((re⁡(p))2+(im⁡(p))2)2- \frac{5 \pi^{2} \left(\operatorname{re}{\left(p\right)} - i \operatorname{im}{\left(p\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} + \left(\operatorname{im}{\left(p\right)}\right)^{2}\right)^{2}} 
-5*pi^2*(-i*im(p) + re(p))^2/(im(p)^2 + re(p)^2)^2
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