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sin(x)^(2)-sin(x)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 sin (x) - sin(x) = 0
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
cambiamos
$$\left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
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
$$a = 1$$
$$b = -1$$
$$c = 0$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = 1$$
$$w_{2} = 0$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
cambiamos
$$\left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = 0$$
, 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
$$w_{1} = 1$$
$$w_{2} = 0$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
Respuesta rápida
[src]
x1 = 0
$$x_{1} = 0$$
pi
x2 = --
2
$$x_{2} = \frac{\pi}{2}$$
x3 = pi
$$x_{3} = \pi$$
x3 = pi
Suma y producto de raíces
[src]
suma
pi -- + pi 2
$$\frac{\pi}{2} + \pi$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
producto
pi 0*--*pi 2
$$\pi 0 \frac{\pi}{2}$$
=
0
$$0$$
0
Respuesta numérica
[src]
x1 = 76.969019673036
x2 = -62.8318530717959
x3 = -73.8274272802392
x4 = -54.9778709863297
x5 = -4.7123888305818
x6 = 18.8495559215388
x7 = -92.6769832292373
x8 = -10.9955747331165
x9 = 100.530964914873
x10 = -84.8230016469244
x11 = -34.5575191894877
x12 = 83.2522058001693
x13 = -18.8495559215388
x14 = 9.42477796076938
x15 = -81.6814089933346
x16 = 65.9734457253857
x17 = 39.2699080280542
x18 = -29.8451300972765
x19 = -43.9822971502571
x20 = 120.951318648179
x21 = 78.5398163397448
x22 = -86.3937977915432
x23 = -31.4159265358979
x24 = -54.9778717129156
x25 = 21.9911485751286
x26 = -53.4070751110265
x27 = -23.5619450064001
x28 = -87.9645943005142
x29 = -1564.51314148772
x30 = 51.8362788966528
x31 = 94.2477796076938
x32 = -10.9955739732138
x33 = 39.2699086388565
x34 = 12.5663706143592
x35 = 53.4070751110265
x36 = -48.6946861243056
x37 = 70.6858345286456
x38 = -21.9911485751286
x39 = -15.707963267949
x40 = -69.1150383789755
x41 = -59.6902604182061
x42 = -80.1106125810393
x43 = 28.2743338823081
x44 = -75.398223686155
x45 = -9.42477796076938
x46 = 97.3893722612836
x47 = 1.57079651244662
x48 = 45.553093663481
x49 = 14.1371670985871
x50 = -65.9734457253857
x51 = 47.1238898038469
x52 = 15.707963267949
x53 = 56.5486677646163
x54 = 34.5575191894877
x55 = 25.1327412287183
x56 = 72.2566310325652
x57 = 0.0
x58 = -61.2610569243204
x59 = 89.5353908137952
x60 = 7.85398173796495
x61 = -42.4115006392452
x62 = 43.9822971502571
x63 = -97.3893722612836
x64 = -25.1327412287183
x65 = 64.4026493102586
x66 = 87.9645943005142
x67 = 32.986723044911
x68 = 76.9690200976964
x69 = 58.1194647431527
x70 = 59.6902604182061
x71 = -72.2566310325652
x72 = -37.6991118430775
x73 = -28.2743338823081
x74 = 91.106186954104
x75 = 26.7035373768773
x76 = 83.2522050600807
x77 = 69.1150383789755
x78 = 32.9867223690379
x79 = -4.71238903613963
x80 = -17.2787597741434
x81 = -94.2477796076938
x82 = 37.6991118430775
x83 = -50.2654824574367
x84 = 81.6814089933346
x85 = 3.14159265358979
x86 = 50.2654824574367
x87 = -6.28318530717959
x88 = 62.8318530717959
x89 = -78.5398163397448
x90 = 6.28318530717959
x91 = -36.1283154212439
x92 = 20.42035215177
x93 = -67.5442421642546
x94 = -40.8407044966673
x95 = 95.8185760548644
x95 = 95.8185760548644
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