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Expresiones idénticas
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Expresiones semejantes
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5*sin^2x+21*sinx+4=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 5*sin (x) + 21*sin(x) + 4 = 0
$$\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 0$$
cambiamos
$$5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)} + 4 = 0$$
$$\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 0$$
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
$$a = 5$$
$$b = 21$$
$$c = 4$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{5}$$
$$w_{2} = -4$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{5} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-4 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(4 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{5} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(-4 \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(4 \right)}$$
$$\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 0$$
cambiamos
$$5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)} + 4 = 0$$
$$\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 0$$
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
$$a = 5$$
$$b = 21$$
$$c = 4$$
, entonces
D = b^2 - 4 * a * c =
(21)^2 - 4 * (5) * (4) = 361
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{5}$$
$$w_{2} = -4$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{5} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-4 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(4 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{5} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(-4 \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(4 \right)}$$
Respuesta rápida
[src]
x1 = pi + asin(1/5)
$$x_{1} = \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
x2 = -asin(1/5)
$$x_{2} = - \operatorname{asin}{\left(\frac{1}{5} \right)}$$
x3 = pi + I*im(asin(4)) + re(asin(4))
$$x_{3} = \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}$$
x4 = -re(asin(4)) - I*im(asin(4))
$$x_{4} = - \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}$$
x4 = -re(asin(4)) - i*im(asin(4))
Suma y producto de raíces
[src]
suma
pi + asin(1/5) - asin(1/5) + pi + I*im(asin(4)) + re(asin(4)) + -re(asin(4)) - I*im(asin(4))
$$\left(\left(- \operatorname{asin}{\left(\frac{1}{5} \right)} + \left(\operatorname{asin}{\left(\frac{1}{5} \right)} + \pi\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right)$$
=
2*pi
$$2 \pi$$
producto
(pi + asin(1/5))*(-asin(1/5))*(pi + I*im(asin(4)) + re(asin(4)))*(-re(asin(4)) - I*im(asin(4)))
$$\left(\operatorname{asin}{\left(\frac{1}{5} \right)} + \pi\right) \left(- \operatorname{asin}{\left(\frac{1}{5} \right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right)$$
=
(pi + asin(1/5))*(I*im(asin(4)) + re(asin(4)))*(pi + I*im(asin(4)) + re(asin(4)))*asin(1/5)
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) \left(\operatorname{asin}{\left(\frac{1}{5} \right)} + \pi\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) \operatorname{asin}{\left(\frac{1}{5} \right)}$$
(pi + asin(1/5))*(i*im(asin(4)) + re(asin(4)))*(pi + i*im(asin(4)) + re(asin(4)))*asin(1/5)
Respuesta numérica
[src]
x1 = -75.5995816069454
x2 = 100.329606994083
x3 = -88.1659522213045
x4 = 66.174803646176
x5 = -34.3561612686974
x6 = 15.9093211887393
x7 = 75.1968657653647
x8 = 97.5907301820739
x9 = -9.22342003997905
x10 = 6.08182738638926
x11 = -21.7897906543382
x12 = -78.3384584189545
x13 = -72.0552731117749
x14 = -50.466840378227
x15 = -94.4491375284841
x16 = -40.639346575877
x17 = -53.2057171902362
x18 = 348.918142469257
x19 = 9.62613588155971
x20 = 59.8916183389964
x21 = -100.732322835664
x22 = -12.7677285351495
x23 = 62.6304951510055
x24 = -44.1836550710474
x25 = 28.4756918030985
x26 = 81.4800510725443
x27 = 3.34295057438012
x28 = -15.5066053471586
x29 = 85.0243595677148
x30 = -358.342920430027
x31 = 12.3650126935688
x32 = 72.4579889533556
x33 = 31.2145686151076
x34 = -19.0509138423291
x35 = 47.3252477246372
x36 = -63.0332109925862
x37 = -37.9004697638678
x38 = -56.7500256854066
x39 = 68.9136804581851
x40 = 91.3075448748943
x41 = -84.6216437261341
x42 = 78.7411742605352
x43 = -65.7720878045953
x44 = -59.4889024974157
x45 = -6.48454322796992
x46 = -25.3340991495087
x47 = -13788.2487986848
x48 = 37.4977539222872
x49 = -2.94023473279946
x50 = 41.0420624174576
x51 = 53.6084330318168
x52 = -0.201357920790331
x53 = 43.7809392294668
x54 = -90.9048290333137
x55 = 18.6481980007484
x56 = -46.9225318830566
x57 = -97.1880143404933
x58 = 24.931383307928
x59 = 22.1925064959189
x60 = 94.0464216869035
x61 = -69.3163962997658
x62 = -31.6172844566883
x63 = 56.3473098438259
x64 = 87.7632363797239
x65 = -81.882766914125
x66 = 34.7588771102781
x67 = -163.56417590746
x68 = -28.0729759615178
x69 = 50.0641245366464
x69 = 50.0641245366464