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log(13)*2*cos(x)^(2)+3*cos(x)-1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
             2                      
log(13)*2*cos (x) + 3*cos(x) - 1 = 0
(2log⁡(13)cos⁡2(x)+3cos⁡(x))−1=0\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(2log⁡(13)cos⁡2(x)+3cos⁡(x))−1=0\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 0
cambiamos
log⁡(169)cos⁡2(x)+3cos⁡(x)−1=0\log{\left(169 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1 = 0
(2log⁡(13)cos⁡2(x)+3cos⁡(x))−1=0\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 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=2log⁡(13)a = 2 \log{\left(13 \right)}
b=3b = 3
c=−1c = -1
, entonces
D = b^2 - 4 * a * c = 

(3)^2 - 4 * (2*log(13)) * (-1) = 9 + 8*log(13)

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

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

o
w1=−3+9+8log⁡(13)4log⁡(13)w_{1} = \frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}}
w2=−9+8log⁡(13)−34log⁡(13)w_{2} = \frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}}
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⁡(−3+9+8log⁡(13)4log⁡(13))x_{1} = \pi n + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}
x1=πn+acos⁡(−3+9+8log⁡(13)4log⁡(13))x_{1} = \pi n + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(−9+8log⁡(13)−34log⁡(13))x_{2} = \pi n + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}
x2=πn+acos⁡(−9+8log⁡(13)−34log⁡(13))x_{2} = \pi n + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(−3+9+8log⁡(13)4log⁡(13))x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}
x3=πn−π+acos⁡(−3+9+8log⁡(13)4log⁡(13))x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(−9+8log⁡(13)−34log⁡(13))x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}
x4=πn−π+acos⁡(−9+8log⁡(13)−34log⁡(13))x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}
Gráfica
0-80-60-40-2020406080-100100-1010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
           /                _______________\       
           |      3       \/ 9 + 8*log(13) |       
x1 = - acos|- --------- - -----------------| + 2*pi
           \  4*log(13)       4*log(13)    /
x1=−acos⁡(−9+8log⁡(13)4log⁡(13)−34log⁡(13))+2πx_{1} = - \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + 2 \pi 
           /                _______________\       
           |      3       \/ 9 + 8*log(13) |       
x2 = - acos|- --------- + -----------------| + 2*pi
           \  4*log(13)       4*log(13)    /
x2=−acos⁡(−34log⁡(13)+9+8log⁡(13)4log⁡(13))+2πx_{2} = - \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi 
         /                _______________\
         |      3       \/ 9 + 8*log(13) |
x3 = acos|- --------- - -----------------|
         \  4*log(13)       4*log(13)    /
x3=acos⁡(−9+8log⁡(13)4log⁡(13)−34log⁡(13))x_{3} = \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} 
         /                _______________\
         |      3       \/ 9 + 8*log(13) |
x4 = acos|- --------- + -----------------|
         \  4*log(13)       4*log(13)    /
x4=acos⁡(−34log⁡(13)+9+8log⁡(13)4log⁡(13))x_{4} = \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} 
x4 = acos(-3/(4*log(13)) + sqrt(9 + 8*log(13))/(4*log(13)))
Suma y producto de raíces [src] 
suma
      /                _______________\                /                _______________\              /                _______________\       /                _______________\
      |      3       \/ 9 + 8*log(13) |                |      3       \/ 9 + 8*log(13) |              |      3       \/ 9 + 8*log(13) |       |      3       \/ 9 + 8*log(13) |
- acos|- --------- - -----------------| + 2*pi + - acos|- --------- + -----------------| + 2*pi + acos|- --------- - -----------------| + acos|- --------- + -----------------|
      \  4*log(13)       4*log(13)    /                \  4*log(13)       4*log(13)    /              \  4*log(13)       4*log(13)    /       \  4*log(13)       4*log(13)    /
acos⁡(−34log⁡(13)+9+8log⁡(13)4log⁡(13))+(acos⁡(−9+8log⁡(13)4log⁡(13)−34log⁡(13))+((−acos⁡(−9+8log⁡(13)4log⁡(13)−34log⁡(13))+2π)+(−acos⁡(−34log⁡(13)+9+8log⁡(13)4log⁡(13))+2π)))\operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + \left(\operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + \left(\left(- \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) + \left(- \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right)\right)\right) 
=
4*pi
4π4 \pi 
producto
