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tgx+ctgx=3/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
tan(x) + cot(x) = 3/2
tan⁡(x)+cot⁡(x)=32\tan{\left(x \right)} + \cot{\left(x \right)} = \frac{3}{2}
Simplificar
Solución detallada
Tenemos la ecuación
tan⁡(x)+cot⁡(x)=32\tan{\left(x \right)} + \cot{\left(x \right)} = \frac{3}{2}
cambiamos
tan⁡(x)+cot⁡(x)−52=0\tan{\left(x \right)} + \cot{\left(x \right)} - \frac{5}{2} = 0
tan⁡(x)+cot⁡(x)−52=0\tan{\left(x \right)} + \cot{\left(x \right)} - \frac{5}{2} = 0
Sustituimos
w=cot⁡(x)w = \cot{\left(x \right)}
Tenemos la ecuación:
w+tan⁡(x)−52=0w + \tan{\left(x \right)} - \frac{5}{2} = 0
cambiamos:
w+tan⁡(x)−52=0w + \tan{\left(x \right)} - \frac{5}{2} = 0
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-5/2 + w + tanx = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w+tan⁡(x)=52w + \tan{\left(x \right)} = \frac{5}{2}
Dividamos ambos miembros de la ecuación en (w + tan(x))/w
w = 5/2 / ((w + tan(x))/w)

Obtenemos la respuesta: w = 5/2 - tan(x)
hacemos cambio inverso
cot⁡(x)=w\cot{\left(x \right)} = w
sustituimos w:
Gráfica
0-80-60-40-2020406080-100100-2000020000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
         /    /        ___\\     /    /        ___\\
         |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||
x1 = I*im|atan|- - -------|| + re|atan|- - -------||
         \    \4      4   //     \    \4      4   //
x1=re⁡(atan⁡(34−7i4))+iim⁡(atan⁡(34−7i4))x_{1} = \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} 
         /    /        ___\\     /    /        ___\\
         |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||
x2 = I*im|atan|- + -------|| + re|atan|- + -------||
         \    \4      4   //     \    \4      4   //
x2=re⁡(atan⁡(34+7i4))+iim⁡(atan⁡(34+7i4))x_{2} = \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} 
x2 = re(atan(3/4 + sqrt(7)*i/4)) + i*im(atan(3/4 + sqrt(7)*i/4))
Suma y producto de raíces [src] 
suma
    /    /        ___\\     /    /        ___\\       /    /        ___\\     /    /        ___\\
    |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||       |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||
I*im|atan|- - -------|| + re|atan|- - -------|| + I*im|atan|- + -------|| + re|atan|- + -------||
    \    \4      4   //     \    \4      4   //       \    \4      4   //     \    \4      4   //
(re⁡(atan⁡(34−7i4))+iim⁡(atan⁡(34−7i4)))+(re⁡(atan⁡(34+7i4))+iim⁡(atan⁡(34+7i4)))\left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right) 
=
    /    /        ___\\       /    /        ___\\     /    /        ___\\     /    /        ___\\
    |    |3   I*\/ 7 ||       |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||
I*im|atan|- - -------|| + I*im|atan|- + -------|| + re|atan|- - -------|| + re|atan|- + -------||
    \    \4      4   //       \    \4      4   //     \    \4      4   //     \    \4      4   //
re⁡(atan⁡(34−7i4))+re⁡(atan⁡(34+7i4))+iim⁡(atan⁡(34−7i4))+iim⁡(atan⁡(34+7i4))\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} 
producto
/    /    /        ___\\     /    /        ___\\\ /    /    /        ___\\     /    /        ___\\\
|    |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||| |    |    |3   I*\/ 7 ||     |    |3   I*\/ 7 |||
|I*im|atan|- - -------|| + re|atan|- - -------|||*|I*im|atan|- + -------|| + re|atan|- + -------|||
\    \    \4      4   //     \    \4      4   /// \    \    \4      4   //     \    \4      4   ///
(re⁡(atan⁡(34−7i4))+iim⁡(atan⁡(34−7i4)))(re⁡(atan⁡(34+7i4))+iim⁡(atan⁡(34+7i4)))\left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right) 
=
/    /    /        ___\\     /    /        ___\\\ /    /    /        ___\\     /    /        ___\\\
|    |    |3   I*\/ 7 ||     |    |3   I*\/ 7 ||| |    |    |3   I*\/ 7 ||     |    |3   I*\/ 7 |||
|I*im|atan|- - -------|| + re|atan|- - -------|||*|I*im|atan|- + -------|| + re|atan|- + -------|||
\    \    \4      4   //     \    \4      4   /// \    \    \4      4   //     \    \4      4   ///
(re⁡(atan⁡(34−7i4))+iim⁡(atan⁡(34−7i4)))(re⁡(atan⁡(34+7i4))+iim⁡(atan⁡(34+7i4)))\left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right) 
(i*im(atan(3/4 - i*sqrt(7)/4)) + re(atan(3/4 - i*sqrt(7)/4)))*(i*im(atan(3/4 + i*sqrt(7)/4)) + re(atan(3/4 + i*sqrt(7)/4)))
Respuesta numérica [src] 
x1 = 0.785398163397448 - 0.397682730611953*i
x2 = 0.785398163397448 + 0.397682730611953*i
x2 = 0.785398163397448 + 0.397682730611953*i
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