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Expresiones idénticas
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- Sinx menos 2 seno de ²x es igual a 0
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Expresiones semejantes
- Sinx+2sin²x=0
Sinx-2sin²x=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 sin(x) - 2*sin (x) = 0
$$- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} = 0$$
cambiamos
$$\sin{\left(x \right)} + \cos{\left(2 x \right)} - 1 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} = 0$$
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
$$a = -2$$
$$b = 1$$
$$c = 0$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = 0$$
$$w_{2} = \frac{1}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{6}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{5 \pi}{6}$$
$$- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} = 0$$
cambiamos
$$\sin{\left(x \right)} + \cos{\left(2 x \right)} - 1 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} = 0$$
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
$$a = -2$$
$$b = 1$$
$$c = 0$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-2) * (0) = 1
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = 0$$
$$w_{2} = \frac{1}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{6}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{5 \pi}{6}$$
Suma y producto de raíces
[src]
suma
pi 5*pi -- + ---- + pi 6 6
$$\left(\frac{\pi}{6} + \frac{5 \pi}{6}\right) + \pi$$
=
2*pi
$$2 \pi$$
producto
pi 5*pi 0*--*----*pi 6 6
$$\pi \frac{5 \pi}{6} \cdot 0 \frac{\pi}{6}$$
=
0
$$0$$
0
Respuesta rápida
[src]
x1 = 0
$$x_{1} = 0$$
pi
x2 = --
6
$$x_{2} = \frac{\pi}{6}$$
5*pi
x3 = ----
6
$$x_{3} = \frac{5 \pi}{6}$$
x4 = pi
$$x_{4} = \pi$$
x4 = pi
Respuesta numérica
[src]
x1 = 84.2994028713261
x2 = -93.7241808320955
x3 = 62.8318530717959
x4 = -476.99848457005
x5 = 90.5825881785057
x6 = -34.5575191894877
x7 = 40.317105721069
x8 = -43.9822971502571
x9 = 28.2743338823081
x10 = 96.8657734856853
x11 = -68.5914396033772
x12 = 69.6386371545737
x13 = 59.6902604182061
x14 = -40.8407044966673
x15 = 56.5486677646163
x16 = 100.530964914873
x17 = -84.8230016469244
x18 = 25.6563400043166
x19 = 15.707963267949
x20 = -18.3259571459405
x21 = -15.707963267949
x22 = -37.6991118430775
x23 = 25.1327412287183
x24 = -72.2566310325652
x25 = -5.75958653158129
x26 = -97.9129710368819
x27 = -18.8495559215388
x28 = 53.4070751110265
x29 = -3.66519142918809
x30 = 47.1238898038469
x31 = 65.9734457253857
x32 = -24.60914245312
x33 = 128.281700021583
x34 = -49.7418836818384
x35 = 12.5663706143592
x36 = 21.9911485751286
x37 = -25.1327412287183
x38 = -65.9734457253857
x39 = -106.290551446455
x40 = 34.0339204138894
x41 = -69.1150383789755
x42 = -62.3082542961976
x43 = 50.789081233035
x44 = 97.3893722612836
x45 = 2.61799387799149
x46 = -41.3643032722656
x47 = -21.9911485751286
x48 = 3.14159265358979
x49 = 69.1150383789755
x50 = -9.94837673636768
x51 = 78.0162175641465
x52 = -53.4070751110265
x53 = -56.025068989018
x54 = 5931.32692997753
x55 = -12.0427718387609
x56 = -62.8318530717959
x57 = -59.6902604182061
x58 = -74.8746249105567
x59 = 31.9395253114962
x60 = -87.9645943005142
x61 = -28.2743338823081
x62 = 46.6002910282486
x63 = 6.28318530717959
x64 = 0.0
x65 = 87.9645943005142
x66 = 43.9822971502571
x67 = -143.989663289532
x68 = -91.6297857297023
x69 = -47.6474885794452
x70 = -75.398223686155
x71 = 75.9218224617533
x72 = 82.2050077689329
x73 = -53.9306738866248
x74 = -31.4159265358979
x75 = 72.2566310325652
x76 = 19.3731546971371
x77 = 94.2477796076938
x78 = 91.106186954104
x79 = 9.42477796076938
x80 = -78.5398163397448
x81 = -81.6814089933346
x82 = -100.007366139275
x83 = 18.8495559215388
x84 = -85.3466004225227
x85 = 50.2654824574367
x86 = 38.2227106186758
x86 = 38.2227106186758