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cos(x)^(2)-sqrt(2)/2*cos(x)=0 la ecuación

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Solución numérica:

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Solución

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

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

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

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

o
$$w_{1} = \frac{\sqrt{2}}{2}$$
$$w_{2} = 0$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{2}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{3} = \pi n - \frac{3 \pi}{4}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
$$x_{4} = \pi n - \frac{\pi}{2}$$
Gráfica
Respuesta rápida [src]
     pi
x1 = --
     4 
$$x_{1} = \frac{\pi}{4}$$
     pi
x2 = --
     2 
$$x_{2} = \frac{\pi}{2}$$
     3*pi
x3 = ----
      2  
$$x_{3} = \frac{3 \pi}{2}$$
     7*pi
x4 = ----
      4  
$$x_{4} = \frac{7 \pi}{4}$$
x4 = 7*pi/4
Suma y producto de raíces [src]
suma
pi   pi   3*pi   7*pi
-- + -- + ---- + ----
4    2     2      4  
$$\frac{7 \pi}{4} + \left(\left(\frac{\pi}{4} + \frac{\pi}{2}\right) + \frac{3 \pi}{2}\right)$$
=
4*pi
$$4 \pi$$
producto
pi pi 3*pi 7*pi
--*--*----*----
4  2   2    4  
$$\frac{7 \pi}{4} \frac{3 \pi}{2} \frac{\pi}{4} \frac{\pi}{2}$$
=
     4
21*pi 
------
  64  
$$\frac{21 \pi^{4}}{64}$$
21*pi^4/64
Respuesta numérica [src]
x1 = 25.9181393921158
x2 = 89.5353906273091
x3 = 61.261056745001
x4 = 42.4115008234622
x5 = -99.7455667514759
x6 = -42.4115008234622
x7 = -29.845130209103
x8 = 17.2787595947439
x9 = 18.0641577581413
x10 = -36.1283155162826
x11 = 86.3937979737193
x12 = -10.9955742875643
x13 = -98.9601685880785
x14 = 82.4668071567321
x15 = 38.484510006475
x16 = -95.8185759344887
x17 = -76.1836218495525
x18 = -14.1371669411541
x19 = -5.49778714378214
x20 = 76.1836218495525
x21 = 55.7632696012188
x22 = 58.1194640914112
x23 = 10.9955742875643
x24 = -62.0464549083984
x25 = 120.16591899981
x26 = -92.6769832808989
x27 = -48.6946861306418
x28 = -23.5619449019235
x29 = -86.3937979737193
x30 = -26.7035375555132
x31 = -19.6349540849362
x32 = -67.5442420521806
x33 = -80.1106126665397
x34 = -1.5707963267949
x35 = 92.6769832808989
x36 = 36.1283155162826
x37 = -39.2699081698724
x38 = 4.71238898038469
x39 = -69.9004365423729
x40 = -76.9690200129499
x41 = 48.6946861306418
x42 = -32.2013246992954
x43 = 88.7499924639117
x44 = 76.9690200129499
x45 = 98.9601685880785
x46 = 83.2522053201295
x47 = -45.553093477052
x48 = -89.5353906273091
x49 = -70.6858347057703
x50 = -82.4668071567321
x51 = 73.8274273593601
x52 = -44.7676953136546
x53 = 11.7809724509617
x54 = -25.9181393921158
x55 = 20.4203522483337
x56 = 68.329640215578
x57 = 1.5707963267949
x58 = 45.553093477052
x59 = 95.8185759344887
x60 = -755.553033188345
x61 = -58.1194640914112
x62 = -93.4623814442964
x63 = -55.7632696012188
x64 = 24.3473430653209
x65 = 80.1106126665397
x66 = 7.85398163397448
x67 = 26.7035375555132
x68 = 29.845130209103
x69 = -54.9778714378214
x70 = -11.7809724509617
x71 = -63.6172512351933
x72 = 14.1371669411541
x73 = 39.2699081698724
x74 = -83.2522053201295
x75 = -4.71238898038469
x76 = 51.8362787842316
x77 = -88.7499924639117
x78 = 64.4026493985908
x79 = -73.8274273593601
x80 = 32.9867228626928
x81 = -32.9867228626928
x82 = 32.2013246992954
x83 = -49.4800842940392
x84 = -51.8362787842316
x85 = 67.5442420521806
x86 = -7.85398163397448
x87 = 69.9004365423729
x88 = 62.0464549083984
x89 = 54.9778714378214
x90 = 99.7455667514759
x91 = -18.0641577581413
x91 = -18.0641577581413