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sqrt(cos(x)^2-cos(x))/5-1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   __________________        
  /    2                     
\/  cos (x) - cos(x)         
--------------------- - 1 = 0
          5
cos⁡2(x)−cos⁡(x)5−1=0\frac{\sqrt{\cos^{2}{\left(x \right)} - \cos{\left(x \right)}}}{5} - 1 = 0
Simplificar
Solución detallada
Tenemos la ecuación
cos⁡2(x)−cos⁡(x)5−1=0\frac{\sqrt{\cos^{2}{\left(x \right)} - \cos{\left(x \right)}}}{5} - 1 = 0
cambiamos
(cos⁡(x)−1)cos⁡(x)5−1=0\frac{\sqrt{\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)}}}{5} - 1 = 0
cos⁡2(x)−cos⁡(x)5−1=0\frac{\sqrt{\cos^{2}{\left(x \right)} - \cos{\left(x \right)}}}{5} - 1 = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
w2−w5=1\frac{\sqrt{w^{2} - w}}{5} = 1
Elevemos las dos partes de la ecuación a la potencia 2
w225−w25=1\frac{w^{2}}{25} - \frac{w}{25} = 1
w225−w25=1\frac{w^{2}}{25} - \frac{w}{25} = 1
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
w225−w25−1=0\frac{w^{2}}{25} - \frac{w}{25} - 1 = 0
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=125a = \frac{1}{25}
b=−125b = - \frac{1}{25}
c=−1c = -1
, entonces
D = b^2 - 4 * a * c = 

(-1/25)^2 - 4 * (1/25) * (-1) = 101/625

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

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

o
w1=12+1012w_{1} = \frac{1}{2} + \frac{\sqrt{101}}{2}
w2=12−1012w_{2} = \frac{1}{2} - \frac{\sqrt{101}}{2}

