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3(tg^2(x)-1)*sqrt(-5cosx)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
  /   2       \   ___________    
3*\tan (x) - 1/*\/ -5*cos(x)  = 0
$$\sqrt{- 5 \cos{\left(x \right)}} 3 \left(\tan^{2}{\left(x \right)} - 1\right) = 0$$
Solución detallada
Tenemos la ecuación
$$\sqrt{- 5 \cos{\left(x \right)}} 3 \left(\tan^{2}{\left(x \right)} - 1\right) = 0$$
cambiamos
$$3 \sqrt{5} \sqrt{- \cos{\left(x \right)}} \left(\tan^{2}{\left(x \right)} - 1\right) = 0$$
$$\sqrt{- 5 \cos{\left(x \right)}} 3 \left(\tan^{2}{\left(x \right)} - 1\right) = 0$$
Sustituimos
$$w = \tan{\left(x \right)}$$
Abramos la expresión en la ecuación
$$\sqrt{5} \sqrt{- \cos{\left(x \right)}} \left(3 w^{2} - 3\right) = 0$$
Obtenemos la ecuación cuadrática
$$3 \sqrt{5} w^{2} \sqrt{- \cos{\left(x \right)}} - 3 \sqrt{5} \sqrt{- \cos{\left(x \right)}} = 0$$
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
$$a = 3 \sqrt{5} \sqrt{- \cos{\left(x \right)}}$$
$$b = 0$$
$$c = - 3 \sqrt{5} \sqrt{- \cos{\left(x \right)}}$$
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (3*sqrt(5)*sqrt(-cos(x))) * (-3*sqrt(5)*sqrt(-cos(x))) = 180*(-cos(x))

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

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

o
$$w_{1} = 1$$
$$w_{2} = -1$$
hacemos cambio inverso
$$\tan{\left(x \right)} = w$$
Tenemos la ecuación
$$\tan{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
O
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(1 \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(-1 \right)}$$
$$x_{2} = \pi n - \frac{\pi}{4}$$
Gráfica
Suma y producto de raíces [src]
suma
  pi   pi
- -- + --
  4    4 
$$- \frac{\pi}{4} + \frac{\pi}{4}$$
=
0
$$0$$
producto
-pi  pi
----*--
 4   4 
$$- \frac{\pi}{4} \frac{\pi}{4}$$
=
   2 
-pi  
-----
  16 
$$- \frac{\pi^{2}}{16}$$
-pi^2/16
Respuesta rápida [src]
     -pi 
x1 = ----
      4  
$$x_{1} = - \frac{\pi}{4}$$
     pi
x2 = --
     4 
$$x_{2} = \frac{\pi}{4}$$
x2 = pi/4
Respuesta numérica [src]
x1 = -96.6039740978861
x2 = 32.2013246992954
x3 = -27.4889357189107
x4 = 10.2101761241668
x5 = -32.2013246992954
x6 = -41.6261026600648
x7 = 55.7632696012188
x8 = 30.6305283725005
x9 = -19.6349540849362
x10 = 90.3207887907066
x11 = -3.92699081698724
x12 = 82.4668071567321
x13 = -69.9004365423729
x14 = 52.621676947629
x15 = -77.7544181763474
x16 = 16.4933614313464
x17 = 41.6261026600648
x18 = 96.6039740978861
x19 = -91.8915851175014
x20 = -46.3384916404494
x21 = -5.49778714378214
x22 = -62.0464549083984
x23 = -84.037603483527
x24 = 91.8915851175014
x25 = -55.7632696012188
x26 = 77.7544181763474
x27 = 84.037603483527
x28 = -71.4712328691678
x29 = 8.63937979737193
x30 = -76.1836218495525
x31 = 60.4756585816035
x32 = 3.92699081698724
x33 = 38.484510006475
x34 = -90.3207887907066
x35 = 58.9048622548086
x36 = -79.3252145031423
x37 = 98.174770424681
x38 = -18.0641577581413
x39 = 76.1836218495525
x40 = -13.3517687777566
x41 = -74.6128255227576
x42 = 24.3473430653209
x43 = -40.0553063332699
x44 = -98.174770424681
x45 = -68.329640215578
x46 = 46.3384916404494
x47 = 40.0553063332699
x48 = -33.7721210260903
x49 = -25.9181393921158
x50 = -52.621676947629
x51 = -49.4800842940392
x52 = -57.3340659280137
x53 = 35.3429173528852
x54 = 25.9181393921158
x55 = 85.6083998103219
x56 = -10.2101761241668
x57 = -35.3429173528852
x58 = -99.7455667514759
x59 = 80.8960108299372
x60 = 74.6128255227576
x61 = -8.63937979737193
x62 = 62.0464549083984
x63 = 54.1924732744239
x64 = -24.3473430653209
x65 = 68.329640215578
x66 = 63.6172512351933
x67 = 19.6349540849362
x68 = -85.6083998103219
x69 = 18.0641577581413
x70 = -93.4623814442964
x71 = 14.9225651045515
x72 = -54.1924732744239
x73 = 2.35619449019234
x74 = 99.7455667514759
x75 = -11.7809724509617
x76 = 11.7809724509617
x77 = -30.6305283725005
x78 = 69.9004365423729
x79 = -63.6172512351933
x80 = -2.35619449019234
x81 = 33.7721210260903
x82 = 47.9092879672443
x83 = -47.9092879672443
x83 = -47.9092879672443