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cos(x)^2-3cos(x)=-2 la ecuación

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Solución numérica:

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Solución

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

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

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

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

o
$$w_{1} = 2$$
$$w_{2} = 1$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(2 \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(2 \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(2 \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(2 \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
Gráfica
Suma y producto de raíces [src]
suma
2*pi + 2*pi - I*im(acos(2)) + I*im(acos(2)) + re(acos(2))
$$\left(2 \pi + \left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)$$
=
4*pi + re(acos(2))
$$\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + 4 \pi$$
producto
0*2*pi*(2*pi - I*im(acos(2)))*(I*im(acos(2)) + re(acos(2)))
$$0 \cdot 2 \pi \left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)$$
=
0
$$0$$
0
Respuesta rápida [src]
x1 = 0
$$x_{1} = 0$$
x2 = 2*pi
$$x_{2} = 2 \pi$$
x3 = 2*pi - I*im(acos(2))
$$x_{3} = 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}$$
x4 = I*im(acos(2)) + re(acos(2))
$$x_{4} = \operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}$$
x4 = re(acos(2)) + i*im(acos(2))
Respuesta numérica [src]
x1 = 50.2654837284661
x2 = -25.1327409215642
x3 = -100.530964552074
x4 = -50.2654813807249
x5 = 6.28318657126167
x6 = 12.5663696338539
x7 = 100.530963867505
x8 = -12.5663714609397
x9 = -1.24358883358213e-6
x10 = 6.28318403478594
x11 = -5215.04380391355
x12 = 56.5486667492183
x13 = -69.1150375400315
x14 = 62.8318539562208
x15 = 62.8318532824014
x16 = -81.6814077446239
x17 = 43.9822959141556
x18 = 56.5486675871477
x19 = 6.28318528412712
x20 = -87.9645943583137
x21 = -25.1327403745683
x22 = 12.5663717119664
x23 = 31.4159269373448
x24 = -25.1327416132853
x25 = -56.548667601399
x26 = 31.4159257044419
x27 = -62.8318524749096
x28 = -69.1150387763928
x29 = 94.2477808851498
x30 = 69.1150378026136
x31 = 87.9645943361855
x32 = 25.1327418552092
x33 = 31.4159264886041
x34 = -75.3982226148333
x35 = -56.5486686262791
x36 = 87.9645955876053
x37 = 69.1150390203837
x38 = -12.566370225933
x39 = -31.4159274784001
x40 = -81.6814090386471
x41 = 25.1327406377965
x42 = 43.9822971695364
x43 = 94.2477796093521
x44 = -75.3982238951945
x45 = -37.6991105864418
x46 = 62.8318531044175
x47 = 62.8318527124984
x48 = 18.8495560791567
x49 = 50.2654824463236
x50 = 12.5663704264737
x51 = 18.8495567908524
x52 = 50.2654811803455
x53 = -37.6991131096277
x54 = -87.9645955574957
x55 = 0.0
x56 = -100.530964660003
x57 = -6.28318510897268
x58 = 81.6814099712889
x59 = 94.2477783269892
x60 = -43.9822984008141
x61 = -6.28318638269977
x62 = 43.9822984432902
x63 = 37.6991107806887
x64 = 87.9645930723958
x65 = -62.8318536920999
x66 = -56.5486673890729
x67 = -75.3982245873506
x68 = 18.8495555496789
x69 = -94.2477785033657
x70 = 31.4159261902552
x71 = -37.6991118774752
x72 = -6.2831842606194
x73 = -94.2477794313305
x74 = -50.2654822701809
x75 = 75.3982235113583
x76 = -18.8495565271099
x77 = 81.6814079442934
x78 = 56.5486688750198
x79 = -31.4159254513224
x80 = -87.9645929959187
x81 = 37.6991120523088
x82 = -50.2654835458582
x83 = 81.6814092136757
x84 = 1.29789534933499e-6
x85 = 100.530964747725
x86 = -81.6814102528289
x87 = -69.1150378835413
x88 = -18.8495553098906
x89 = -94.2477807087722
x90 = 37.6991128567789
x91 = -43.9822958485246
x92 = -31.4159267343047
x93 = -12.5663705508209
x94 = 75.398222869856
x95 = 75.3982241007593
x96 = -43.9822971744631
x96 = -43.9822971744631