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cos(5*x)^2=4 la ecuación

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Solución

Ha introducido [src]
   2         
cos (5*x) = 4
cos2(5x)=4\cos^{2}{\left(5 x \right)} = 4
Solución detallada
Tenemos la ecuación
cos2(5x)=4\cos^{2}{\left(5 x \right)} = 4
cambiamos
cos2(5x)4=0\cos^{2}{\left(5 x \right)} - 4 = 0
cos2(5x)4=0\cos^{2}{\left(5 x \right)} - 4 = 0
Sustituimos
w=cos(5x)w = \cos{\left(5 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=Db2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=Db2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=0b = 0
c=4c = -4
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-4) = 16

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

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

o
w1=2w_{1} = 2
w2=2w_{2} = -2
hacemos cambio inverso
cos(5x)=w\cos{\left(5 x \right)} = w
Tenemos la ecuación
cos(5x)=w\cos{\left(5 x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
5x=πn+acos(w)5 x = \pi n + \operatorname{acos}{\left(w \right)}
5x=πn+acos(w)π5 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
5x=πn+acos(w)5 x = \pi n + \operatorname{acos}{\left(w \right)}
5x=πn+acos(w)π5 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
55
sustituimos w:
x1=πn5+acos(w1)5x_{1} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(w_{1} \right)}}{5}
x1=πn5+acos(2)5x_{1} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(2 \right)}}{5}
x1=πn5+acos(2)5x_{1} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(2 \right)}}{5}
x2=πn5+acos(w2)5x_{2} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(w_{2} \right)}}{5}
x2=πn5+acos(2)5x_{2} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(-2 \right)}}{5}
x2=πn5+acos(2)5x_{2} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(-2 \right)}}{5}
x3=πn5+acos(w1)5π5x_{3} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(w_{1} \right)}}{5} - \frac{\pi}{5}
x3=πn5π5+acos(2)5x_{3} = \frac{\pi n}{5} - \frac{\pi}{5} + \frac{\operatorname{acos}{\left(2 \right)}}{5}
x3=πn5π5+acos(2)5x_{3} = \frac{\pi n}{5} - \frac{\pi}{5} + \frac{\operatorname{acos}{\left(2 \right)}}{5}
x4=πn5+acos(w2)5π5x_{4} = \frac{\pi n}{5} + \frac{\operatorname{acos}{\left(w_{2} \right)}}{5} - \frac{\pi}{5}
x4=πn5π5+acos(2)5x_{4} = \frac{\pi n}{5} - \frac{\pi}{5} + \frac{\operatorname{acos}{\left(-2 \right)}}{5}
x4=πn5π5+acos(2)5x_{4} = \frac{\pi n}{5} - \frac{\pi}{5} + \frac{\operatorname{acos}{\left(-2 \right)}}{5}
Gráfica
0-80-60-40-2020406080-10010005
Respuesta rápida [src]
       re(acos(-2))   2*pi   I*im(acos(-2))
x1 = - ------------ + ---- - --------------
            5          5           5       
x1=re(acos(2))5+2π5iim(acos(2))5x_{1} = - \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5} + \frac{2 \pi}{5} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5}
     2*pi   I*im(acos(2))
x2 = ---- - -------------
      5           5      
x2=2π5iim(acos(2))5x_{2} = \frac{2 \pi}{5} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{5}
     re(acos(-2))   I*im(acos(-2))
x3 = ------------ + --------------
          5               5       
x3=re(acos(2))5+iim(acos(2))5x_{3} = \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5}
     I*im(acos(2))
x4 = -------------
           5      
x4=iim(acos(2))5x_{4} = \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{5}
x4 = i*im(acos(2))/5
Suma y producto de raíces [src]
suma
  re(acos(-2))   2*pi   I*im(acos(-2))   2*pi   I*im(acos(2))   re(acos(-2))   I*im(acos(-2))   I*im(acos(2))
- ------------ + ---- - -------------- + ---- - ------------- + ------------ + -------------- + -------------
       5          5           5           5           5              5               5                5      
((re(acos(2))5+iim(acos(2))5)+((2π5iim(acos(2))5)+(re(acos(2))5+2π5iim(acos(2))5)))+iim(acos(2))5\left(\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5}\right) + \left(\left(\frac{2 \pi}{5} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{5}\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5} + \frac{2 \pi}{5} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5}\right)\right)\right) + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{5}
=
4*pi
----
 5  
4π5\frac{4 \pi}{5}
producto
/  re(acos(-2))   2*pi   I*im(acos(-2))\ /2*pi   I*im(acos(2))\ /re(acos(-2))   I*im(acos(-2))\ I*im(acos(2))
|- ------------ + ---- - --------------|*|---- - -------------|*|------------ + --------------|*-------------
\       5          5           5       / \ 5           5      / \     5               5       /       5      
iim(acos(2))5(2π5iim(acos(2))5)(re(acos(2))5+2π5iim(acos(2))5)(re(acos(2))5+iim(acos(2))5)\frac{i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{5} \left(\frac{2 \pi}{5} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{5}\right) \left(- \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5} + \frac{2 \pi}{5} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5}\right) \left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-2 \right)}\right)}}{5}\right)
=
-I*(2*pi - I*im(acos(2)))*(I*im(acos(-2)) + re(acos(-2)))*(-2*pi + I*im(acos(-2)) + re(acos(-2)))*im(acos(2)) 
--------------------------------------------------------------------------------------------------------------
                                                     625                                                      
i(2πiim(acos(2)))(re(acos(2))+iim(acos(2)))(2π+re(acos(2))+iim(acos(2)))im(acos(2))625- \frac{i \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) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(-2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-2 \right)}\right)}\right) \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{625}
-i*(2*pi - i*im(acos(2)))*(i*im(acos(-2)) + re(acos(-2)))*(-2*pi + i*im(acos(-2)) + re(acos(-2)))*im(acos(2))/625
Respuesta numérica [src]
x1 = 0.628318530717959 + 0.263391579384963*i
x2 = 1.25663706143592 - 0.263391579384963*i
x3 = 0.628318530717959 - 0.263391579384963*i
x4 = 0.263391579384963*i
x4 = 0.263391579384963*i