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sin(ax/5)+(a/5)*cos(ax/5)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   /a*x\   a    /a*x\    
sin|---| + -*cos|---| = 0
   \ 5 /   5    \ 5 /
a5cos⁡(ax5)+sin⁡(ax5)=0\frac{a}{5} \cos{\left(\frac{a x}{5} \right)} + \sin{\left(\frac{a x}{5} \right)} = 0
Simplificar
Solución detallada
Tenemos la ecuación
a5cos⁡(ax5)+sin⁡(ax5)=0\frac{a}{5} \cos{\left(\frac{a x}{5} \right)} + \sin{\left(\frac{a x}{5} \right)} = 0
cambiamos:
sin⁡(ax5)cos⁡(ax5)=−a5\frac{\sin{\left(\frac{a x}{5} \right)}}{\cos{\left(\frac{a x}{5} \right)}} = - \frac{a}{5}
o
tan⁡(ax5)=−a5\tan{\left(\frac{a x}{5} \right)} = - \frac{a}{5}
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
ax5=πn+atan⁡(a5)\frac{a x}{5} = \pi n + \operatorname{atan}{\left(\frac{a}{5} \right)}
O
ax5=πn+atan⁡(a5)\frac{a x}{5} = \pi n + \operatorname{atan}{\left(\frac{a}{5} \right)}
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
a5\frac{a}{5}
obtenemos la respuesta:
x1=5(πn+atan⁡(a5))ax_{1} = \frac{5 \left(\pi n + \operatorname{atan}{\left(\frac{a}{5} \right)}\right)}{a}
Gráfica
Suma y producto de raíces [src] 
suma
       /    /        _________\\          /    /        _________\\        /    /       _________\\          /    /       _________\\
       |    |       /       2 ||          |    |       /       2 ||        |    |      /       2 ||          |    |      /       2 ||
       |    |-5 + \/  25 + a  ||          |    |-5 + \/  25 + a  ||        |    |5 + \/  25 + a  ||          |    |5 + \/  25 + a  ||
       |atan|-----------------||          |atan|-----------------||        |atan|----------------||          |atan|----------------||
       |    \        a        /|          |    \        a        /|        |    \       a        /|          |    \       a        /|
- 10*re|-----------------------| - 10*I*im|-----------------------| + 10*re|----------------------| + 10*I*im|----------------------|
       \           a           /          \           a           /        \          a           /          \          a           /
(−10re⁡(atan⁡(a2+25−5a)a)−10iim⁡(atan⁡(a2+25−5a)a))+(10re⁡(atan⁡(a2+25+5a)a)+10iim⁡(atan⁡(a2+25+5a)a))\left(- 10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)} - 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)}\right) + \left(10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)} + 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)}\right) 
=
       /    /        _________\\        /    /       _________\\          /    /        _________\\          /    /       _________\\
       |    |       /       2 ||        |    |      /       2 ||          |    |       /       2 ||          |    |      /       2 ||
       |    |-5 + \/  25 + a  ||        |    |5 + \/  25 + a  ||          |    |-5 + \/  25 + a  ||          |    |5 + \/  25 + a  ||
       |atan|-----------------||        |atan|----------------||          |atan|-----------------||          |atan|----------------||
       |    \        a        /|        |    \       a        /|          |    \        a        /|          |    \       a        /|
- 10*re|-----------------------| + 10*re|----------------------| - 10*I*im|-----------------------| + 10*I*im|----------------------|
       \           a           /        \          a           /          \           a           /          \          a           /
−10re⁡(atan⁡(a2+25−5a)a)+10re⁡(atan⁡(a2+25+5a)a)−10iim⁡(atan⁡(a2+25−5a)a)+10iim⁡(atan⁡(a2+25+5a)a)- 10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)} + 10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)} - 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)} + 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)} 
producto
/       /    /        _________\\          /    /        _________\\\ /     /    /       _________\\          /    /       _________\\\
|       |    |       /       2 ||          |    |       /       2 ||| |     |    |      /       2 ||          |    |      /       2 |||
