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cos(2x)-3*cos(x)+2=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
cos(2*x) - 3*cos(x) + 2 = 0
$$\left(- 3 \cos{\left(x \right)} + \cos{\left(2 x \right)}\right) + 2 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(- 3 \cos{\left(x \right)} + \cos{\left(2 x \right)}\right) + 2 = 0$$
cambiamos
$$- 3 \cos{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0$$
$$2 \cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 1 = 0$$
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
$$a = 2$$
$$b = -3$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c = 

(-3)^2 - 4 * (2) * (1) = 1

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

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

o
$$w_{1} = 1$$
$$w_{2} = \frac{1}{2}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{2} = \pi n + \frac{\pi}{3}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{3} = \pi n - \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{4} = \pi n - \frac{2 \pi}{3}$$
Gráfica
Respuesta rápida [src]
x1 = 0
$$x_{1} = 0$$
     -pi 
x2 = ----
      3  
$$x_{2} = - \frac{\pi}{3}$$
     pi
x3 = --
     3 
$$x_{3} = \frac{\pi}{3}$$
x3 = pi/3
Suma y producto de raíces [src]
suma
  pi   pi
- -- + --
  3    3 
$$- \frac{\pi}{3} + \frac{\pi}{3}$$
=
0
$$0$$
producto
  -pi  pi
0*----*--
   3   3 
$$\frac{\pi}{3} \cdot 0 \left(- \frac{\pi}{3}\right)$$
=
0
$$0$$
0
Respuesta numérica [src]
x1 = 45.0294947014537
x2 = -13.6135681655558
x3 = -87.9645943592724
x4 = -93.2005820564972
x5 = 50.2654824463735
x6 = -94.2477794807135
x7 = -57.5958653158129
x8 = -36.6519142918809
x9 = 113.097336365294
x10 = 99.4837673636768
x11 = 63.8790506229925
x12 = -24.0855436775217
x13 = -1.0471975511966
x14 = -32.4631240870945
x15 = -25.1327426219818
x16 = 95.2949771588904
x17 = 42.9350995990605
x18 = 12.5663704906642
x19 = -75.3982238054715
x20 = 80.634211442138
x21 = 70.162235930172
x22 = 1.0471975511966
x23 = -19.8967534727354
x24 = -86.9173967493176
x25 = -42.9350995990605
x26 = -38.7463093942741
x27 = -81.6814090372847
x28 = -80.634211442138
x29 = 62.8318525639961
x30 = 86.9173967493176
x31 = 26.1799387799149
x32 = -182.212372564603
x33 = 7.33038285837618
x34 = 43.9822971693062
x35 = 57.5958653158129
x36 = -68.0678408277789
x37 = -49.2182849062401
x38 = 145.560459616327
x39 = -74.3510261349584
x40 = -37.6991118767899
x41 = 17.8023583703422
x42 = -50.2654823282564
x43 = -63.8790506229925
x44 = 6.2831852843889
x45 = -76.4454212373516
x46 = 24.0855436775217
x47 = -5.23598775598299
x48 = -7.33038285837618
x49 = 32.4631240870945
x50 = -13691.0607854089
x51 = 19.8967534727354
x52 = 49.2182849062401
x53 = -95.2949771588904
x54 = 74.3510261349584
x55 = 51.3126800086333
x56 = 100.530964913886
x57 = 93.2005820564972
x58 = -61.7846555205993
x59 = 56.5486676412843
x60 = 38.7463093942741
x61 = -82.7286065445312
x62 = 94.2477796093529
x63 = -31.4159266583933
x64 = -89.0117918517108
x65 = -17.8023583703422
x66 = 36.6519142918809
x67 = 81.6814091225562
x68 = 126.710903694788
x69 = -6.28318517679891
x70 = -99.4837673636768
x71 = 0.0
x72 = 87.9645943352828
x73 = 30.3687289847013
x74 = 68.0678408277789
x75 = 61.7846555205993
x76 = 89.0117918517108
x77 = 37.6991119735314
x78 = 82.7286065445312
x79 = -55.5014702134197
x80 = 100.530964793178
x81 = -12.5663692278075
x82 = -69.1150383715098
x83 = -45.0294947014537
x84 = -11.5191730631626
x85 = -51.3126800086333
x86 = 55.5014702134197
x87 = 13.6135681655558
x88 = -30.3687289847013
x89 = 11.5191730631626
x90 = -43.9822971747058
x91 = 5.23598775598299
x91 = 5.23598775598299