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Cosx4-cos2x=1 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
cos(x)*4 - cos(2*x) = 1
4cos⁡(x)−cos⁡(2x)=14 \cos{\left(x \right)} - \cos{\left(2 x \right)} = 1
Simplificar
Solución detallada
Tenemos la ecuación
4cos⁡(x)−cos⁡(2x)=14 \cos{\left(x \right)} - \cos{\left(2 x \right)} = 1
cambiamos
−2(cos⁡(x)−1)2=0- 2 \left(\cos{\left(x \right)} - 1\right)^{2} = 0
−2cos⁡2(x)+4cos⁡(x)−2=0- 2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 2 = 0
Sustituimos
w=cos⁡(x)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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−2a = -2
b=4b = 4
c=−2c = -2
, entonces
D = b^2 - 4 * a * c = 

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

Como D = 0 hay sólo una raíz.
w = -b/2a = -4/2/(-2)

w1=1w_{1} = 1
hacemos cambio inverso
cos⁡(x)=w\cos{\left(x \right)} = w
Tenemos la ecuación
cos⁡(x)=w\cos{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
sustituimos w:
x1=πn+acos⁡(w1)x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}
x1=πn+acos⁡(1)x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}
x1=πnx_{1} = \pi n
x2=πn+acos⁡(w1)−πx_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x2=πn−π+acos⁡(1)x_{2} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}
x2=πn−πx_{2} = \pi n - \pi
Gráfica
0-80-60-40-2020406080-100100-1010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
  pi   pi        /      ___\        /      ___\
- -- + -- - I*log\2 - \/ 3 / - I*log\2 + \/ 3 /
  2    2
−ilog⁡(3+2)+((−π2+π2)−ilog⁡(2−3))- i \log{\left(\sqrt{3} + 2 \right)} + \left(\left(- \frac{\pi}{2} + \frac{\pi}{2}\right) - i \log{\left(2 - \sqrt{3} \right)}\right) 
=
       /      ___\        /      ___\
- I*log\2 + \/ 3 / - I*log\2 - \/ 3 /
−ilog⁡(3+2)−ilog⁡(2−3)- i \log{\left(\sqrt{3} + 2 \right)} - i \log{\left(2 - \sqrt{3} \right)} 
producto
-pi  pi /      /      ___\\ /      /      ___\\
----*--*\-I*log\2 - \/ 3 //*\-I*log\2 + \/ 3 //
 2   2
−ilog⁡(3+2)−ilog⁡(2−3)−π2π2- i \log{\left(\sqrt{3} + 2 \right)} - i \log{\left(2 - \sqrt{3} \right)} - \frac{\pi}{2} \frac{\pi}{2} 
=
  2    /      ___\    /      ___\
pi *log\2 + \/ 3 /*log\2 - \/ 3 /
---------------------------------
                4
π2log⁡(2−3)log⁡(3+2)4\frac{\pi^{2} \log{\left(2 - \sqrt{3} \right)} \log{\left(\sqrt{3} + 2 \right)}}{4} 
pi^2*log(2 + sqrt(3))*log(2 - sqrt(3))/4
Respuesta rápida [src] 
     -pi 
x1 = ----
      2
x1=−π2x_{1} = - \frac{\pi}{2} 
     pi
x2 = --
     2
x2=π2x_{2} = \frac{\pi}{2} 
           /      ___\
x3 = -I*log\2 - \/ 3 /
x3=−ilog⁡(2−3)x_{3} = - i \log{\left(2 - \sqrt{3} \right)} 
           /      ___\
x4 = -I*log\2 + \/ 3 /
x4=−ilog⁡(3+2)x_{4} = - i \log{\left(\sqrt{3} + 2 \right)} 
x4 = -i*log(sqrt(3) + 2)
Respuesta numérica [src] 
x1 = 7.85398163397448
x2 = -86.3937979737193
x3 = 58.1194640914112
x4 = 23.5619449019235
x5 = -67.5442420521806
x6 = -4.71238898038469
x7 = -20.4203522483337
x8 = -733.561884613217
x9 = 83.2522053201295
x10 = -29.845130209103
x11 = -39.2699081698724
x12 = -98.9601685880785
x13 = 98.9601685880785
x14 = 86.3937979737193
x15 = 26.7035375555132
x16 = -48.6946861306418
x17 = -89.5353906273091
x18 = -23997.0554844456
x19 = -17.2787595947439
x20 = -2560688.60999614
x21 = 20.4203522483337
x22 = 48.6946861306418
x23 = -64.4026493985908
x24 = 67.5442420521806
x25 = 14.1371669411541
x26 = -26.7035375555132
x27 = 42.4115008234622
x28 = -70.6858347057703
x29 = -32.9867228626928
x30 = 39.2699081698724
x31 = 4.71238898038469
x32 = 73.8274273593601
x33 = 89.5353906273091
x34 = 45.553093477052
x35 = 70.6858347057703
x36 = -95.8185759344887
x37 = -7.85398163397448
x38 = 120.951317163207
x39 = 76.9690200129499
x40 = 32.9867228626928
x41 = -23.5619449019235
x42 = 64.4026493985908
x43 = -36.1283155162826
x44 = -83.2522053201295
x45 = -1.5707963267949
x46 = -58.1194640914112
x47 = -10.9955742875643
x48 = 1.5707963267949
x49 = 29.845130209103
x50 = -73.8274273593601
x51 = -92.6769832808989
x52 = -54.9778714378214
x53 = 80.1106126665397
x54 = 54.9778714378214
x55 = -196.349540849362
x56 = -76.9690200129499
x57 = 36.1283155162826
x58 = 61.261056745001
x59 = 92.6769832808989
x60 = -61.261056745001
x61 = 17.2787595947439
x62 = 10.9955742875643
x63 = -51.8362787842316
x64 = -45.553093477052
x65 = -42.4115008234622
x66 = -80.1106126665397
x67 = 51.8362787842316
x68 = 95.8185759344887
x69 = -14.1371669411541
x69 = -14.1371669411541
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