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2(tgx-ctgx)=3^1/2((tgx)^2+(ctgx)^2)-2*3^1/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
                        ___ /   2         2   \       ___
2*(tan(x) - cot(x)) = \/ 3 *\tan (x) + cot (x)/ - 2*\/ 3 
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) = \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 2 \sqrt{3}$$
Solución detallada
Tenemos la ecuación
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) = \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 2 \sqrt{3}$$
cambiamos
$$- \sqrt{3} \left(\tan^{2}{\left(x \right)} + \frac{1}{\tan^{2}{\left(x \right)}}\right) + 2 \tan{\left(x \right)} - 2 \cot{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) - \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 1 + 2 \sqrt{3} = 0$$
Sustituimos
$$w = \cot{\left(x \right)}$$
Abramos la expresión en la ecuación
$$- 2 w - \sqrt{3} \left(w^{2} + \tan^{2}{\left(x \right)}\right) + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
Obtenemos la ecuación cuadrática
$$- \sqrt{3} w^{2} - 2 w - \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = - \sqrt{3}$$
$$b = -2$$
$$c = - \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}$$
, entonces
D = b^2 - 4 * a * c = 

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

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

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

o
$$w_{1} = - \frac{\sqrt{3} \left(\sqrt{4 \sqrt{3} \left(- \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}\right) + 4} + 2\right)}{6}$$
$$w_{2} = - \frac{\sqrt{3} \left(2 - \sqrt{4 \sqrt{3} \left(- \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}\right) + 4}\right)}{6}$$
hacemos cambio inverso
$$\cot{\left(x \right)} = w$$
sustituimos w:
Gráfica
Suma y producto de raíces [src]
suma
  pi   pi   pi   pi
- -- - -- + -- + --
  4    6    4    3 
$$\left(\left(- \frac{\pi}{4} - \frac{\pi}{6}\right) + \frac{\pi}{4}\right) + \frac{\pi}{3}$$
=
pi
--
6 
$$\frac{\pi}{6}$$
producto
-pi  -pi  pi pi
----*----*--*--
 4    6   4  3 
$$\frac{\pi}{3} \frac{\pi}{4} \cdot - \frac{\pi}{4} \left(- \frac{\pi}{6}\right)$$
=
  4
pi 
---
288
$$\frac{\pi^{4}}{288}$$
pi^4/288
Respuesta rápida [src]
     -pi 
x1 = ----
      4  
$$x_{1} = - \frac{\pi}{4}$$
     -pi 
x2 = ----
      6  
$$x_{2} = - \frac{\pi}{6}$$
     pi
x3 = --
     4 
$$x_{3} = \frac{\pi}{4}$$
     pi
x4 = --
     3 
$$x_{4} = \frac{\pi}{3}$$
x4 = pi/3
Respuesta numérica [src]
x1 = 55.7632696012188
x2 = -33.7721210260903
x3 = -43.1968989868597
x4 = -47.9092879672443
x5 = 46.3384916404494
x6 = 16.4933614313464
x7 = 90.3207887907066
x8 = 70.162235930172
x9 = 60.4756585816035
x10 = -99.7455667514759
x11 = -8.37758040957278
x12 = -96.342174710087
x13 = -85.6083998103219
x14 = -55.7632696012188
x15 = 32.2013246992954
x16 = 18.0641577581413
x17 = 35.6047167406843
x18 = -46.0766922526503
x19 = 63.8790506229925
x20 = -91.8915851175014
x21 = 41.8879020478639
x22 = -82.2050077689329
x23 = 48.1710873550435
x24 = -13.3517687777566
x25 = 13.6135681655558
x26 = -3.92699081698724
x27 = -71.4712328691678
x28 = -97.9129710368819
x29 = 56.025068989018
x30 = -75.9218224617533
x31 = 40.0553063332699
x32 = -25.9181393921158
x33 = -31.9395253114962
x34 = 5.75958653158129
x35 = 2.35619449019234
x36 = 85.870199198121
x37 = 96.6039740978861
x38 = -11.7809724509617
x39 = -16.2315620435473
x40 = 4.18879020478639
x41 = 82.4668071567321
x42 = -74.3510261349584
x43 = 54.1924732744239
x44 = -17.8023583703422
x45 = -49.4800842940392
x46 = 34.0339204138894
x47 = -2.0943951023932
x48 = 12.0427718387609
x49 = 49.7418836818384
x50 = 84.037603483527
x51 = 88.7499924639117
x52 = -77.7544181763474
x53 = 24.3473430653209
x54 = -87.1791961371168
x55 = 27.7507351067098
x56 = -24.0855436775217
x57 = 22.776546738526
x58 = 26.1799387799149
x59 = 44.7676953136546
x60 = -9.94837673636768
x61 = 19.8967534727354
x62 = 78.0162175641465
x63 = -57.3340659280137
x64 = -90.0589894029074
x65 = 76.1836218495525
x66 = 8.63937979737193
x67 = -69.9004365423729
x68 = 68.329640215578
x69 = -63.6172512351933
x70 = 98.174770424681
x71 = -19.6349540849362
x72 = -52.3598775598299
x73 = -93.4623814442964
x74 = -41.6261026600648
x75 = -27.4889357189107
x76 = -60.2138591938044
x77 = -61.7846555205993
x78 = 30.6305283725005
x79 = 93.7241808320955
x80 = 10.2101761241668
x81 = -68.0678408277789
x82 = 100.007366139275
x83 = 74.6128255227576
x84 = 81.1578102177363
x85 = -30.3687289847013
x86 = -53.9306738866248
x87 = -65.1880475619882
x88 = -79.3252145031423
x89 = -38.2227106186758
x90 = 52.621676947629
x91 = -83.7758040957278
x92 = 71.733032256967
x93 = 38.484510006475
x94 = 92.1533845053006
x95 = -5.49778714378214
x96 = -35.3429173528852
x97 = 62.3082542961976
x98 = -39.7935069454707
x99 = -21.2057504117311
x100 = 66.7588438887831
x100 = 66.7588438887831