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cos(2x)+cos(x)=0 la ecuación

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Solución numérica:

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Solución

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

(1)^2 - 4 * (2) * (-1) = 9

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

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

o
$$w_{1} = \frac{1}{2}$$
$$w_{2} = -1$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(-1 \right)}$$
$$x_{2} = \pi n + \pi$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{3} = \pi n - \frac{2 \pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}$$
$$x_{4} = \pi n$$
Gráfica
Suma y producto de raíces [src]
suma
  5*pi        pi   pi        5*pi
- ---- - pi - -- + -- + pi + ----
   3          3    3          3  
$$\left(\left(\left(\left(- \frac{5 \pi}{3} - \pi\right) - \frac{\pi}{3}\right) + \frac{\pi}{3}\right) + \pi\right) + \frac{5 \pi}{3}$$
=
0
$$0$$
producto
-5*pi       -pi  pi    5*pi
-----*(-pi)*----*--*pi*----
  3          3   3      3  
$$\frac{5 \pi}{3} \pi \frac{\pi}{3} \cdot - \frac{\pi}{3} \cdot - \frac{5 \pi}{3} \left(- \pi\right)$$
=
      6
-25*pi 
-------
   81  
$$- \frac{25 \pi^{6}}{81}$$
-25*pi^6/81
Respuesta rápida [src]
     -5*pi
x1 = -----
       3  
$$x_{1} = - \frac{5 \pi}{3}$$
x2 = -pi
$$x_{2} = - \pi$$
     -pi 
x3 = ----
      3  
$$x_{3} = - \frac{\pi}{3}$$
     pi
x4 = --
     3 
$$x_{4} = \frac{\pi}{3}$$
x5 = pi
$$x_{5} = \pi$$
     5*pi
x6 = ----
      3  
$$x_{6} = \frac{5 \pi}{3}$$
x6 = 5*pi/3
Respuesta numérica [src]
x1 = -646.120889088301
x2 = -28.2743337200245
x3 = 3.14159271706432
x4 = 74.3510261349584
x5 = -84.8230022421807
x6 = -72.256630877064
x7 = 65.9734457528689
x8 = -15.7079632965016
x9 = 55.5014702134197
x10 = -68.0678408277789
x11 = -65.9734457650482
x12 = -99.4837673636768
x13 = 97.389372486408
x14 = -34.5575189638817
x15 = -91.1061871711313
x16 = 47.1238898268985
x17 = -84.8230015251551
x18 = 21.9911485973609
x19 = 3.14159267447126
x20 = -40.8407049942712
x21 = -40.8407047547408
x22 = 70.162235930172
x23 = 57.5958653158129
x24 = -47.1238900222279
x25 = -74.3510261349584
x26 = 47.1238897752019
x27 = -70.162235930172
x28 = -13.6135681655558
x29 = -47.1238905036874
x30 = 107.86134777325
x31 = 84.8230014287926
x32 = -17.8023583703422
x33 = 21.9911485851931
x34 = 82.7286065445312
x35 = -55.5014702134197
x36 = 63.8790506229925
x37 = -84.8230014829768
x38 = 53.4070753369186
x39 = 80.634211442138
x40 = -30.3687289847013
x41 = 36.6519142918809
x42 = 15.7079634367135
x43 = -19.8967534727354
x44 = -32.4631240870945
x45 = 91.1061868116125
x46 = 40.8407042778045
x47 = -11.5191730631626
x48 = -61.7846555205993
x49 = -93.2005820564972
x50 = 59.6902605931502
x51 = -21.9911485864549
x52 = -24.0855436775217
x53 = -9.42477812311019
x54 = 47.1238894268221
x55 = 91.1061868861836
x56 = -5.23598775598299
x57 = -78.5398161151012
x58 = -3.14159287255706
x59 = 78.5398161904624
x60 = -95.2949771588904
x61 = 17.8023583703422
x62 = -97.3893724356252
x63 = -76.4454212373516
x64 = -51.3126800086333
x65 = -26.1799387799149
x66 = 34.5575190335478
x67 = 3.14159276530697
x68 = 24.0855436775217
x69 = 13.6135681655558
x70 = -7.33038285837618
x71 = -40.8407044128941
x72 = -53.4070752795041
x73 = 91.1061863890352
x74 = 99.4837673636768
x75 = 3.14159322994749
x76 = 72.2566310277195
x77 = 47.1238901206303
x78 = 9.42477818680547
x79 = 28.2743338652086
x80 = 30.3687289847013
x81 = 38.7463093942741
x82 = 91.1061869261407
x83 = 32.4631240870945
x84 = 61.7846555205993
x85 = -40.8407044009017
x86 = -57.5958653158129
x87 = -59.690260457585
x88 = -63.8790506229925
x89 = 11.5191730631626
x90 = 68.0678408277789
x91 = -49.2182849062401
x92 = 26.1799387799149
x93 = 76.4454212373516
x94 = 19.8967534727354
x95 = -719.424718069224
x95 = -719.424718069224