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2*sin(x)^(3)-2sin(x)+cos(x)^(2)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
     3                    2       
2*sin (x) - 2*sin(x) + cos (x) = 0
$$\left(2 \sin^{3}{\left(x \right)} - 2 \sin{\left(x \right)}\right) + \cos^{2}{\left(x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\left(2 \sin^{3}{\left(x \right)} - 2 \sin{\left(x \right)}\right) + \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$\left(1 - 2 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)} = 0$$
$$2 \sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} + 1 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Tenemos la ecuación:
$$2 w^{3} - w^{2} - 2 w + 1 = 0$$
cambiamos
$$\left(- 2 w + \left(\left(- w^{2} + \left(2 w^{3} - 2\right)\right) + 1\right)\right) + 2 = 0$$
o
$$\left(- 2 w + \left(\left(- w^{2} + \left(2 w^{3} - 2 \cdot 1^{3}\right)\right) + 1^{2}\right)\right) + 2 = 0$$
$$- 2 \left(w - 1\right) + \left(- (w^{2} - 1^{2}) + 2 \left(w^{3} - 1^{3}\right)\right) = 0$$
$$- 2 \left(w - 1\right) + \left(- (w - 1) \left(w + 1\right) + 2 \left(w - 1\right) \left(\left(w^{2} + w\right) + 1^{2}\right)\right) = 0$$
Saquemos el factor común -1 + w fuera de paréntesis
obtendremos:
$$\left(w - 1\right) \left(\left(- (w + 1) + 2 \left(\left(w^{2} + w\right) + 1^{2}\right)\right) - 2\right) = 0$$
o
$$\left(w - 1\right) \left(2 w^{2} + w - 1\right) = 0$$
entonces:
$$w_{1} = 1$$
y además
obtenemos la ecuación
$$2 w^{2} + w - 1 = 0$$
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:
$$w_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{3} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 2$$
$$b = 1$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c = 

(1)^2 - 4 * (2) * (-1) = 9

Como D > 0 la ecuación tiene dos raíces.
w2 = (-b + sqrt(D)) / (2*a)

w3 = (-b - sqrt(D)) / (2*a)

o
$$w_{2} = \frac{1}{2}$$
$$w_{3} = -1$$
Entonces la respuesta definitiva es para 1 - sin(x)^2 - 2*sin(x) + 2*sin(x)^3 = 0:
$$w_{1} = 1$$
$$w_{2} = \frac{1}{2}$$
$$w_{3} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{6}$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(w_{3} \right)}$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{3} = 2 \pi n - \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
$$x_{5} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{5} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{5} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{6} = 2 \pi n - \operatorname{asin}{\left(w_{3} \right)} + \pi$$
$$x_{6} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{6} = 2 \pi n + \frac{3 \pi}{2}$$
Gráfica
Respuesta rápida [src]
     -pi 
x1 = ----
      2  
$$x_{1} = - \frac{\pi}{2}$$
     pi
x2 = --
     6 
$$x_{2} = \frac{\pi}{6}$$
     pi
x3 = --
     2 
$$x_{3} = \frac{\pi}{2}$$
     5*pi
x4 = ----
      6  
$$x_{4} = \frac{5 \pi}{6}$$
x4 = 5*pi/6
Suma y producto de raíces [src]
suma
  pi   pi   pi   5*pi
- -- + -- + -- + ----
  2    6    2     6  
$$\left(\left(- \frac{\pi}{2} + \frac{\pi}{6}\right) + \frac{\pi}{2}\right) + \frac{5 \pi}{6}$$
=
pi
$$\pi$$
producto
-pi  pi pi 5*pi
----*--*--*----
 2   6  2   6  
$$\frac{5 \pi}{6} \frac{\pi}{2} \cdot - \frac{\pi}{2} \frac{\pi}{6}$$
=
     4
-5*pi 
------
 144  
$$- \frac{5 \pi^{4}}{144}$$
-5*pi^4/144
Respuesta numérica [src]
x1 = 82.2050077689329
x2 = 4.71238880691387
x3 = 44.5058959258554
x4 = -23.5619449976123
x5 = 21.4675497995303
x6 = 73.8274274747537
x7 = -26.7035375499817
x8 = -49.7418836818384
x9 = -1.57079642667187
x10 = 61.2610566319796
x11 = -32.9867226238685
x12 = 45.5530933503559
x13 = 51.8362788859895
x14 = -47.6474885794452
x15 = 71.733032256967
x16 = 70.6858347691939
x17 = -36.1283154275804
x18 = -83.2522054927418
x19 = 92.6769831096667
x20 = 20.4203521585441
x21 = 38.2227106186758
x22 = 6.80678408277789
x23 = -86.3937979167537
x24 = 88.4881930761125
x25 = 34.0339204138894
x26 = -42.4115008014227
x27 = 67.5442422323769
x28 = -20.4203520689893
x29 = 64.4026493153646
x30 = -16.2315620435473
x31 = -17.2787592932965
x32 = -93.7241808320955
x33 = 17.278759548867
x34 = -148.178453494319
x35 = -32.9867229252256
x36 = 75.9218224617533
x37 = -37.1755130674792
x38 = -64.4026492206202
x39 = -70.685834660727
x40 = -60.2138591938044
x41 = 86.3937978900306
x42 = -89.5353907420184
x43 = 29.8451303165131
x44 = -43.4586983746588
x45 = -87.4409955249159
x46 = -91.6297857297023
x47 = 95.8185760414926
x48 = 50.789081233035
x49 = 10.9955744307638
x50 = -22.5147473507269
x51 = 27.7507351067098
x52 = 7.85398192170864
x53 = -95.8185758682353
x54 = -12.0427718387609
x55 = -1205.84798020288
x56 = -56.025068989018
x57 = 78.0162175641465
x58 = 80.1106131512283
x59 = 7.85398172963525
x60 = -51.8362786903221
x61 = 1.57079629276184
x62 = 94.7713783832921
x63 = -9.94837673636768
x64 = -100.007366139275
x65 = 14.1371670730688
x66 = -81.1578102177363
x67 = 48.6946859579429
x68 = -45.553093584514
x69 = -58.1194640015777
x70 = -73.8274272808552
x71 = 40.317105721069
x72 = -97.9129710368819
x73 = 45.553093243161
x74 = 42.4115005752109
x75 = -80.1106125856939
x76 = 42.4115007315081
x77 = -53.9306738866248
x78 = 26.7035377309766
x79 = 31.9395253114962
x80 = -3.66519142918809
x81 = 58.1194642502768
x82 = 0.523598775598299
x83 = -67.5442421530539
x84 = 23.561945082026
x85 = 84.2994028713261
x86 = 65.4498469497874
x87 = -66.497044500984
x88 = -39.2699083433452
x89 = 89.5353903604948
x90 = -76.969019659253
x91 = 182.735972683806
x92 = -29.8451301001406
x93 = -7.85398150123285
x94 = 54.977871506785
x95 = 98.9601686002212
x96 = 39.2699073274867
x97 = -5.75958653158129
x98 = -14.1371668422459
x98 = -14.1371668422459