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2sin^2x+sinx=1 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
     2                
2*sin (x) + sin(x) = 1
$$2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} = 1$$
Solución detallada
Tenemos la ecuación
$$2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} = 1$$
cambiamos
$$\sin{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$\left(2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) - 1 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
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.
w1 = (-b + sqrt(D)) / (2*a)

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

o
$$w_{1} = \frac{1}{2}$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
Gráfica
Respuesta rápida [src]
     -pi 
x1 = ----
      2  
$$x_{1} = - \frac{\pi}{2}$$
     pi
x2 = --
     6 
$$x_{2} = \frac{\pi}{6}$$
     5*pi
x3 = ----
      6  
$$x_{3} = \frac{5 \pi}{6}$$
     3*pi
x4 = ----
      2  
$$x_{4} = \frac{3 \pi}{2}$$
x4 = 3*pi/2
Suma y producto de raíces [src]
suma
  pi   pi   5*pi   3*pi
- -- + -- + ---- + ----
  2    6     6      2  
$$\left(\left(- \frac{\pi}{2} + \frac{\pi}{6}\right) + \frac{5 \pi}{6}\right) + \frac{3 \pi}{2}$$
=
2*pi
$$2 \pi$$
producto
-pi  pi 5*pi 3*pi
----*--*----*----
 2   6   6    2  
$$\frac{3 \pi}{2} \frac{5 \pi}{6} \cdot - \frac{\pi}{2} \frac{\pi}{6}$$
=
     4
-5*pi 
------
  48  
$$- \frac{5 \pi^{4}}{48}$$
-5*pi^4/48
Respuesta numérica [src]
x1 = -62.3082542961976
x2 = -53.9306738866248
x3 = -49.7418836818384
x4 = -32.9867230405965
x5 = -100.007366139275
x6 = 10.995574056153
x7 = 23.5619451122289
x8 = -87.4409955249159
x9 = -95.8185758681551
x10 = -85.3466004225227
x11 = 63.3554518473942
x12 = 52.8834763354282
x13 = -47.6474885794452
x14 = 34.0339204138894
x15 = 46.6002910282486
x16 = -41.3643032722656
x17 = 80.1106131458253
x18 = -12.0427718387609
x19 = -93.7241808320955
x20 = 2.61799387799149
x21 = -56.025068989018
x22 = -91.6297857297023
x23 = -9.94837673636768
x24 = 86.393797888715
x25 = 25.6563400043166
x26 = 92.6769830871924
x27 = 82.2050077689329
x28 = -66.497044500984
x29 = 31.9395253114962
x30 = 92.6769826185806
x31 = 78.0162175641465
x32 = -32.98672341235
x33 = -79.0634151153431
x34 = 98.9601683847854
x35 = -45.5530935873709
x36 = 90.5825881785057
x37 = -51.8362786898924
x38 = -76.9690201780717
x39 = 8.90117918517108
x40 = -68.5914396033772
x41 = -43.4586983746588
x42 = 69.6386371545737
x43 = 75.9218224617533
x44 = -7.85398149924071
x45 = -14.1371668400256
x46 = -64.4026491963026
x47 = 42.4115007297604
x48 = 61.2610569380464
x49 = -97.9129710368819
x50 = -76.9690204511548
x51 = -22.5147473507269
x52 = -70.6858344924983
x53 = 88.4881930761125
x54 = -26.7035373476123
x55 = -58.1194639999037
x56 = 48.6946859325274
x57 = 10.9955740992967
x58 = -24.60914245312
x59 = 54.9778712411975
x60 = 96.8657734856853
x61 = 44.5058959258554
x62 = -83.2522055292846
x63 = 29.8451303193672
x64 = 73.8274274783337
x65 = 36.1283159916529
x66 = -76.9690198122422
x67 = 94.7713783832921
x68 = -5.75958653158129
x69 = 17.2787597959772
x70 = -89.5353907455655
x71 = -3.66519142918809
x72 = -20.4203520418601
x73 = 19.3731546971371
x74 = 40.317105721069
x75 = 84.2994028713261
x76 = 27.7507351067098
x77 = -16.2315620435473
x78 = -39.2699083757319
x79 = 4.71238877821279
x80 = -1.57079642893127
x81 = -95.8185760435073
x82 = 71.733032256967
x83 = -35.081117965086
x84 = 0.523598775598299
x85 = 38.2227106186758
x86 = -60.2138591938044
x87 = -18.3259571459405
x88 = 67.5442422659503
x88 = 67.5442422659503