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3*cos(2*x)+cos(x)-4=0

3*cos(2*x)+cos(x)-4=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
3*cos(2*x) + cos(x) - 4 = 0
(cos⁡(x)+3cos⁡(2x))−4=0\left(\cos{\left(x \right)} + 3 \cos{\left(2 x \right)}\right) - 4 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(cos⁡(x)+3cos⁡(2x))−4=0\left(\cos{\left(x \right)} + 3 \cos{\left(2 x \right)}\right) - 4 = 0
cambiamos
6cos⁡2(x)+cos⁡(x)−5=06 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - 5 = 0
6cos⁡2(x)+cos⁡(x)−5=06 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - 5 = 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=6a = 6
b=1b = 1
c=−5c = -5
, entonces
D = b^2 - 4 * a * c = 

(1)^2 - 4 * (6) * (-5) = 121

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

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

o
w1=56w_{1} = \frac{5}{6}
w2=−1w_{2} = -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⁡(56)x_{1} = \pi n + \operatorname{acos}{\left(\frac{5}{6} \right)}
x1=πn+acos⁡(56)x_{1} = \pi n + \operatorname{acos}{\left(\frac{5}{6} \right)}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(−1)x_{2} = \pi n + \operatorname{acos}{\left(-1 \right)}
x2=πn+πx_{2} = \pi n + \pi
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(56)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{5}{6} \right)}
x3=πn−π+acos⁡(56)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{5}{6} \right)}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(−1)x_{4} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}
x4=πnx_{4} = \pi n
Gráfica
0-80-60-40-2020406080-100100-1010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = 0
x1=0x_{1} = 0 
               /      ____\
               |7   \/ 13 |
x2 = pi + I*log|- - ------|
               \6     6   /
x2=π+ilog⁡(76−136)x_{2} = \pi + i \log{\left(\frac{7}{6} - \frac{\sqrt{13}}{6} \right)} 
               /      ____\
               |7   \/ 13 |
x3 = pi + I*log|- + ------|
               \6     6   /
x3=π+ilog⁡(136+76)x_{3} = \pi + i \log{\left(\frac{\sqrt{13}}{6} + \frac{7}{6} \right)} 
x3 = pi + i*log(sqrt(13)/6 + 7/6)
Suma y producto de raíces [src] 
suma
          /      ____\             /      ____\
          |7   \/ 13 |             |7   \/ 13 |
pi + I*log|- - ------| + pi + I*log|- + ------|
          \6     6   /             \6     6   /
(π+ilog⁡(76−136))+(π+ilog⁡(136+76))\left(\pi + i \log{\left(\frac{7}{6} - \frac{\sqrt{13}}{6} \right)}\right) + \left(\pi + i \log{\left(\frac{\sqrt{13}}{6} + \frac{7}{6} \right)}\right) 
=
            /      ____\        /      ____\
            |7   \/ 13 |        |7   \/ 13 |
2*pi + I*log|- - ------| + I*log|- + ------|
            \6     6   /        \6     6   /
2π+ilog⁡(76−136)+ilog⁡(136+76)2 \pi + i \log{\left(\frac{7}{6} - \frac{\sqrt{13}}{6} \right)} + i \log{\left(\frac{\sqrt{13}}{6} + \frac{7}{6} \right)} 
producto
  /          /      ____\\ /          /      ____\\
  |          |7   \/ 13 || |          |7   \/ 13 ||
0*|pi + I*log|- - ------||*|pi + I*log|- + ------||
  \          \6     6   // \          \6     6   //
0(π+ilog⁡(76−136))(π+ilog⁡(136+76))0 \left(\pi + i \log{\left(\frac{7}{6} - \frac{\sqrt{13}}{6} \right)}\right) \left(\pi + i \log{\left(\frac{\sqrt{13}}{6} + \frac{7}{6} \right)}\right) 
=
0
00 
0
Respuesta numérica [src] 
x1 = 18.8495556748901
x2 = -62.8318528192025
x3 = -37.6991118771601
x4 = 87.9645943357691
x5 = -12.5663702112782
x6 = 100.530964766054
x7 = -87.9645943759068
x8 = 25.132741471072
x9 = -25.1327413703358
x10 = 69.1150381381372
x11 = -62.8318532907913
x12 = -69.1150386310748
x13 = -50.2654822946212
x14 = 18.8495557023389
x15 = 56.5486676084927
x16 = -56.5486675148031
x17 = -18.8495557368747
x18 = -31.4159267060472
x19 = 75.3982239445512
x20 = 25.1327409939176
x21 = 0.0
x22 = 94.2477796093524
x23 = -75.3982238630204
x24 = -6.28318513710845
x25 = -69.1150384554896
x26 = 31.4159267917545
x27 = 62.8318528283756
x28 = -320.442451006313
x29 = -94.2477794523777
x30 = 81.6814091772056
x31 = -18.8495556736075
x32 = -25.1327414782656
x33 = 37.6991120201661
x34 = -87.9645943587609
x35 = 69.1150386147231
x36 = 12.5663704511249
x37 = 62.831852761042
x38 = 31.4159268347111
x39 = 43.9822971694299
x40 = -81.6814090380234
x41 = -12.5663703613676
x42 = -100.530964698576
x43 = -43.9822971745758
x44 = -18.849556149113
x45 = 50.2654824463467
x46 = -100.530964668541
x47 = 6.28318528424792
x48 = 75.3982238949822
x48 = 75.3982238949822
Gráfico
3*cos(2*x)+cos(x)-4=0 la ecuación
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