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√(1-2cos(x))*log(-2sin(x),5)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
  ______________                      
\/ 1 - 2*cos(x) *log(-2*sin(x), 5) = 0
$$\sqrt{1 - 2 \cos{\left(x \right)}} \log{\left(- 2 \sin{\left(x \right)} \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\sqrt{1 - 2 \cos{\left(x \right)}} \log{\left(- 2 \sin{\left(x \right)} \right)} = 0$$
cambiamos
$$\frac{\sqrt{1 - 2 \cos{\left(x \right)}} \log{\left(- 2 \sin{\left(x \right)} \right)}}{\log{\left(5 \right)}} = 0$$
$$\sqrt{1 - 2 \cos{\left(x \right)}} \log{\left(- 2 \sin{\left(x \right)} \right)} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Tenemos la ecuación
$$\frac{\sqrt{1 - 2 \cos{\left(x \right)}} \log{\left(- 2 w \right)}}{\log{\left(5 \right)}} = 0$$
$$\frac{\sqrt{1 - 2 \cos{\left(x \right)}} \log{\left(- 2 w \right)}}{\log{\left(5 \right)}} = 0$$
Devidimos ambás partes de la ecuación por el multiplicador de log =sqrt(1 - 2*cos(x))/log(5)
$$\log{\left(- 2 w \right)} = 0$$
Es la ecuación de la forma:
log(v)=p

Por definición log
v=e^p

entonces
$$- 2 w = e^{\frac{0}{\sqrt{1 - 2 \cos{\left(x \right)}} \frac{1}{\log{\left(5 \right)}}}}$$
simplificamos
$$- 2 w = 1$$
$$w = - \frac{1}{2}$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
Gráfica
Respuesta rápida [src]
     -pi 
x1 = ----
      6  
$$x_{1} = - \frac{\pi}{6}$$
     pi
x2 = --
     3 
$$x_{2} = \frac{\pi}{3}$$
     7*pi
x3 = ----
      6  
$$x_{3} = \frac{7 \pi}{6}$$
     5*pi
x4 = ----
      3  
$$x_{4} = \frac{5 \pi}{3}$$
x4 = 5*pi/3
Suma y producto de raíces [src]
suma
  pi   pi   7*pi   5*pi
- -- + -- + ---- + ----
  6    3     6      3  
$$\left(\left(- \frac{\pi}{6} + \frac{\pi}{3}\right) + \frac{7 \pi}{6}\right) + \frac{5 \pi}{3}$$
=
3*pi
$$3 \pi$$
producto
-pi  pi 7*pi 5*pi
----*--*----*----
 6   3   6    3  
$$\frac{5 \pi}{3} \frac{7 \pi}{6} \cdot - \frac{\pi}{6} \frac{\pi}{3}$$
=
      4
-35*pi 
-------
  324  
$$- \frac{35 \pi^{4}}{324}$$
-35*pi^4/324
Respuesta numérica [src]
x1 = -15.1843644923507
x2 = -38.2227106186758
x3 = 1287.52938919622
x4 = -31.9395253114962
x5 = -84.2994028713261
x6 = -69.6386371545737
x7 = 62.3082542961976
x8 = -0.523598775598299
x9 = 49.7418836818384
x10 = 91.6297857297023
x11 = -90.5825881785057
x12 = -25.6563400043166
x13 = -82.2050077689329
x14 = 93.7241808320955
x15 = 72.7802298081635
x16 = -40.317105721069
x17 = -34.0339204138894
x18 = 9.94837673636768
x19 = 81.1578102177363
x20 = 87.4409955249159
x21 = 43.4586983746588
x22 = 22.5147473507269
x23 = -2.61799387799149
x24 = 5.75958653158129
x25 = -94.7713783832921
x26 = 35.081117965086
x27 = -6.80678408277789
x28 = 56.025068989018
x29 = 97.9129710368819
x30 = -65.4498469497874
x31 = 3.66519142918809
x32 = 100.007366139275
x33 = 12.0427718387609
x34 = -88.4881930761125
x35 = 66.497044500984
x36 = -78.0162175641465
x37 = 47.6474885794452
x38 = -44.5058959258554
x39 = 37.1755130674792
x40 = -71.733032256967
x41 = -27.7507351067098
x42 = 53.9306738866248
x43 = -75.9218224617533
x44 = -59.1666616426078
x45 = 85.3466004225227
x46 = 28.7979326579064
x47 = -50.789081233035
x48 = 60.2138591938044
x49 = 41.3643032722656
x50 = 16.2315620435473
x51 = -46.6002910282486
x52 = 18.3259571459405
x53 = -21.4675497995303
x53 = -21.4675497995303