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cospi(2x+36)/4=-sqrt(2)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
                         ___ 
cos(p)*I*(2*x + 36)   -\/ 2  
------------------- = -------
         4               2   
$$\frac{i \cos{\left(p \right)} \left(2 x + 36\right)}{4} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Solución detallada
Tenemos la ecuación
$$\frac{i \cos{\left(p \right)} \left(2 x + 36\right)}{4} = \frac{\left(-1\right) \sqrt{2}}{2}$$
cambiamos
$$\frac{i \left(x + 18\right) \cos{\left(p \right)}}{2} - 1 + \frac{\sqrt{2}}{2} = 0$$
$$\frac{i \cos{\left(p \right)} \left(2 x + 36\right)}{4} - 1 - \frac{\left(-1\right) \sqrt{2}}{2} = 0$$
Sustituimos
$$w = \cos{\left(p \right)}$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 - -sqrt+2)/2 + i*w2*x/4+36/4 = 0

Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
-1 + sqrt(2)/2 + i*w*(36 + 2*x)/4 = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$\frac{i w \left(2 x + 36\right)}{4} + \frac{\sqrt{2}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (sqrt(2)/2 + i*w*(36 + 2*x)/4)/w
w = 1 / ((sqrt(2)/2 + i*w*(36 + 2*x)/4)/w)

Obtenemos la respuesta: w = i*(-2 + sqrt(2))/(18 + x)
hacemos cambio inverso
$$\cos{\left(p \right)} = w$$
sustituimos w:
Gráfica
Respuesta rápida [src]
                         ___                                                    ___                                 
                       \/ 2 *sin(re(p))*sinh(im(p))                         I*\/ 2 *cos(re(p))*cosh(im(p))          
x1 = -18 - --------------------------------------------------- + ---------------------------------------------------
              2            2             2            2             2            2             2            2       
           cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))   cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))
$$x_{1} = -18 - \frac{\sqrt{2} \sin{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}} + \frac{\sqrt{2} i \cos{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}}$$
x1 = -18 - sqrt(2)*sin(re(p))*sinh(im(p))/(sin(re(p))^2*sinh(im(p))^2 + cos(re(p))^2*cosh(im(p))^2) + sqrt(2)*i*cos(re(p))*cosh(im(p))/(sin(re(p))^2*sinh(im(p))^2 + cos(re(p))^2*cosh(im(p))^2)
Suma y producto de raíces [src]
suma
                    ___                                                    ___                                 
                  \/ 2 *sin(re(p))*sinh(im(p))                         I*\/ 2 *cos(re(p))*cosh(im(p))          
-18 - --------------------------------------------------- + ---------------------------------------------------
         2            2             2            2             2            2             2            2       
      cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))   cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))
$$-18 - \frac{\sqrt{2} \sin{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}} + \frac{\sqrt{2} i \cos{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}}$$
=
                    ___                                                    ___                                 
                  \/ 2 *sin(re(p))*sinh(im(p))                         I*\/ 2 *cos(re(p))*cosh(im(p))          
-18 - --------------------------------------------------- + ---------------------------------------------------
         2            2             2            2             2            2             2            2       
      cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))   cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))
$$-18 - \frac{\sqrt{2} \sin{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}} + \frac{\sqrt{2} i \cos{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}}$$
producto
                    ___                                                    ___                                 
                  \/ 2 *sin(re(p))*sinh(im(p))                         I*\/ 2 *cos(re(p))*cosh(im(p))          
-18 - --------------------------------------------------- + ---------------------------------------------------
         2            2             2            2             2            2             2            2       
      cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))   cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))
$$-18 - \frac{\sqrt{2} \sin{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}} + \frac{\sqrt{2} i \cos{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}}$$
=
 /  /  ___                            \                            \ 
-\I*\\/ 2  + 18*sin(re(p))*sinh(im(p))/ - 18*cos(re(p))*cosh(im(p))/ 
---------------------------------------------------------------------
          -cos(re(p))*cosh(im(p)) + I*sin(re(p))*sinh(im(p))         
$$- \frac{i \left(18 \sin{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)} + \sqrt{2}\right) - 18 \cos{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)} - \cos{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}$$
-(i*(sqrt(2) + 18*sin(re(p))*sinh(im(p))) - 18*cos(re(p))*cosh(im(p)))/(-cos(re(p))*cosh(im(p)) + i*sin(re(p))*sinh(im(p)))