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b=m*cos(t*w) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
b = m*cos(t*w)
b=mcos⁡(tw)b = m \cos{\left(t w \right)}
Simplificar
Solución detallada
Tenemos la ecuación
b=mcos⁡(tw)b = m \cos{\left(t w \right)}
es la ecuación trigonométrica más simple
Dividamos ambos miembros de la ecuación en -m

La ecuación se convierte en
cos⁡(tw)=bm\cos{\left(t w \right)} = \frac{b}{m}
Esta ecuación se reorganiza en
tw=πn+acos⁡(bm)t w = \pi n + \operatorname{acos}{\left(\frac{b}{m} \right)}
tw=πn+acos⁡(bm)−πt w = \pi n + \operatorname{acos}{\left(\frac{b}{m} \right)} - \pi
O
tw=πn+acos⁡(bm)t w = \pi n + \operatorname{acos}{\left(\frac{b}{m} \right)}
tw=πn+acos⁡(bm)−πt w = \pi n + \operatorname{acos}{\left(\frac{b}{m} \right)} - \pi
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
tt
obtenemos la respuesta:
w1=πn+acos⁡(bm)tw_{1} = \frac{\pi n + \operatorname{acos}{\left(\frac{b}{m} \right)}}{t}
w2=πn+acos⁡(bm)−πtw_{2} = \frac{\pi n + \operatorname{acos}{\left(\frac{b}{m} \right)} - \pi}{t}
Gráfica
Respuesta rápida [src] 
       /  /    /    /b\\       \           /    /b\\      \   /    /    /b\\       \                 /    /b\\
       |  |- re|acos|-|| + 2*pi|*im(t)   im|acos|-||*re(t)|   |- re|acos|-|| + 2*pi|*re(t)   im(t)*im|acos|-||
       |  \    \    \m//       /           \    \m//      |   \    \    \m//       /                 \    \m//
w1 = I*|- ---------------------------- - -----------------| + ---------------------------- - -----------------
       |          2        2                2        2    |           2        2                2        2    
       \        im (t) + re (t)           im (t) + re (t) /         im (t) + re (t)           im (t) + re (t)
w1=i(−(−re⁡(acos⁡(bm))+2π)im⁡(t)(re⁡(t))2+(im⁡(t))2−re⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2)+(−re⁡(acos⁡(bm))+2π)re⁡(t)(re⁡(t))2+(im⁡(t))2−im⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2w_{1} = i \left(- \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{im}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{re}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}}\right) + \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{re}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{im}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} 
         /    /b\\     /    /b\\
         |acos|-||     |acos|-||
         |    \m/|     |    \m/|
w2 = I*im|-------| + re|-------|
         \   t   /     \   t   /
w2=re⁡(acos⁡(bm)t)+iim⁡(acos⁡(bm)t)w_{2} = \operatorname{re}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)} + i \operatorname{im}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)} 
w2 = re(acos(b/m)/t) + i*im(acos(b/m)/t)
Suma y producto de raíces [src] 
suma
  /  /    /    /b\\       \           /    /b\\      \   /    /    /b\\       \                 /    /b\\       /    /b\\     /    /b\\
  |  |- re|acos|-|| + 2*pi|*im(t)   im|acos|-||*re(t)|   |- re|acos|-|| + 2*pi|*re(t)   im(t)*im|acos|-||       |acos|-||     |acos|-||
  |  \    \    \m//       /           \    \m//      |   \    \    \m//       /                 \    \m//       |    \m/|     |    \m/|
I*|- ---------------------------- - -----------------| + ---------------------------- - ----------------- + I*im|-------| + re|-------|
  |          2        2                2        2    |           2        2                2        2           \   t   /     \   t   /
  \        im (t) + re (t)           im (t) + re (t) /         im (t) + re (t)           im (t) + re (t)
(re⁡(acos⁡(bm)t)+iim⁡(acos⁡(bm)t))+(i(−(−re⁡(acos⁡(bm))+2π)im⁡(t)(re⁡(t))2+(im⁡(t))2−re⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2)+(−re⁡(acos⁡(bm))+2π)re⁡(t)(re⁡(t))2+(im⁡(t))2−im⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2)\left(\operatorname{re}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)} + i \operatorname{im}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)}\right) + \left(i \left(- \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{im}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{re}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}}\right) + \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{re}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{im}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}}\right) 
=
  /  /    /    /b\\       \           /    /b\\      \       /    /b\\   /    /    /b\\       \                 /    /b\\     /    /b\\
  |  |- re|acos|-|| + 2*pi|*im(t)   im|acos|-||*re(t)|       |acos|-||   |- re|acos|-|| + 2*pi|*re(t)   im(t)*im|acos|-||     |acos|-||
  |  \    \    \m//       /           \    \m//      |       |    \m/|   \    \    \m//       /                 \    \m//     |    \m/|
