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5*sin^2x+21*sinx+4=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2                       
5*sin (x) + 21*sin(x) + 4 = 0
(5sin⁡2(x)+21sin⁡(x))+4=0\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(5sin⁡2(x)+21sin⁡(x))+4=0\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 0
cambiamos
5sin⁡2(x)+21sin⁡(x)+4=05 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)} + 4 = 0
(5sin⁡2(x)+21sin⁡(x))+4=0\left(5 \sin^{2}{\left(x \right)} + 21 \sin{\left(x \right)}\right) + 4 = 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=5a = 5
b=21b = 21
c=4c = 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
w1=−15w_{1} = - \frac{1}{5}
w2=−4w_{2} = -4
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⁡(−15)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{5} \right)}
x1=2πn−asin⁡(15)x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(−4)x_{2} = 2 \pi n + \operatorname{asin}{\left(-4 \right)}
x2=2πn−asin⁡(4)x_{2} = 2 \pi n - \operatorname{asin}{\left(4 \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(−15)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{5} \right)} + \pi
x3=2πn+asin⁡(15)+πx_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(−4)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(-4 \right)}
x4=2πn+π+asin⁡(4)x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(4 \right)}
Gráfica
0-80-60-40-2020406080-100100-5050
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = pi + asin(1/5)
x1=asin⁡(15)+πx_{1} = \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi 
x2 = -asin(1/5)
x2=−asin⁡(15)x_{2} = - \operatorname{asin}{\left(\frac{1}{5} \right)} 
x3 = pi + I*im(asin(4)) + re(asin(4))
x3=re⁡(asin⁡(4))+π+iim⁡(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))
x4=−re⁡(asin⁡(4))−iim⁡(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))
((−asin⁡(15)+(asin⁡(15)+π))+(re⁡(asin⁡(4))+π+iim⁡(asin⁡(4))))+(−re⁡(asin⁡(4))−iim⁡(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
2π2 \pi 
producto
(pi + asin(1/5))*(-asin(1/5))*(pi + I*im(asin(4)) + re(asin(4)))*(-re(asin(4)) - I*im(asin(4)))
(asin⁡(15)+π)(−asin⁡(15))(re⁡(asin⁡(4))+π+iim⁡(asin⁡(4)))(−re⁡(asin⁡(4))−iim⁡(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)
(re⁡(asin⁡(4))+iim⁡(asin⁡(4)))(asin⁡(15)+π)(re⁡(asin⁡(4))+π+iim⁡(asin⁡(4)))asin⁡(15)\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
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