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tg*π(2x−21)/6=√3/3 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
                        ___
tan(p)*I*(2*x - 21)   \/ 3 
------------------- = -----
         6              3  
$$\frac{i \tan{\left(p \right)} \left(2 x - 21\right)}{6} = \frac{\sqrt{3}}{3}$$
Solución detallada
Tenemos la ecuación
$$\frac{i \tan{\left(p \right)} \left(2 x - 21\right)}{6} = \frac{\sqrt{3}}{3}$$
cambiamos
$$\frac{i \left(2 x - 21\right) \tan{\left(p \right)}}{6} - 1 - \frac{\sqrt{3}}{3} = 0$$
$$\frac{i \tan{\left(p \right)} \left(2 x - 21\right)}{6} - 1 - \frac{\sqrt{3}}{3} = 0$$
Sustituimos
$$w = \tan{\left(p \right)}$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 - sqrt3/3 + i*w2*x/6+21/6 = 0

Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
-1 - sqrt(3)/3 + i*w*(-21 + 2*x)/6 = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$\frac{i w \left(2 x - 21\right)}{6} - \frac{\sqrt{3}}{3} = 1$$
Dividamos ambos miembros de la ecuación en (-sqrt(3)/3 + i*w*(-21 + 2*x)/6)/w
w = 1 / ((-sqrt(3)/3 + i*w*(-21 + 2*x)/6)/w)

Obtenemos la respuesta: w = -2*i*(3 + sqrt(3))/(-21 + 2*x)
hacemos cambio inverso
$$\tan{\left(p \right)} = w$$
sustituimos w:
Gráfica
Suma y producto de raíces [src]
suma
                                              ___                                                                                                    ___                                                    
21                                          \/ 3 *sinh(2*im(p))                                                                                  I*\/ 3 *sin(2*re(p))                                       
-- - -------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------
2    /            2                                  2                 \                                  /            2                                  2                 \                               
     |         sin (2*re(p))                     sinh (2*im(p))        |                                  |         sin (2*re(p))                     sinh (2*im(p))        |                               
     |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))   |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))
     |                              2                                 2|                                  |                              2                                 2|                               
     \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                                  \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                               
$$\frac{21}{2} - \frac{\sqrt{3} i \sin{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)} - \frac{\sqrt{3} \sinh{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)}$$
=
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21                                          \/ 3 *sinh(2*im(p))                                                                                  I*\/ 3 *sin(2*re(p))                                       
-- - -------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------
2    /            2                                  2                 \                                  /            2                                  2                 \                               
     |         sin (2*re(p))                     sinh (2*im(p))        |                                  |         sin (2*re(p))                     sinh (2*im(p))        |                               
     |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))   |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))
     |                              2                                 2|                                  |                              2                                 2|                               
     \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                                  \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                               
$$\frac{21}{2} - \frac{\sqrt{3} i \sin{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)} - \frac{\sqrt{3} \sinh{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)}$$
producto
                                              ___                                                                                                    ___                                                    
21                                          \/ 3 *sinh(2*im(p))                                                                                  I*\/ 3 *sin(2*re(p))                                       
-- - -------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------
2    /            2                                  2                 \                                  /            2                                  2                 \                               
     |         sin (2*re(p))                     sinh (2*im(p))        |                                  |         sin (2*re(p))                     sinh (2*im(p))        |                               
     |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))   |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))
     |                              2                                 2|                                  |                              2                                 2|                               
     \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                                  \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                               
$$\frac{21}{2} - \frac{\sqrt{3} i \sin{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)} - \frac{\sqrt{3} \sinh{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)}$$
=
 /                     /                        ___                    ___              \\ 
-\-21*sin(2*re(p)) + I*\-21*sinh(2*im(p)) + 2*\/ 3 *cos(2*re(p)) + 2*\/ 3 *cosh(2*im(p))// 
-------------------------------------------------------------------------------------------
                             2*sin(2*re(p)) + 2*I*sinh(2*im(p))                            
$$- \frac{i \left(2 \sqrt{3} \cos{\left(2 \operatorname{re}{\left(p\right)} \right)} - 21 \sinh{\left(2 \operatorname{im}{\left(p\right)} \right)} + 2 \sqrt{3} \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right) - 21 \sin{\left(2 \operatorname{re}{\left(p\right)} \right)}}{2 \sin{\left(2 \operatorname{re}{\left(p\right)} \right)} + 2 i \sinh{\left(2 \operatorname{im}{\left(p\right)} \right)}}$$
-(-21*sin(2*re(p)) + i*(-21*sinh(2*im(p)) + 2*sqrt(3)*cos(2*re(p)) + 2*sqrt(3)*cosh(2*im(p))))/(2*sin(2*re(p)) + 2*i*sinh(2*im(p)))
Respuesta rápida [src]
                                                   ___                                                                                                    ___                                                    
     21                                          \/ 3 *sinh(2*im(p))                                                                                  I*\/ 3 *sin(2*re(p))                                       
x1 = -- - -------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------
     2    /            2                                  2                 \                                  /            2                                  2                 \                               
          |         sin (2*re(p))                     sinh (2*im(p))        |                                  |         sin (2*re(p))                     sinh (2*im(p))        |                               
          |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))   |------------------------------- + -------------------------------|*(cos(2*re(p)) + cosh(2*im(p)))
          |                              2                                 2|                                  |                              2                                 2|                               
          \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                                  \(cos(2*re(p)) + cosh(2*im(p)))    (cos(2*re(p)) + cosh(2*im(p))) /                               
$$x_{1} = \frac{21}{2} - \frac{\sqrt{3} i \sin{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)} - \frac{\sqrt{3} \sinh{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\frac{\sin^{2}{\left(2 \operatorname{re}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}} + \frac{\sinh^{2}{\left(2 \operatorname{im}{\left(p\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)^{2}}\right) \left(\cos{\left(2 \operatorname{re}{\left(p\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(p\right)} \right)}\right)}$$
x1 = 21/2 - sqrt(3)*i*sin(2*re(p))/((sin(2*re(p))^2/(cos(2*re(p)) + cosh(2*im(p)))^2 + sinh(2*im(p))^2/(cos(2*re(p)) + cosh(2*im(p)))^2)*(cos(2*re(p)) + cosh(2*im(p)))) - sqrt(3)*sinh(2*im(p))/((sin(2*re(p))^2/(cos(2*re(p)) + cosh(2*im(p)))^2 + sinh(2*im(p))^2/(cos(2*re(p)) + cosh(2*im(p)))^2)*(cos(2*re(p)) + cosh(2*im(p))))