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Expresiones semejantes
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Expresiones con funciones
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3*sin(x)^(2)-5*sin(x)-2=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 3*sin (x) - 5*sin(x) - 2 = 0
$$\left(3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
cambiamos
$$3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2 = 0$$
$$\left(3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
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
$$a = 3$$
$$b = -5$$
$$c = -2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = 2$$
$$w_{2} = - \frac{1}{3}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{3} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
$$\left(3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
cambiamos
$$3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2 = 0$$
$$\left(3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
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
$$a = 3$$
$$b = -5$$
$$c = -2$$
, entonces
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (3) * (-2) = 49
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = 2$$
$$w_{2} = - \frac{1}{3}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{3} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
Suma y producto de raíces
[src]
suma
pi + asin(1/3) - asin(1/3) + pi - re(asin(2)) - I*im(asin(2)) + I*im(asin(2)) + re(asin(2))
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) + \left(\left(- \operatorname{asin}{\left(\frac{1}{3} \right)} + \left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)\right)$$
=
2*pi
$$2 \pi$$
producto
(pi + asin(1/3))*(-asin(1/3))*(pi - re(asin(2)) - I*im(asin(2)))*(I*im(asin(2)) + re(asin(2)))
$$\left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right) \left(- \operatorname{asin}{\left(\frac{1}{3} \right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)$$
=
(pi + asin(1/3))*(I*im(asin(2)) + re(asin(2)))*(-pi + I*im(asin(2)) + re(asin(2)))*asin(1/3)
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \operatorname{asin}{\left(\frac{1}{3} \right)}$$
(pi + asin(1/3))*(i*im(asin(2)) + re(asin(2)))*(-pi + i*im(asin(2)) + re(asin(2)))*asin(1/3)
Respuesta rápida
[src]
x1 = pi + asin(1/3)
$$x_{1} = \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
x2 = -asin(1/3)
$$x_{2} = - \operatorname{asin}{\left(\frac{1}{3} \right)}$$
x3 = pi - re(asin(2)) - I*im(asin(2))
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}$$
x4 = I*im(asin(2)) + re(asin(2))
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}$$
x4 = re(asin(2)) + i*im(asin(2))
Respuesta numérica
[src]
x1 = 43.642460240803
x2 = 4335.05802504446
x3 = 66.3132826348398
x4 = -46.7840528943928
x5 = -9.08494105131526
x6 = -44.3221340597112
x7 = -0.339836909454122
x8 = 81.3415720838805
x9 = 97.7292091707377
x10 = 68.7752014695213
x11 = 18.5097190120846
x12 = -6117020.52838282
x13 = 100.191128005419
x14 = -69.4548752884296
x15 = -19.1893928309929
x16 = -56.8885046740704
x17 = 24.7929043192642
x18 = 31.0760896264438
x19 = 3.48142956304392
x20 = 28.6141707917623
x21 = 16.0478001774031
x22 = 85.1628385563785
x23 = 5.94334839772546
x24 = -53.0672382015724
x25 = -15.3681263584948
x26 = 62.4920161623417
x27 = -25.4725781381725
x28 = 91.4460238635581
x29 = -40.5008675872132
x30 = -75.7380605956092
x31 = -82.0212459027887
x32 = -34.2176822800336
x33 = 60.0300973276602
x34 = -78.1999794302907
x35 = 47.463726713301
x36 = 37.3592749336234
x37 = 53.7469120204806
x38 = -97.0495353518295
x39 = 78.879653249199
x40 = 75.0583867767009
x41 = 2512.93428596238
x42 = -1194.14504527358
x43 = -6.62302221663371
x44 = 49.9256455479826
x45 = 34.8973560989418
x46 = 41.1805414061214
x47 = -94.5876165171479
x48 = 12.2265337049051
x49 = -12.9062075238133
x50 = 87.6247573910601
x51 = -27.934496972854
x52 = -65.6336088159315
x53 = 56.2088308551622
x54 = 175.589351691574
x55 = -71.9167941231111
x56 = 72.5964679420194
x57 = -63.17168998125
x58 = 22.3309854845827
x59 = -166.164573730805
x60 = -38.0389487525316
x61 = -2.80175574413567
x62 = -21.6513116656744
x63 = -84.4831647374703
x64 = -103.332720659009
x65 = -50.6053193668908
x66 = -31.7557634453521
x67 = 93.9079426982397
x68 = -88.3044312099683
x69 = 9.7646148702235
x70 = -59.350423508752
x71 = -90.7663500446499
x71 = -90.7663500446499
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