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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
$$\left(4 \sqrt{3} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right) - 7 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(4 \sqrt{3} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right) - 7 = 0$$
cambiamos
$$4 \cos^{2}{\left(x \right)} + 4 \sqrt{3} \cos{\left(x \right)} - 8 = 0$$
$$4 \cos^{2}{\left(x \right)} + 4 \sqrt{3} \cos{\left(x \right)} - 8 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 4$$
$$b = 4 \sqrt{3}$$
$$c = -8$$
, 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
$$w_{1} = - \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2}$$
$$w_{2} = - \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{11}}{2} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{11}}{2} - \frac{\sqrt{3}}{2} \right)}$$
Gráfica
Respuesta rápida [src]
     -pi 
x1 = ----
      6  
$$x_{1} = - \frac{\pi}{6}$$
     pi
x2 = --
     6 
$$x_{2} = \frac{\pi}{6}$$
               /      -2        \
x3 = pi + I*log|----------------|
               |  ____       ___|
               \\/ 23  - 3*\/ 3 /
$$x_{3} = \pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}$$
               /       2        \
x4 = pi + I*log|----------------|
               |  ____       ___|
               \\/ 23  + 3*\/ 3 /
$$x_{4} = \pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)}$$
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 /
$$\left(\pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)}\right) + \left(\left(- \frac{\pi}{6} + \frac{\pi}{6}\right) + \left(\pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}\right)\right)$$
=
            /      -2        \        /       2        \
2*pi + I*log|----------------| + I*log|----------------|
            |  ____       ___|        |  ____       ___|
            \\/ 23  - 3*\/ 3 /        \\/ 23  + 3*\/ 3 /
$$2 \pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)} + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}$$
producto
-pi  pi /          /      -2        \\ /          /       2        \\
----*--*|pi + I*log|----------------||*|pi + I*log|----------------||
 6   6  |          |  ____       ___|| |          |  ____       ___||
        \          \\/ 23  - 3*\/ 3 // \          \\/ 23  + 3*\/ 3 //
$$- \frac{\pi}{6} \frac{\pi}{6} \left(\pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}\right) \left(\pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)}\right)$$
=
   2 /          /      -2        \\ /          /       2        \\ 
-pi *|pi + I*log|----------------||*|pi + I*log|----------------|| 
     |          |  ____       ___|| |          |  ____       ___|| 
     \          \\/ 23  - 3*\/ 3 // \          \\/ 23  + 3*\/ 3 // 
-------------------------------------------------------------------
                                 36                                
$$- \frac{\pi^{2} \left(\pi + i \log{\left(\frac{2}{\sqrt{23} + 3 \sqrt{3}} \right)}\right) \left(\pi + i \log{\left(- \frac{2}{- 3 \sqrt{3} + \sqrt{23}} \right)}\right)}{36}$$
-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