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tg^2(x)+ctg^2(x)+3*(tg(x)+ctg(x))+4=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2         2                                 
tan (x) + cot (x) + 3*(tan(x) + cot(x)) + 4 = 0
(3(tan⁡(x)+cot⁡(x))+(tan⁡2(x)+cot⁡2(x)))+4=0\left(3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right)\right) + 4 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(3(tan⁡(x)+cot⁡(x))+(tan⁡2(x)+cot⁡2(x)))+4=0\left(3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right)\right) + 4 = 0
cambiamos
tan⁡2(x)+3tan⁡(x)+cot⁡2(x)+3cot⁡(x)+3=0\tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 \cot{\left(x \right)} + 3 = 0
3(tan⁡(x)+cot⁡(x))+tan⁡2(x)+cot⁡2(x)+3=03 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 = 0
Sustituimos
w=cot⁡(x)w = \cot{\left(x \right)}
Es la ecuación de la forma
a*w^2 + b*w + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=3b = 3
c=tan⁡2(x)+3tan⁡(x)+3c = \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + 3
, entonces
D = b^2 - 4 * a * c = 

(3)^2 - 4 * (1) * (3 + tan(x)^2 + 3*tan(x)) = -3 - 12*tan(x) - 4*tan(x)^2

La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

o
w1=−4tan⁡2(x)−12tan⁡(x)−32−32w_{1} = \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}
w2=−−4tan⁡2(x)−12tan⁡(x)−32−32w_{2} = - \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}
hacemos cambio inverso
cot⁡(x)=w\cot{\left(x \right)} = w
sustituimos w:
Gráfica
0-80-60-40-2020406080-1001000250000000000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
           /    /        ___\\       /    /        ___\\       /    /        ___\\       /    /        ___\\
  pi       |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||
- -- + - re|atan|- - -------|| - I*im|atan|- - -------|| + - re|atan|- + -------|| - I*im|atan|- + -------||
  4        \    \2      2   //       \    \2      2   //       \    \2      2   //       \    \2      2   //
(−re⁡(atan⁡(12+3i2))−iim⁡(atan⁡(12+3i2)))+(−π4+(−re⁡(atan⁡(12−3i2))−iim⁡(atan⁡(12−3i2))))\left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) + \left(- \frac{\pi}{4} + \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right)\right) 
=
    /    /        ___\\     /    /        ___\\            /    /        ___\\       /    /        ___\\
    |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||   pi       |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||
- re|atan|- + -------|| - re|atan|- - -------|| - -- - I*im|atan|- + -------|| - I*im|atan|- - -------||
    \    \2      2   //     \    \2      2   //   4        \    \2      2   //       \    \2      2   //
−re⁡(atan⁡(12+3i2))−re⁡(atan⁡(12−3i2))−π4−iim⁡(atan⁡(12+3i2))−iim⁡(atan⁡(12−3i2))- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - \frac{\pi}{4} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} 
producto
     /    /    /        ___\\       /    /        ___\\\ /    /    /        ___\\       /    /        ___\\\
-pi  |    |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||| |    |    |1   I*\/ 3 ||       |    |1   I*\/ 3 |||
----*|- re|atan|- - -------|| - I*im|atan|- - -------|||*|- re|atan|- + -------|| - I*im|atan|- + -------|||
 4   \    \    \2      2   //       \    \2      2   /// \    \    \2      2   //       \    \2      2   ///
−π4(−re⁡(atan⁡(12−3i2))−iim⁡(atan⁡(12−3i2)))(−re⁡(atan⁡(12+3i2))−iim⁡(atan⁡(12+3i2)))- \frac{\pi}{4} \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) 
=
    /    /    /        ___\\     /    /        ___\\\ /    /    /        ___\\     /    /        ___\\\ 
    |    |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||| |    |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||| 
-pi*|I*im|atan|- + -------|| + re|atan|- + -------|||*|I*im|atan|- - -------|| + re|atan|- - -------||| 
    \    \    \2      2   //     \    \2      2   /// \    \    \2      2   //     \    \2      2   /// 
--------------------------------------------------------------------------------------------------------
                                                   4
−π(re⁡(atan⁡(12−3i2))+iim⁡(atan⁡(12−3i2)))(re⁡(atan⁡(12+3i2))+iim⁡(atan⁡(12+3i2)))4- \frac{\pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)}{4} 
-pi*(i*im(atan(1/2 + i*sqrt(3)/2)) + re(atan(1/2 + i*sqrt(3)/2)))*(i*im(atan(1/2 - i*sqrt(3)/2)) + re(atan(1/2 - i*sqrt(3)/2)))/4
Respuesta rápida [src] 
     -pi 
x1 = ----
      4
x1=−π4x_{1} = - \frac{\pi}{4} 
         /    /        ___\\       /    /        ___\\
         |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||
x2 = - re|atan|- - -------|| - I*im|atan|- - -------||
         \    \2      2   //       \    \2      2   //
x2=−re⁡(atan⁡(12−3i2))−iim⁡(atan⁡(12−3i2))x_{2} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} 
         /    /        ___\\       /    /        ___\\
         |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||
x3 = - re|atan|- + -------|| - I*im|atan|- + -------||
         \    \2      2   //       \    \2      2   //
x3=−re⁡(atan⁡(12+3i2))−iim⁡(atan⁡(12+3i2))x_{3} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} 
x3 = -re(atan(1/2 + sqrt(3)*i/2)) - i*im(atan(1/2 + sqrt(3)*i/2))
Respuesta numérica [src] 
x1 = 40.0553062162188
x2 = -76.1836216304923
x3 = -54.1924731017967
x4 = -32.2013244485502
x5 = -98.1747706322284
x6 = 2.35619441168939
x7 = 30.6305285608463
x8 = 68.3296401581437
x9 = 18.0641575857363
x10 = -54.192473039953
x11 = -98.1747702202347
x12 = -63.6172513370425
x13 = 33.7721213315842
x14 = -41.6261027538328
x15 = 55.7632699272178
x16 = -91.8915846481544
x17 = 99.7455671224518
x18 = -3.92699077227342
x19 = -3.92699077276863
x20 = -25.9181399071463
x21 = 24.3473429940575
x22 = 46.3384915761999
x23 = -76.1836218812413
x24 = 14.9225649923453
x25 = 77.7544185241376
x26 = 90.3207887399165
x27 = -85.6083999205894
x28 = 62.0464548249302
x29 = 93.4623811044448
x30 = 96.6039735663169
x31 = -25.918139886171
x32 = -10.2101758562123
x33 = 84.0376034236404
x34 = -47.9092883924126
x35 = 11.7809727371379
x36 = -3.92699110792993
x37 = 74.6128249560098
x38 = -0.785398151967561
x39 = -19.6349541709315
x39 = -19.6349541709315
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