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Sin^2x-sinx=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2                
sin (x) - sin(x) = 0
sin⁡2(x)−sin⁡(x)=0\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0
Simplificar
Solución detallada
Tenemos la ecuación
sin⁡2(x)−sin⁡(x)=0\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0
cambiamos
(sin⁡(x)−1)sin⁡(x)=0\left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0
sin⁡2(x)−sin⁡(x)=0\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0
Sustituimos
w=sin⁡(x)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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=−1b = -1
c=0c = 0
, entonces
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (1) * (0) = 1

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

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

o
w1=1w_{1} = 1
w2=0w_{2} = 0
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:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(1)x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}
x1=2πn+π2x_{1} = 2 \pi n + \frac{\pi}{2}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(0)x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}
x2=2πnx_{2} = 2 \pi n
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi
x3=2πn+π2x_{3} = 2 \pi n + \frac{\pi}{2}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(0)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi
x4=2πn+πx_{4} = 2 \pi n + \pi
Gráfica
0-80-60-40-2020406080-1001002.5-2.5
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
pi     
-- + pi
2
π2+π\frac{\pi}{2} + \pi 
=
3*pi
----
 2
3π2\frac{3 \pi}{2} 
producto
  pi   
0*--*pi
  2
π0π2\pi 0 \frac{\pi}{2} 
=
0
00 
0
Respuesta rápida [src] 
x1 = 0
x1=0x_{1} = 0 
     pi
x2 = --
     2
x2=π2x_{2} = \frac{\pi}{2} 
x3 = pi
x3=πx_{3} = \pi 
x3 = pi
Respuesta numérica [src] 
x1 = 32.986723044911
x2 = 65.9734457253857
x3 = 69.1150383789755
x4 = 7.85398173796495
x5 = -21.9911485751286
x6 = 21.9911485751286
x7 = 83.2522058001693
x8 = -15.707963267949
x9 = -17.2787597741434
x10 = 9.42477796076938
x11 = 97.3893722612836
x12 = 53.4070751110265
x13 = 58.1194647431527
x14 = 6.28318530717959
x15 = -34.5575191894877
x16 = 76.969019673036
x17 = -4.7123888305818
x18 = 32.9867223690379
x19 = 28.2743338823081
x20 = -94.2477796076938
x21 = 45.553093663481
x22 = -4.71238903613963
x23 = -61.2610569243204
x24 = -92.6769832292373
x25 = 95.8185760548644
x26 = -36.1283154212439
x27 = -25.1327412287183
x28 = -53.4070751110265
x29 = 59.6902604182061
x30 = -67.5442421642546
x31 = 56.5486677646163
x32 = 20.42035215177
x33 = 25.1327412287183
x34 = 64.4026493102586
x35 = 14.1371670985871
x36 = 72.2566310325652
x37 = 26.7035373768773
x38 = -50.2654824574367
x39 = -10.9955747331165
x40 = 78.5398163397448
x41 = -87.9645943005142
x42 = 37.6991118430775
x43 = -6.28318530717959
x44 = -73.8274272802392
x45 = -37.6991118430775
x46 = -43.9822971502571
x47 = 47.1238898038469
x48 = -86.3937977915432
x49 = 83.2522050600807
x50 = -1564.51314148772
x51 = 3.14159265358979
x52 = -72.2566310325652
x53 = -81.6814089933346
x54 = -40.8407044966673
x55 = -65.9734457253857
x56 = -29.8451300972765
x57 = 120.951318648179
x58 = 0.0
x59 = -28.2743338823081
x60 = 91.106186954104
x61 = -48.6946861243056
x62 = -42.4115006392452
x63 = 70.6858345286456
x64 = 43.9822971502571
x65 = 100.530964914873
x66 = -97.3893722612836
x67 = 89.5353908137952
x68 = 81.6814089933346
x69 = -75.398223686155
x70 = -23.5619450064001
x71 = -78.5398163397448
x72 = 51.8362788966528
x73 = 50.2654824574367
x74 = 94.2477796076938
x75 = -84.8230016469244
x76 = -59.6902604182061
x77 = 12.5663706143592
x78 = -69.1150383789755
x79 = 39.2699086388565
x80 = 18.8495559215388
x81 = 34.5575191894877
x82 = -54.9778717129156
x83 = -62.8318530717959
x84 = 15.707963267949
x85 = -18.8495559215388
x86 = 87.9645943005142
x87 = 76.9690200976964
x88 = 1.57079651244662
x89 = -54.9778709863297
x90 = -9.42477796076938
x91 = -80.1106125810393
x92 = 62.8318530717959
x93 = -31.4159265358979
x94 = -10.9955739732138
x95 = 39.2699080280542
x95 = 39.2699080280542
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