Como
w2−w=5\sqrt{w^{2} - w} = 5
y
w2−w≥0\sqrt{w^{2} - w} \geq 0
entonces
5≥05 \geq 0
Entonces la respuesta definitiva es:
w1=12+1012w_{1} = \frac{1}{2} + \frac{\sqrt{101}}{2}
w2=12−1012w_{2} = \frac{1}{2} - \frac{\sqrt{101}}{2}
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⁡(12+1012)x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}
x1=πn+acos⁡(12+1012)x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(12−1012)x_{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}
x2=πn+acos⁡(12−1012)x_{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(12+1012)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}
x3=πn−π+acos⁡(12+1012)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(12−1012)x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}
x4=πn−π+acos⁡(12−1012)x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}
Gráfica
0-80-60-40-2020406080-1001001-2
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
    /    /      _____\\              /    /      _____\\              /    /      _____\\       /    /      _____\\     /    /      _____\\       /    /      _____\\     /    /      _____\\
    |    |1   \/ 101 ||              |    |1   \/ 101 ||              |    |1   \/ 101 ||       |    |1   \/ 101 ||     |    |1   \/ 101 ||       |    |1   \/ 101 ||     |    |1   \/ 101 ||
- re|acos|- - -------|| + 2*pi - I*im|acos|- - -------|| + 2*pi - I*im|acos|- + -------|| + I*im|acos|- - -------|| + re|acos|- - -------|| + I*im|acos|- + -------|| + re|acos|- + -------||
    \    \2      2   //              \    \2      2   //              \    \2      2   //       \    \2      2   //     \    \2      2   //       \    \2      2   //     \    \2      2   //
((re⁡(acos⁡(12−1012))+iim⁡(acos⁡(12−1012)))+((2π−iim⁡(acos⁡(12+1012)))+(−re⁡(acos⁡(12−1012))+2π−iim⁡(acos⁡(12−1012)))))+(re⁡(acos⁡(12+1012))+iim⁡(acos⁡(12+1012)))\left(\left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)}\right) + \left(\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)}\right)\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)}\right) 
=
         /    /      _____\\
         |    |1   \/ 101 ||
4*pi + re|acos|- + -------||
         \    \2      2   //
re⁡(acos⁡(12+1012))+4π\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)} + 4 \pi 
producto
/    /    /      _____\\              /    /      _____\\\ /           /    /      _____\\\ /    /    /      _____\\     /    /      _____\\\ /    /    /      _____\\     /    /      _____\\\
|    |    |1   \/ 101 ||              |    |1   \/ 101 ||| |           |    |1   \/ 101 ||| |    |    |1   \/ 101 ||     |    |1   \/ 101 ||| |    |    |1   \/ 101 ||     |    |1   \/ 101 |||
|- re|acos|- - -------|| + 2*pi - I*im|acos|- - -------|||*|2*pi - I*im|acos|- + -------|||*|I*im|acos|- - -------|| + re|acos|- - -------|||*|I*im|acos|- + -------|| + re|acos|- + -------|||
\    \    \2      2   //              \    \2      2   /// \           \    \2      2   /// \    \    \2      2   //     \    \2      2   /// \    \    \2      2   //     \    \2      2   ///
(2π−iim⁡(acos⁡(12+1012)))(−re⁡(acos⁡(12−1012))+2π−iim⁡(acos⁡(12−1012)))(re⁡(acos⁡(12−1012))+iim⁡(acos⁡(12−1012)))(re⁡(acos⁡(12+1012))+iim⁡(acos⁡(12+1012)))\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)}\right) 
=
 /           /    /      _____\\\ /    /    /      _____\\     /    /      _____\\\ /    /    /      _____\\     /    /      _____\\\ /            /    /      _____\\     /    /      _____\\\
 |           |    |1   \/ 101 ||| |    |    |1   \/ 101 ||     |    |1   \/ 101 ||| |    |    |1   \/ 101 ||     |    |1   \/ 101 ||| |            |    |1   \/ 101 ||     |    |1   \/ 101 |||
-|2*pi - I*im|acos|- + -------|||*|I*im|acos|- + -------|| + re|acos|- + -------|||*|I*im|acos|- - -------|| + re|acos|- - -------|||*|-2*pi + I*im|acos|- - -------|| + re|acos|- - -------|||
 \           \    \2      2   /// \    \    \2      2   //     \    \2      2   /// \    \    \2      2   //     \    \2      2   /// \            \    \2      2   //     \    \2      2   ///
−(2π−iim⁡(acos⁡(12+1012)))(re⁡(acos⁡(12−1012))+iim⁡(acos⁡(12−1012)))(re⁡(acos⁡(12+1012))+iim⁡(acos⁡(12+1012)))(−2π+re⁡(acos⁡(12−1012))+iim⁡(acos⁡(12−1012)))- \left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)}\right) 
-(2*pi - i*im(acos(1/2 + sqrt(101)/2)))*(i*im(acos(1/2 + sqrt(101)/2)) + re(acos(1/2 + sqrt(101)/2)))*(i*im(acos(1/2 - sqrt(101)/2)) + re(acos(1/2 - sqrt(101)/2)))*(-2*pi + i*im(acos(1/2 - sqrt(101)/2)) + re(acos(1/2 - sqrt(101)/2)))
Respuesta rápida [src] 
         /    /      _____\\              /    /      _____\\
         |    |1   \/ 101 ||              |    |1   \/ 101 ||
x1 = - re|acos|- - -------|| + 2*pi - I*im|acos|- - -------||
         \    \2      2   //              \    \2      2   //
x1=−re⁡(acos⁡(12−1012))+2π−iim⁡(acos⁡(12−1012))x_{1} = - \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} 
                /    /      _____\\
                |    |1   \/ 101 ||
x2 = 2*pi - I*im|acos|- + -------||
                \    \2      2   //
x2=2π−iim⁡(acos⁡(12+1012))x_{2} = 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)} 
         /    /      _____\\     /    /      _____\\
         |    |1   \/ 101 ||     |    |1   \/ 101 ||
x3 = I*im|acos|- - -------|| + re|acos|- - -------||
         \    \2      2   //     \    \2      2   //
x3=re⁡(acos⁡(12−1012))+iim⁡(acos⁡(12−1012))x_{3} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} - \frac{\sqrt{101}}{2} \right)}\right)} 
         /    /      _____\\     /    /      _____\\
         |    |1   \/ 101 ||     |    |1   \/ 101 ||
x4 = I*im|acos|- + -------|| + re|acos|- + -------||
         \    \2      2   //     \    \2      2   //
x4=re⁡(acos⁡(12+1012))+iim⁡(acos⁡(12+1012))x_{4} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{2} + \frac{\sqrt{101}}{2} \right)}\right)} 
x4 = re(acos(1/2 + sqrt(101)/2)) + i*im(acos(1/2 + sqrt(101)/2))
Respuesta numérica [src] 
x1 = -21.9911485751286 + 2.19031114446532*i
x2 = -84.8230016469244 + 2.19031114446532*i
x3 = 59.6902604182061 + 2.19031114446532*i
x4 = 94.2477796076938 + 2.3941266607411*i
x5 = 15.707963267949 + 2.19031114446532*i
x6 = -3.14159265358979 + 2.19031114446532*i
x7 = 34.5575191894877 - 2.19031114446532*i
x8 = -97.3893722612836 - 2.19031114446532*i
x9 = -43.9822971502571 + 2.3941266607411*i
x10 = 12.5663706143592 - 2.3941266607411*i
x11 = -53.4070751110265 - 2.19031114446532*i
x12 = -28.2743338823081 + 2.19031114446532*i
x13 = -9.42477796076938 - 2.19031114446532*i
x14 = -37.6991118430775 + 2.3941266607411*i
x15 = 75.398223686155 - 2.3941266607411*i
x16 = 50.2654824574367 - 2.3941266607411*i
x17 = -81.6814089933346 + 2.3941266607411*i
x18 = 6.28318530717959 + 2.3941266607411*i
x19 = 72.2566310325652 + 2.19031114446532*i
x20 = 25.1327412287183 + 2.3941266607411*i
x21 = 31.4159265358979 - 2.3941266607411*i
x22 = 25.1327412287183 - 2.3941266607411*i
x23 = -62.8318530717959 + 2.3941266607411*i
x24 = -84.8230016469244 - 2.19031114446532*i
x25 = -72.2566310325652 + 2.19031114446532*i
x26 = 78.5398163397448 - 2.19031114446532*i
x27 = 50.2654824574367 + 2.3941266607411*i
x27 = 50.2654824574367 + 2.3941266607411*i
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