/      /                _______________\       \ /      /                _______________\       \     /                _______________\     /                _______________\
|      |      3       \/ 9 + 8*log(13) |       | |      |      3       \/ 9 + 8*log(13) |       |     |      3       \/ 9 + 8*log(13) |     |      3       \/ 9 + 8*log(13) |
|- acos|- --------- - -----------------| + 2*pi|*|- acos|- --------- + -----------------| + 2*pi|*acos|- --------- - -----------------|*acos|- --------- + -----------------|
\      \  4*log(13)       4*log(13)    /       / \      \  4*log(13)       4*log(13)    /       /     \  4*log(13)       4*log(13)    /     \  4*log(13)       4*log(13)    /
(−acos⁡(−9+8log⁡(13)4log⁡(13)−34log⁡(13))+2π)(−acos⁡(−34log⁡(13)+9+8log⁡(13)4log⁡(13))+2π)acos⁡(−9+8log⁡(13)4log⁡(13)−34log⁡(13))acos⁡(−34log⁡(13)+9+8log⁡(13)4log⁡(13))\left(- \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \left(- \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} 
=
/      /       ____________________\       \ /      /       ____________________\       \     /       ____________________\     /       ____________________\
|      |-3 + \/ 9 + log(815730721) |       | |      |-3 - \/ 9 + log(815730721) |       |     |-3 + \/ 9 + log(815730721) |     |-3 - \/ 9 + log(815730721) |
|- acos|---------------------------| + 2*pi|*|- acos|---------------------------| + 2*pi|*acos|---------------------------|*acos|---------------------------|
\      \         4*log(13)         /       / \      \         4*log(13)         /       /     \         4*log(13)         /     \         4*log(13)         /
(−acos⁡(−3+9+log⁡(815730721)4log⁡(13))+2π)(−acos⁡(−9+log⁡(815730721)−34log⁡(13))+2π)acos⁡(−3+9+log⁡(815730721)4log⁡(13))acos⁡(−9+log⁡(815730721)−34log⁡(13))\left(- \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + \log{\left(815730721 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \left(- \operatorname{acos}{\left(\frac{- \sqrt{9 + \log{\left(815730721 \right)}} - 3}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + \log{\left(815730721 \right)}}}{4 \log{\left(13 \right)}} \right)} \operatorname{acos}{\left(\frac{- \sqrt{9 + \log{\left(815730721 \right)}} - 3}{4 \log{\left(13 \right)}} \right)} 
(-acos((-3 + sqrt(9 + log(815730721)))/(4*log(13))) + 2*pi)*(-acos((-3 - sqrt(9 + log(815730721)))/(4*log(13))) + 2*pi)*acos((-3 + sqrt(9 + log(815730721)))/(4*log(13)))*acos((-3 - sqrt(9 + log(815730721)))/(4*log(13)))
Respuesta numérica [src] 
x1 = -33.951575403966
x2 = 30.0845696266773
x3 = -17.5181990123182
x4 = -91.7121307396257
x5 = -54.0130188965482
x6 = -26.4640981379389
x7 = -3.74753643911152
x8 = 128.19935501166
x9 = 84.2170578614027
x10 = -11.2350137051386
x11 = -92.9164226984732
x12 = 61.5004961625753
x13 = -30.0845696266773
x14 = -39.0304687522981
x15 = 99.1996080056528
x16 = 13.8977275235798
x17 = -41.446648282189
x18 = -99.1996080056528
x19 = -60.2962042037278
x20 = 16.3139070534707
x21 = 32.7472834451185
x22 = 40.2347607111456
x23 = 26.4640981379389
x24 = -61.5004961625753
x25 = 27.6683900967864
x26 = 97.9953160468053
x27 = 66.5793895109074
x28 = 22.5970923606503
x29 = -80.350052084114
x30 = -70.4463952881961
x31 = 3.74753643911152
x32 = -55.2173108553957
x33 = 57.8800246738369
x34 = -57.8800246738369
x35 = 11.2350137051386
x36 = 70.4463952881961
x37 = -64.1632099810165
x38 = 46.5179460183252
x39 = 418.437766712964
x40 = -151226.239622066
x41 = 80.350052084114
x42 = -51.5968393666573
x43 = 55.2173108553957
x44 = -32.7472834451185
x45 = -40.2347607111456
x46 = -21.3852047896068
x47 = 64.1632099810165
x48 = 36.3677549338569
x49 = 39.0304687522981
x50 = -97.9953160468053
x51 = -10.0307217462911
x52 = 74.0668667769344
x53 = 76.7295805953756
x54 = -4.95182839795898
x55 = -27.6683900967864
x56 = 2.53564886806806
x57 = -13.8977275235798
x58 = 77.9338725542231
x59 = -74.0668667769344
x60 = -95.5791365169144
x61 = -48.9341255482161
x62 = 10.0307217462911
x63 = -77.9338725542231
x64 = -47.7298335893686
x65 = 91.7121307396257
x66 = -76.7295805953756
x67 = -67.7836814697549
x68 = 60.2962042037278
x69 = -71.6506872470435
x70 = 90.5002431685823
x71 = 23.8013843194977
x72 = -16.3139070534707
x73 = -7.61454221640019
x74 = -20.1809128307594
x75 = 67.7836814697549
x76 = 54.0130188965482
x77 = 17.5181990123182
x78 = 20.1809128307594
x79 = -23.8013843194977
x80 = 71.6506872470435
x81 = -84.2170578614027
x82 = 83.0127659025552
x83 = -85.4289454324461
x84 = -90.5002431685823
x85 = 47.7298335893686
x86 = 33.951575403966
x86 = 33.951575403966
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