I*|- ---------------------------- - -----------------| + I*im|-------| + ---------------------------- - ----------------- + re|-------|
  |          2        2                2        2    |       \   t   /           2        2                2        2         \   t   /
  \        im (t) + re (t)           im (t) + re (t) /                         im (t) + re (t)           im (t) + re (t)
i(−(−re⁡(acos⁡(bm))+2π)im⁡(t)(re⁡(t))2+(im⁡(t))2−re⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2)+re⁡(acos⁡(bm)t)+iim⁡(acos⁡(bm)t)+(−re⁡(acos⁡(bm))+2π)re⁡(t)(re⁡(t))2+(im⁡(t))2−im⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2i \left(- \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{im}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{re}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}}\right) + \operatorname{re}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)} + i \operatorname{im}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)} + \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{re}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{im}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} 
producto
/  /  /    /    /b\\       \           /    /b\\      \   /    /    /b\\       \                 /    /b\\\ /    /    /b\\     /    /b\\\
|  |  |- re|acos|-|| + 2*pi|*im(t)   im|acos|-||*re(t)|   |- re|acos|-|| + 2*pi|*re(t)   im(t)*im|acos|-||| |    |acos|-||     |acos|-|||
|  |  \    \    \m//       /           \    \m//      |   \    \    \m//       /                 \    \m//| |    |    \m/|     |    \m/||
|I*|- ---------------------------- - -----------------| + ---------------------------- - -----------------|*|I*im|-------| + re|-------||
|  |          2        2                2        2    |           2        2                2        2    | \    \   t   /     \   t   //
\  \        im (t) + re (t)           im (t) + re (t) /         im (t) + re (t)           im (t) + re (t) /
(re⁡(acos⁡(bm)t)+iim⁡(acos⁡(bm)t))(i(−(−re⁡(acos⁡(bm))+2π)im⁡(t)(re⁡(t))2+(im⁡(t))2−re⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2)+(−re⁡(acos⁡(bm))+2π)re⁡(t)(re⁡(t))2+(im⁡(t))2−im⁡(t)im⁡(acos⁡(bm))(re⁡(t))2+(im⁡(t))2)\left(\operatorname{re}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)} + i \operatorname{im}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)}\right) \left(i \left(- \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{im}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{re}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}}\right) + \frac{\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} + 2 \pi\right) \operatorname{re}{\left(t\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\operatorname{im}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}}\right) 
=
 /    /    /b\\     /    /b\\\                                                                                                         
 |    |acos|-||     |acos|-|||                                                                                                         
 |    |    \m/|     |    \m/|| //          /    /b\\\                 /    /b\\     //          /    /b\\\           /    /b\\      \\ 
-|I*im|-------| + re|-------||*||-2*pi + re|acos|-|||*re(t) + im(t)*im|acos|-|| - I*||-2*pi + re|acos|-|||*im(t) - im|acos|-||*re(t)|| 
 \    \   t   /     \   t   // \\          \    \m///                 \    \m//     \\          \    \m///           \    \m//      // 
---------------------------------------------------------------------------------------------------------------------------------------
                                                              2        2                                                               
                                                            im (t) + re (t)
−(re⁡(acos⁡(bm)t)+iim⁡(acos⁡(bm)t))(−i((re⁡(acos⁡(bm))−2π)im⁡(t)−re⁡(t)im⁡(acos⁡(bm)))+(re⁡(acos⁡(bm))−2π)re⁡(t)+im⁡(t)im⁡(acos⁡(bm)))(re⁡(t))2+(im⁡(t))2- \frac{\left(\operatorname{re}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)} + i \operatorname{im}{\left(\frac{\operatorname{acos}{\left(\frac{b}{m} \right)}}{t}\right)}\right) \left(- i \left(\left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} - 2 \pi\right) \operatorname{im}{\left(t\right)} - \operatorname{re}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)} - 2 \pi\right) \operatorname{re}{\left(t\right)} + \operatorname{im}{\left(t\right)} \operatorname{im}{\left(\operatorname{acos}{\left(\frac{b}{m} \right)}\right)}\right)}{\left(\operatorname{re}{\left(t\right)}\right)^{2} + \left(\operatorname{im}{\left(t\right)}\right)^{2}} 
-(i*im(acos(b/m)/t) + re(acos(b/m)/t))*((-2*pi + re(acos(b/m)))*re(t) + im(t)*im(acos(b/m)) - i*((-2*pi + re(acos(b/m)))*im(t) - im(acos(b/m))*re(t)))/(im(t)^2 + re(t)^2)
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