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log2cos(x)=0.2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
log(2*cos(x)) = 1/5
log⁡(2cos⁡(x))=15\log{\left(2 \cos{\left(x \right)} \right)} = \frac{1}{5}
Simplificar
Solución detallada
Tenemos la ecuación
log⁡(2cos⁡(x))=15\log{\left(2 \cos{\left(x \right)} \right)} = \frac{1}{5}
cambiamos
log⁡(2cos⁡(x))−15=0\log{\left(2 \cos{\left(x \right)} \right)} - \frac{1}{5} = 0
log⁡(2cos⁡(x))−15=0\log{\left(2 \cos{\left(x \right)} \right)} - \frac{1}{5} = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
Tenemos la ecuación
log⁡(2w)−15=0\log{\left(2 w \right)} - \frac{1}{5} = 0
log⁡(2w)=15\log{\left(2 w \right)} = \frac{1}{5}
Es la ecuación de la forma:
log(v)=p

Por definición log
v=e^p

entonces
2w=e152 w = e^{\frac{1}{5}}
simplificamos
2w=e152 w = e^{\frac{1}{5}}
w=e152w = \frac{e^{\frac{1}{5}}}{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:
Gráfica
0-80-60-40-2020406080-100100-1010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
      / 1/5\              / 1/5\
      |e   |              |e   |
- acos|----| + 2*pi + acos|----|
      \ 2  /              \ 2  /
acos⁡(e152)+(−acos⁡(e152)+2π)\operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + \left(- \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi\right) 
=
2*pi
2π2 \pi 
producto
/      / 1/5\       \     / 1/5\
|      |e   |       |     |e   |
|- acos|----| + 2*pi|*acos|----|
\      \ 2  /       /     \ 2  /
(−acos⁡(e152)+2π)acos⁡(e152)\left(- \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} 
=
/      / 1/5\       \     / 1/5\
|      |e   |       |     |e   |
|- acos|----| + 2*pi|*acos|----|
\      \ 2  /       /     \ 2  /
(−acos⁡(e152)+2π)acos⁡(e152)\left(- \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} 
(-acos(exp(1/5)/2) + 2*pi)*acos(exp(1/5)/2)
Respuesta rápida [src] 
           / 1/5\       
           |e   |       
x1 = - acos|----| + 2*pi
           \ 2  /
x1=−acos⁡(e152)+2πx_{1} = - \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi 
         / 1/5\
         |e   |
x2 = acos|----|
         \ 2  /
x2=acos⁡(e152)x_{2} = \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} 
x2 = acos(exp(1/5)/2)
Respuesta numérica [src] 
x1 = 36.7852615401172
x2 = 19.763406224499
x3 = 57.4625180675766
x4 = 24.2188909257581
x5 = -93.3339293047335
x6 = -19.763406224499
x7 = -7.19703561013987
x8 = -55.634817461656
x9 = -95.1616299106541
x10 = -17.9357056185785
x11 = -82.5952592962949
x12 = -11.6525203113989
x13 = -74.4843733831948
x14 = -0.913850302960281
x15 = -51.179332760397
x16 = -38.6129621460378
x17 = -36.7852615401172
x18 = 101.444815217834
x19 = -49.3516321544764
x20 = -80.7675586903743
x21 = 68.2011880760152
x22 = -61.9180027688356
x23 = 82.5952592962949
x24 = 30.5020762329377
x25 = 0.913850302960281
x26 = 76.3120739891153
x27 = -87.0507439975539
x28 = -13.4802209173195
x29 = 32.3297768388582
x30 = -99.6171146119131
x31 = 70.0288886819357
x32 = -68.2011880760152
x33 = -76.3120739891153
x34 = 44.8961474532174
x35 = 49.3516321544764
x36 = 99.6171146119131
x37 = 17.9357056185785
x38 = -44.8961474532174
x39 = -32.3297768388582
x40 = -101.444815217834
x41 = 87.0507439975539
x42 = -5.36933500421931
x43 = 88.8784446034745
x44 = -43.0684468472968
x45 = -30.5020762329377
x46 = 13.4802209173195
x47 = 61.9180027688356
x48 = 80.7675586903743
x49 = 38.6129621460378
x50 = -70.0288886819357
x51 = 11.6525203113989
x52 = -57.4625180675766
x53 = 5.36933500421931
x54 = 7.19703561013987
x55 = 74.4843733831948
x56 = 26.0465915316786
x57 = -26.0465915316786
x58 = 93.3339293047335
x59 = 43.0684468472968
x60 = 63.7457033747561
x61 = -63.7457033747561
x62 = 55.634817461656
x63 = -24.2188909257581
x63 = -24.2188909257581
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