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cospi(4x-7)/6=sqrt(3)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                       ___
cos(p)*I*(4*x - 7)   \/ 3 
------------------ = -----
        6              2
icos⁡(p)(4x−7)6=32\frac{i \cos{\left(p \right)} \left(4 x - 7\right)}{6} = \frac{\sqrt{3}}{2}
Simplificar
Solución detallada
Tenemos la ecuación
icos⁡(p)(4x−7)6=32\frac{i \cos{\left(p \right)} \left(4 x - 7\right)}{6} = \frac{\sqrt{3}}{2}
cambiamos
i(4x−7)cos⁡(p)6−1−32=0\frac{i \left(4 x - 7\right) \cos{\left(p \right)}}{6} - 1 - \frac{\sqrt{3}}{2} = 0
icos⁡(p)(4x−7)6−1−32=0\frac{i \cos{\left(p \right)} \left(4 x - 7\right)}{6} - 1 - \frac{\sqrt{3}}{2} = 0
Sustituimos
w=cos⁡(p)w = \cos{\left(p \right)}
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 - sqrt3/2 + i*w4*x/6+7/6 = 0

Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
-1 - sqrt(3)/2 + i*w*(-7 + 4*x)/6 = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
iw(4x−7)6−32=1\frac{i w \left(4 x - 7\right)}{6} - \frac{\sqrt{3}}{2} = 1
Dividamos ambos miembros de la ecuación en (-sqrt(3)/2 + i*w*(-7 + 4*x)/6)/w
w = 1 / ((-sqrt(3)/2 + i*w*(-7 + 4*x)/6)/w)

Obtenemos la respuesta: w = -3*i*(2 + sqrt(3))/(-7 + 4*x)
hacemos cambio inverso
cos⁡(p)=w\cos{\left(p \right)} = w
sustituimos w:
Gráfica
Respuesta rápida [src] 
                          ___                                                        ___                                  
     7                3*\/ 3 *sin(re(p))*sinh(im(p))                           3*I*\/ 3 *cos(re(p))*cosh(im(p))           
x1 = - + ------------------------------------------------------- - -------------------------------------------------------
     4     /   2            2             2            2       \     /   2            2             2            2       \
         4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/   4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/
x1=74+33sin⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))−33icos⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))x_{1} = \frac{7}{4} + \frac{3 \sqrt{3} \sin{\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)} \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)}\right)} - \frac{3 \sqrt{3} i \cos{\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)} \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)}\right)} 
x1 = 7/4 + 3*sqrt(3)*sin(re(p))*sinh(im(p))/(4*(sin(re(p))^2*sinh(im(p))^2 + cos(re(p))^2*cosh(im(p))^2)) - 3*sqrt(3)*i*cos(re(p))*cosh(im(p))/(4*(sin(re(p))^2*sinh(im(p))^2 + cos(re(p))^2*cosh(im(p))^2))
Suma y producto de raíces [src] 
suma
                     ___                                                        ___                                  
7                3*\/ 3 *sin(re(p))*sinh(im(p))                           3*I*\/ 3 *cos(re(p))*cosh(im(p))           
- + ------------------------------------------------------- - -------------------------------------------------------
4     /   2            2             2            2       \     /   2            2             2            2       \
    4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/   4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/
74+33sin⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))−33icos⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))\frac{7}{4} + \frac{3 \sqrt{3} \sin{\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)} \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)}\right)} - \frac{3 \sqrt{3} i \cos{\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)} \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)}\right)} 
=
                     ___                                                        ___                                  
7                3*\/ 3 *sin(re(p))*sinh(im(p))                           3*I*\/ 3 *cos(re(p))*cosh(im(p))           
- + ------------------------------------------------------- - -------------------------------------------------------
4     /   2            2             2            2       \     /   2            2             2            2       \
    4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/   4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/
74+33sin⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))−33icos⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))\frac{7}{4} + \frac{3 \sqrt{3} \sin{\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)} \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)}\right)} - \frac{3 \sqrt{3} i \cos{\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)} \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)}\right)} 
producto
                     ___                                                        ___                                  
7                3*\/ 3 *sin(re(p))*sinh(im(p))                           3*I*\/ 3 *cos(re(p))*cosh(im(p))           
- + ------------------------------------------------------- - -------------------------------------------------------
4     /   2            2             2            2       \     /   2            2             2            2       \
    4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/   4*\cos (re(p))*cosh (im(p)) + sin (re(p))*sinh (im(p))/
74+33sin⁡(re⁡(p))sinh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))−33icos⁡(re⁡(p))cosh⁡(im⁡(p))4(sin⁡2(re⁡(p))sinh⁡2(im⁡(p))+cos⁡2(re⁡(p))cosh⁡2(im⁡(p)))\frac{7}{4} + \frac{3 \sqrt{3} \sin{\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)} \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)}\right)} - \frac{3 \sqrt{3} i \cos{\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)} \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)}\right)} 
=
  /    ___                           \                           
I*\3*\/ 3  + 7*sin(re(p))*sinh(im(p))/ - 7*cos(re(p))*cosh(im(p))
-----------------------------------------------------------------
      4*(-cos(re(p))*cosh(im(p)) + I*sin(re(p))*sinh(im(p)))
i(7sin⁡(re⁡(p))sinh⁡(im⁡(p))+33)−7cos⁡(re⁡(p))cosh⁡(im⁡(p))4(isin⁡(re⁡(p))sinh⁡(im⁡(p))−cos⁡(re⁡(p))cosh⁡(im⁡(p)))\frac{i \left(7 \sin{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)} + 3 \sqrt{3}\right) - 7 \cos{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{4 \left(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)}\right)} 
(i*(3*sqrt(3) + 7*sin(re(p))*sinh(im(p))) - 7*cos(re(p))*cosh(im(p)))/(4*(-cos(re(p))*cosh(im(p)) + i*sin(re(p))*sinh(im(p))))
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