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ctg^2(x)+8ln(sinx) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2                       
cot (x) + 8*log(sin(x)) = 0
8log⁡(sin⁡(x))+cot⁡2(x)=08 \log{\left(\sin{\left(x \right)} \right)} + \cot^{2}{\left(x \right)} = 0
Simplificar
Solución detallada
Tenemos la ecuación
8log⁡(sin⁡(x))+cot⁡2(x)=08 \log{\left(\sin{\left(x \right)} \right)} + \cot^{2}{\left(x \right)} = 0
cambiamos
8log⁡(sin⁡(x))+cot⁡2(x)=08 \log{\left(\sin{\left(x \right)} \right)} + \cot^{2}{\left(x \right)} = 0
8log⁡(sin⁡(x))+cot⁡2(x)=08 \log{\left(\sin{\left(x \right)} \right)} + \cot^{2}{\left(x \right)} = 0
Sustituimos
w=sin⁡(x)w = \sin{\left(x \right)}
Tenemos la ecuación
8log⁡(w)+cot⁡2(x)=08 \log{\left(w \right)} + \cot^{2}{\left(x \right)} = 0
8log⁡(w)=−cot⁡2(x)8 \log{\left(w \right)} = - \cot^{2}{\left(x \right)}
Devidimos ambás partes de la ecuación por el multiplicador de log =8
log⁡(w)=−cot⁡2(x)8\log{\left(w \right)} = - \frac{\cot^{2}{\left(x \right)}}{8}
Es la ecuación de la forma:
log(v)=p

Por definición log
v=e^p

entonces
w=e(−1)cot⁡2(x)8w = e^{\frac{\left(-1\right) \cot^{2}{\left(x \right)}}{8}}
simplificamos
w=e−cot⁡2(x)8w = e^{- \frac{\cot^{2}{\left(x \right)}}{8}}
hacemos cambio inverso
sin⁡(x)=w\sin{\left(x \right)} = w
Tenemos la ecuación
sin⁡(x)=w\sin{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
Gráfica
0-80-60-40-2020406080-100100-10000000001000000000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta numérica [src] 
x1 = 1.57079657617902
x2 = 51.8362789040308
x3 = -98.9601684962676 - 4.9130793884145e-7*i
x4 = 20.4203521472267
x5 = -60.0063846844947
x6 = 95.8185760640235
x7 = 14.1371671109466
x8 = 64.4026493067856
x9 = -10.9955741133206 - 1.98368139674135e-7*i
x10 = 44.2984214165457
x11 = -9.74090222705799
x12 = -4.71238867872979
x13 = -80.1106125777443
x14 = -97.7054965275722
x15 = -29.8451300952151
x16 = 83.2522047173199 - 1.23909973796456e-7*i
x17 = 32.986722581167
x18 = 45.5530937338517
x19 = 70.6858344772512
x20 = -67.5442421715074
x21 = -42.4115005816937
x22 = 58.119464334569
x23 = -10.9955745537217
x24 = -48.6946858228435
x25 = -61.2610569958328
x26 = -49.9493581911481
x27 = -73.8274272797817
x28 = -86.3937977397458
x29 = -98.9601687905598
x30 = 71.9405067662766
x31 = -23.5619450121989
x32 = -36.1283154167875
x33 = 34.2413949231991
x34 = -54.9778716795764
x35 = 38.0152361093661
x36 = 89.535390891299
x37 = -5.96706104089097
x38 = -16.0240875342376
x39 = 32.9867230934146 - 4.69581599978079e-8*i
x40 = 7.85398174377895
x41 = 26.7035373193711
x42 = -53.7231993773151
x43 = -54.9778713095234 - 3.47033153727537e-7*i
x44 = 83.2522055949645 + 1.56719488268768e-7*i
x45 = 27.9582096160195
x46 = 78.2236920734562
x47 = 39.2699084617611
x48 = -92.6769829696594
x49 = 83.2522055971896
x50 = -17.2787598383324
x51 = -93.9316553414052
x52 = 88.2807185668028
x53 = -87.6484700342256
x54 = 81.9975332596232
x55 = 76.9690197219456
x56 = -43.6661728839685
x56 = -43.6661728839685
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