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sinП(8*x+7):4=-sqrt(2)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                        ___ 
sin(p)*I*(8*x + 7)   -\/ 2  
------------------ = -------
        4               2
isin⁡(p)(8x+7)4=(−1)22\frac{i \sin{\left(p \right)} \left(8 x + 7\right)}{4} = \frac{\left(-1\right) \sqrt{2}}{2}
Simplificar
Solución detallada
Tenemos la ecuación
isin⁡(p)(8x+7)4=(−1)22\frac{i \sin{\left(p \right)} \left(8 x + 7\right)}{4} = \frac{\left(-1\right) \sqrt{2}}{2}
cambiamos
i(8x+7)sin⁡(p)4−1+22=0\frac{i \left(8 x + 7\right) \sin{\left(p \right)}}{4} - 1 + \frac{\sqrt{2}}{2} = 0
isin⁡(p)(8x+7)4−1−(−1)22=0\frac{i \sin{\left(p \right)} \left(8 x + 7\right)}{4} - 1 - \frac{\left(-1\right) \sqrt{2}}{2} = 0
Sustituimos
w=sin⁡(p)w = \sin{\left(p \right)}
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 - -sqrt+2)/2 + i*w8*x/4+7/4 = 0

Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
-1 + sqrt(2)/2 + i*w*(7 + 8*x)/4 = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
iw(8x+7)4+22=1\frac{i w \left(8 x + 7\right)}{4} + \frac{\sqrt{2}}{2} = 1
Dividamos ambos miembros de la ecuación en (sqrt(2)/2 + i*w*(7 + 8*x)/4)/w
w = 1 / ((sqrt(2)/2 + i*w*(7 + 8*x)/4)/w)

Obtenemos la respuesta: w = 2*i*(-2 + sqrt(2))/(7 + 8*x)
hacemos cambio inverso
sin⁡(p)=w\sin{\left(p \right)} = w
sustituimos w:
Gráfica
Suma y producto de raíces [src] 
suma
                      ___                                                        ___                                   
  7                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
- - + ------------------------------------------------------- + -------------------------------------------------------
  8     /   2            2              2           2       \     /   2            2              2           2       \
      4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
−78+2isin⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))- \frac{7}{8} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
=
                      ___                                                        ___                                   
  7                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
- - + ------------------------------------------------------- + -------------------------------------------------------
  8     /   2            2              2           2       \     /   2            2              2           2       \
      4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
−78+2isin⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))- \frac{7}{8} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
producto
                      ___                                                        ___                                   
  7                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
- - + ------------------------------------------------------- + -------------------------------------------------------
  8     /   2            2              2           2       \     /   2            2              2           2       \
      4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
−78+2isin⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))- \frac{7}{8} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
=
 /  /      ___                           \                           \ 
-\I*\- 2*\/ 2  + 7*cos(re(p))*sinh(im(p))/ + 7*cosh(im(p))*sin(re(p))/ 
-----------------------------------------------------------------------
         8*cosh(im(p))*sin(re(p)) + 8*I*cos(re(p))*sinh(im(p))
−i(7cos⁡(re⁡(p))sinh⁡(im⁡(p))−22)+7sin⁡(re⁡(p))cosh⁡(im⁡(p))8sin⁡(re⁡(p))cosh⁡(im⁡(p))+8icos⁡(re⁡(p))sinh⁡(im⁡(p))- \frac{i \left(7 \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)} - 2 \sqrt{2}\right) + 7 \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{8 \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)} + 8 i \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}} 
-(i*(-2*sqrt(2) + 7*cos(re(p))*sinh(im(p))) + 7*cosh(im(p))*sin(re(p)))/(8*cosh(im(p))*sin(re(p)) + 8*i*cos(re(p))*sinh(im(p)))
Respuesta rápida [src] 
                           ___                                                        ___                                   
       7                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
x1 = - - + ------------------------------------------------------- + -------------------------------------------------------
       8     /   2            2              2           2       \     /   2            2              2           2       \
           4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   4*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
x1=−78+2isin⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))x_{1} = - \frac{7}{8} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
x1 = -7/8 + sqrt(2)*i*sin(re(p))*cosh(im(p))/(4*(sin(re(p))^2*cosh(im(p))^2 + cos(re(p))^2*sinh(im(p))^2)) + sqrt(2)*cos(re(p))*sinh(im(p))/(4*(sin(re(p))^2*cosh(im(p))^2 + cos(re(p))^2*sinh(im(p))^2))
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