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2*cos(2*x)+4*sqrt(3)*cos(x)-7=0

2*cos(2*x)+4*sqrt(3)*cos(x)-7=0 la ecuación

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Solución

Ha introducido [src] 
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2*cos(2*x) + 4*\/ 3 *cos(x) - 7 = 0
(43cos⁡(x)+2cos⁡(2x))−7=0\left(4 \sqrt{3} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right) - 7 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(43cos⁡(x)+2cos⁡(2x))−7=0\left(4 \sqrt{3} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right) - 7 = 0
cambiamos
4cos⁡2(x)+43cos⁡(x)−8=04 \cos^{2}{\left(x \right)} + 4 \sqrt{3} \cos{\left(x \right)} - 8 = 0
4cos⁡2(x)+43cos⁡(x)−8=04 \cos^{2}{\left(x \right)} + 4 \sqrt{3} \cos{\left(x \right)} - 8 = 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=4a = 4
b=43b = 4 \sqrt{3}
c=−8c = -8
, entonces
D = b^2 - 4 * a * c = 

(4*sqrt(3))^2 - 4 * (4) * (-8) = 176

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

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

o
w1=−32+112w_{1} = - \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2}
w2=−112−32w_{2} = - \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{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⁡(−32+112)x_{1} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}
x1=πn+acos⁡(−32+112)x_{1} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(−112−32)x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}
x2=πn+acos⁡(−112−32)x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(−32+112)x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}
x3=πn−π+acos⁡(−32+112)x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(−112−32)x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}
x4=πn−π+acos⁡(−112−32)x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}
Gráfica
0-80-60-40-2020406080-100100-2020
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     -pi 
x1 = ----
      6
x1=−π6x_{1} = - \frac{\pi}{6} 
     pi
x2 = --
     6
x2=π6x_{2} = \frac{\pi}{6} 
               /      -2        \
x3 = pi + I*log|----------------|
               |  ____       ___|
               \\/ 23  - 3*\/ 3 /
x3=π+ilog⁡(−2−33+23)x_{3} = \pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)} 
               /       2        \
x4 = pi + I*log|----------------|
               |  ____       ___|
               \\/ 23  + 3*\/ 3 /
x4=π+ilog⁡(223+33)x_{4} = \pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)} 
x4 = pi + i*log(2/(sqrt(23) + 3*sqrt(3)))
Suma y producto de raíces [src] 
suma
  pi   pi             /      -2        \             /       2        \
- -- + -- + pi + I*log|----------------| + pi + I*log|----------------|
  6    6              |  ____       ___|             |  ____       ___|
                      \\/ 23  - 3*\/ 3 /             \\/ 23  + 3*\/ 3 /
(π+ilog⁡(223+33))+((−π6+π6)+(π+ilog⁡(−2−33+23)))\left(\pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)}\right) + \left(\left(- \frac{\pi}{6} + \frac{\pi}{6}\right) + \left(\pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}\right)\right) 
=
            /      -2        \        /       2        \
2*pi + I*log|----------------| + I*log|----------------|
            |  ____       ___|        |  ____       ___|
            \\/ 23  - 3*\/ 3 /        \\/ 23  + 3*\/ 3 /
2π+ilog⁡(223+33)+ilog⁡(−2−33+23)2 \pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)} + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)} 
producto
-pi  pi /          /      -2        \\ /          /       2        \\
----*--*|pi + I*log|----------------||*|pi + I*log|----------------||
 6   6  |          |  ____       ___|| |          |  ____       ___||
        \          \\/ 23  - 3*\/ 3 // \          \\/ 23  + 3*\/ 3 //
−π6π6(π+ilog⁡(−2−33+23))(π+ilog⁡(223+33))- \frac{\pi}{6} \frac{\pi}{6} \left(\pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}\right) \left(\pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)}\right) 
=
   2 /          /      -2        \\ /          /       2        \\ 
-pi *|pi + I*log|----------------||*|pi + I*log|----------------|| 
     |          |  ____       ___|| |          |  ____       ___|| 
     \          \\/ 23  - 3*\/ 3 // \          \\/ 23  + 3*\/ 3 // 
-------------------------------------------------------------------
                                 36
−π2(π+ilog⁡(223+33))(π+ilog⁡(−2−33+23))36- \frac{\pi^{2} \left(\pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)}\right) \left(\pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}\right)}{36} 
-pi^2*(pi + i*log(-2/(sqrt(23) - 3*sqrt(3))))*(pi + i*log(2/(sqrt(23) + 3*sqrt(3))))/36
Respuesta numérica [src] 
x1 = 696.909970321336
x2 = -49.7418836818384
x3 = 18.3259571459405
x4 = 93.7241808320955
x5 = 38.2227106186758
x6 = -74.8746249105567
x7 = -57.0722665402146
x8 = 88.4881930761125
x9 = 31.9395253114962
x10 = -25.6563400043166
x11 = 56.025068989018
x12 = -101.054563690472
x13 = 44.5058959258554
x14 = 5.75958653158129
x15 = -5.75958653158129
x16 = -176.452787376627
x17 = 13.0899693899575
x18 = -82.2050077689329
x19 = -93.7241808320955
x20 = 81.1578102177363
x21 = -68.5914396033772
x22 = 68.5914396033772
x23 = 30.8923277602996
x24 = 100.007366139275
x25 = -56.025068989018
x26 = -30.8923277602996
x27 = -24.60914245312
x28 = -8294.32820425265
x29 = 37.1755130674792
x30 = -63.3554518473942
x31 = 69.6386371545737
x32 = -43.4586983746588
x33 = 63.3554518473942
x34 = 19.3731546971371
x35 = -62.3082542961976
x36 = -18.3259571459405
x37 = -31.9395253114962
x38 = 25.6563400043166
x39 = 169.122404518251
x40 = -88.4881930761125
x41 = -87.4409955249159
x42 = -69.6386371545737
x43 = -50.789081233035
x44 = -6.80678408277789
x45 = -19.3731546971371
x46 = -38.2227106186758
x47 = 62.3082542961976
x48 = -94.7713783832921
x49 = -100.007366139275
x50 = 49.7418836818384
x51 = 74.8746249105567
x52 = 12.0427718387609
x53 = -12.0427718387609
x54 = 82.2050077689329
x55 = 57.0722665402146
x56 = -13.0899693899575
x57 = 24.60914245312
x58 = 0.523598775598299
x59 = 213.104701668508
x60 = -75.9218224617533
x61 = 75.9218224617533
x61 = 75.9218224617533
Gráfico
2*cos(2*x)+4*sqrt(3)*cos(x)-7=0 la ecuación
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