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cos(x)^(2)=cos(x) la ecuación

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Solución numérica:

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Solución

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

(-1)^2 - 4 * (1) * (0) = 1

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

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

o
$$w_{1} = 1$$
$$w_{2} = 0$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{2}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{3} = \pi n - \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
$$x_{4} = \pi n - \frac{\pi}{2}$$
Gráfica
Respuesta rápida [src]
x1 = 0
$$x_{1} = 0$$
     pi
x2 = --
     2 
$$x_{2} = \frac{\pi}{2}$$
     3*pi
x3 = ----
      2  
$$x_{3} = \frac{3 \pi}{2}$$
x4 = 2*pi
$$x_{4} = 2 \pi$$
x4 = 2*pi
Suma y producto de raíces [src]
suma
pi   3*pi       
-- + ---- + 2*pi
2     2         
$$\left(\frac{\pi}{2} + \frac{3 \pi}{2}\right) + 2 \pi$$
=
4*pi
$$4 \pi$$
producto
  pi 3*pi     
0*--*----*2*pi
  2   2       
$$2 \pi \frac{3 \pi}{2} \cdot 0 \frac{\pi}{2}$$
=
0
$$0$$
0
Respuesta numérica [src]
x1 = 1.5707963267949
x2 = 14.1371669411541
x3 = -70.6858347057703
x4 = 32.9867228626928
x5 = 61.261056745001
x6 = -36.1283155162826
x7 = 7.85398163397448
x8 = 42.4115008234622
x9 = -25.1327413641924
x10 = 98.9601685880785
x11 = 83.2522053201295
x12 = -56.5486675907774
x13 = -89.5353906273091
x14 = -100.530964736174
x15 = -54.9778714378214
x16 = -67.5442420521806
x17 = -51.8362787842316
x18 = -83.2522053201295
x19 = -14.1371669411541
x20 = -69.115038497193
x21 = 92.6769832808989
x22 = 50.2654824463558
x23 = -43.9822971746199
x24 = 64.4026493985908
x25 = 48.6946861306418
x26 = 100.530964774136
x27 = 76.9690200129499
x28 = -32.9867228626928
x29 = 23.5619449019235
x30 = -62.8318534973011
x31 = 10.9955742875643
x32 = 69.1150379836781
x33 = 69.1150385134118
x34 = -94.2477794613449
x35 = -37.6991118770355
x36 = -80.1106126665397
x37 = -29.845130209103
x38 = -6.28318514935383
x39 = -86.3937979737193
x40 = 4.71238898038469
x41 = 17.2787595947439
x42 = 25.1327408583892
x43 = 20.4203522483337
x44 = 87.9645943356049
x45 = -76.9690200129499
x46 = 81.6814091609407
x47 = 12.5663704623094
x48 = -31.4159266930206
x49 = -1.5707963267949
x50 = 86.3937979737193
x51 = 94.2477796093526
x52 = 39.2699081698724
x53 = -39.2699081698724
x54 = 36.1283155162826
x55 = -75.3982238479311
x56 = 6.28318528429551
x57 = 51.8362787842316
x58 = 80.1106126665397
x59 = -18.8495562409837
x60 = 25.1327418085792
x61 = -50.2654823051418
x62 = 0.0
x63 = -92.6769832808989
x64 = 89.5353906273091
x65 = 58.1194640914112
x66 = 73.8274273593601
x67 = -58.1194640914112
x68 = -20.4203522483337
x69 = 54.9778714378214
x70 = -95.8185759344887
x71 = 29.845130209103
x72 = 26.7035375555132
x73 = 37.6991120060109
x74 = -23.5619449019235
x75 = -42.4115008234622
x76 = -48.6946861306418
x77 = 70.6858347057703
x78 = -98.9601685880785
x79 = -389.557489134924
x80 = -26.7035375555132
x81 = -10.9955742875643
x82 = 18.8495557729205
x83 = 75.3982238342404
x84 = -87.964594358935
x85 = 56.5486676180351
x86 = -12.5663704469816
x87 = 45.553093477052
x88 = 62.8318529132021
x89 = -73.8274273593601
x90 = 43.9822971693881
x91 = -4.71238898038469
x92 = -45.553093477052
x93 = -81.6814090377756
x94 = -7.85398163397448
x95 = -17.2787595947439
x96 = 95.8185759344887
x97 = 31.4159266948554
x97 = 31.4159266948554