|       |    |-5 + \/  25 + a  ||          |    |-5 + \/  25 + a  ||| |     |    |5 + \/  25 + a  ||          |    |5 + \/  25 + a  |||
|       |atan|-----------------||          |atan|-----------------||| |     |atan|----------------||          |atan|----------------|||
|       |    \        a        /|          |    \        a        /|| |     |    \       a        /|          |    \       a        /||
|- 10*re|-----------------------| - 10*I*im|-----------------------||*|10*re|----------------------| + 10*I*im|----------------------||
\       \           a           /          \           a           // \     \          a           /          \          a           //
(−10re⁡(atan⁡(a2+25−5a)a)−10iim⁡(atan⁡(a2+25−5a)a))(10re⁡(atan⁡(a2+25+5a)a)+10iim⁡(atan⁡(a2+25+5a)a))\left(- 10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)} - 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)}\right) \left(10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)} + 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)}\right) 
=
     /    /    /        _________\\     /    /        _________\\\ /    /    /       _________\\     /    /       _________\\\
     |    |    |       /       2 ||     |    |       /       2 ||| |    |    |      /       2 ||     |    |      /       2 |||
     |    |    |-5 + \/  25 + a  ||     |    |-5 + \/  25 + a  ||| |    |    |5 + \/  25 + a  ||     |    |5 + \/  25 + a  |||
     |    |atan|-----------------||     |atan|-----------------||| |    |atan|----------------||     |atan|----------------|||
     |    |    \        a        /|     |    \        a        /|| |    |    \       a        /|     |    \       a        /||
-100*|I*im|-----------------------| + re|-----------------------||*|I*im|----------------------| + re|----------------------||
     \    \           a           /     \           a           // \    \          a           /     \          a           //
−100(re⁡(atan⁡(a2+25−5a)a)+iim⁡(atan⁡(a2+25−5a)a))(re⁡(atan⁡(a2+25+5a)a)+iim⁡(atan⁡(a2+25+5a)a))- 100 \left(\operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)} + i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)}\right) \left(\operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)} + i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)}\right) 
-100*(i*im(atan((-5 + sqrt(25 + a^2))/a)/a) + re(atan((-5 + sqrt(25 + a^2))/a)/a))*(i*im(atan((5 + sqrt(25 + a^2))/a)/a) + re(atan((5 + sqrt(25 + a^2))/a)/a))
Respuesta rápida [src] 
            /    /        _________\\          /    /        _________\\
            |    |       /       2 ||          |    |       /       2 ||
            |    |-5 + \/  25 + a  ||          |    |-5 + \/  25 + a  ||
            |atan|-----------------||          |atan|-----------------||
            |    \        a        /|          |    \        a        /|
x1 = - 10*re|-----------------------| - 10*I*im|-----------------------|
            \           a           /          \           a           /
x1=−10re⁡(atan⁡(a2+25−5a)a)−10iim⁡(atan⁡(a2+25−5a)a)x_{1} = - 10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)} - 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} - 5}{a} \right)}}{a}\right)} 
          /    /       _________\\          /    /       _________\\
          |    |      /       2 ||          |    |      /       2 ||
          |    |5 + \/  25 + a  ||          |    |5 + \/  25 + a  ||
          |atan|----------------||          |atan|----------------||
          |    \       a        /|          |    \       a        /|
x2 = 10*re|----------------------| + 10*I*im|----------------------|
          \          a           /          \          a           /
x2=10re⁡(atan⁡(a2+25+5a)a)+10iim⁡(atan⁡(a2+25+5a)a)x_{2} = 10 \operatorname{re}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)} + 10 i \operatorname{im}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} + 25} + 5}{a} \right)}}{a}\right)} 
x2 = 10*re(atan((sqrt(a^2 + 25) + 5)/a)/a) + 10*i*im(atan((sqrt(a^2 + 25) + 5)/a)/a)
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