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3*sin^2(x)-11*sin(x)-9=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2                       
3*sin (x) - 11*sin(x) - 9 = 0
(3sin⁡2(x)−11sin⁡(x))−9=0\left(3 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) - 9 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(3sin⁡2(x)−11sin⁡(x))−9=0\left(3 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) - 9 = 0
cambiamos
3sin⁡2(x)−11sin⁡(x)−9=03 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} - 9 = 0
(3sin⁡2(x)−11sin⁡(x))−9=0\left(3 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) - 9 = 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=3a = 3
b=−11b = -11
c=−9c = -9
, entonces
D = b^2 - 4 * a * c = 

(-11)^2 - 4 * (3) * (-9) = 229

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

o
w1=116+2296w_{1} = \frac{11}{6} + \frac{\sqrt{229}}{6}
w2=116−2296w_{2} = \frac{11}{6} - \frac{\sqrt{229}}{6}
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⁡(116+2296)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}
x1=2πn+asin⁡(116+2296)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(116−2296)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)}
x2=2πn+asin⁡(116−2296)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn+π−asin⁡(116+2296)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}
x3=2πn+π−asin⁡(116+2296)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(116−2296)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)} + \pi
x4=2πn−asin⁡(116−2296)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)} + \pi
Gráfica
0-80-60-40-2020406080-100100-2525
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
         /       _____\       /       _____\          /    /       _____\\       /    /       _____\\       /    /       _____\\     /    /       _____\\
         |11   \/ 229 |       |11   \/ 229 |          |    |11   \/ 229 ||       |    |11   \/ 229 ||       |    |11   \/ 229 ||     |    |11   \/ 229 ||
pi - asin|-- - -------| + asin|-- - -------| + pi - re|asin|-- + -------|| - I*im|asin|-- + -------|| + I*im|asin|-- + -------|| + re|asin|-- + -------||
         \6       6   /       \6       6   /          \    \6       6   //       \    \6       6   //       \    \6       6   //     \    \6       6   //
(re⁡(asin⁡(116+2296))+iim⁡(asin⁡(116+2296)))+((asin⁡(116−2296)+(π−asin⁡(116−2296)))+(−re⁡(asin⁡(116+2296))+π−iim⁡(asin⁡(116+2296))))\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)}\right) + \left(\left(\operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)} + \left(\pi - \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)}\right)\right) 
=
2*pi
2π2 \pi 
producto
/         /       _____\\     /       _____\ /       /    /       _____\\       /    /       _____\\\ /    /    /       _____\\     /    /       _____\\\
|         |11   \/ 229 ||     |11   \/ 229 | |       |    |11   \/ 229 ||       |    |11   \/ 229 ||| |    |    |11   \/ 229 ||     |    |11   \/ 229 |||
|pi - asin|-- - -------||*asin|-- - -------|*|pi - re|asin|-- + -------|| - I*im|asin|-- + -------|||*|I*im|asin|-- + -------|| + re|asin|-- + -------|||
\         \6       6   //     \6       6   / \       \    \6       6   //       \    \6       6   /// \    \    \6       6   //     \    \6       6   ///
(π−asin⁡(116−2296))asin⁡(116−2296)(−re⁡(asin⁡(116+2296))+π−iim⁡(asin⁡(116+2296)))(re⁡(asin⁡(116+2296))+iim⁡(asin⁡(116+2296)))\left(\pi - \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)}\right) \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)}\right) 
=
 /         /       _____\\ /    /    /       _____\\     /    /       _____\\\ /          /    /       _____\\     /    /       _____\\\     /       _____\
 |         |11   \/ 229 || |    |    |11   \/ 229 ||     |    |11   \/ 229 ||| |          |    |11   \/ 229 ||     |    |11   \/ 229 |||     |11   \/ 229 |
-|pi - asin|-- - -------||*|I*im|asin|-- + -------|| + re|asin|-- + -------|||*|-pi + I*im|asin|-- + -------|| + re|asin|-- + -------|||*asin|-- - -------|
 \         \6       6   // \    \    \6       6   //     \    \6       6   /// \          \    \6       6   //     \    \6       6   ///     \6       6   /
−(π−asin⁡(116−2296))(re⁡(asin⁡(116+2296))+iim⁡(asin⁡(116+2296)))(−π+re⁡(asin⁡(116+2296))+iim⁡(asin⁡(116+2296)))asin⁡(116−2296)- \left(\pi - \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)}\right) \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)} 
-(pi - asin(11/6 - sqrt(229)/6))*(i*im(asin(11/6 + sqrt(229)/6)) + re(asin(11/6 + sqrt(229)/6)))*(-pi + i*im(asin(11/6 + sqrt(229)/6)) + re(asin(11/6 + sqrt(229)/6)))*asin(11/6 - sqrt(229)/6)
Respuesta rápida [src] 
              /       _____\
              |11   \/ 229 |
x1 = pi - asin|-- - -------|
              \6       6   /
x1=π−asin⁡(116−2296)x_{1} = \pi - \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)} 
         /       _____\
         |11   \/ 229 |
x2 = asin|-- - -------|
         \6       6   /
x2=asin⁡(116−2296)x_{2} = \operatorname{asin}{\left(\frac{11}{6} - \frac{\sqrt{229}}{6} \right)} 
            /    /       _____\\       /    /       _____\\
            |    |11   \/ 229 ||       |    |11   \/ 229 ||
x3 = pi - re|asin|-- + -------|| - I*im|asin|-- + -------||
            \    \6       6   //       \    \6       6   //
x3=−re⁡(asin⁡(116+2296))+π−iim⁡(asin⁡(116+2296))x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} 
         /    /       _____\\     /    /       _____\\
         |    |11   \/ 229 ||     |    |11   \/ 229 ||
x4 = I*im|asin|-- + -------|| + re|asin|-- + -------||
         \    \6       6   //     \    \6       6   //
x4=re⁡(asin⁡(116+2296))+iim⁡(asin⁡(116+2296))x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{11}{6} + \frac{\sqrt{229}}{6} \right)}\right)} 
x4 = re(asin(11/6 + sqrt(229)/6)) + i*im(asin(11/6 + sqrt(229)/6))
Respuesta numérica [src] 
x1 = 54.1668951500192
x2 = -44.7421171892498
x3 = -38.4589318820702
x4 = 292.927936822843
x5 = -84.0631816079317
x6 = 55.7888477256236
x7 = -76.1580437251477
x8 = -46.3640697648542
x9 = 73.0164510715579
x10 = -51.0253024964294
x11 = 5.52336526818691
x12 = -90.3463669151113
x13 = 62.0720330328032
x14 = 36.9392918040848
x15 = -82.4412290323273
x16 = -13.3261906533518
x17 = -19.6093759605314
x18 = 79.2996363787375
x19 = -58.9304403792134
x20 = -71.4968109935726
x21 = -63.5916731107885
x22 = 18.0897358825461
x23 = 74.6384036471624
x24 = 10.1845979997621
x25 = 104.432377607456
x26 = 24.3729211897257
x27 = 41.60052453566
x28 = 85.5828216859171
x29 = 29.0341539213008
x30 = -14.9481432289563
x31 = -2.38177261459712
x32 = -21.2313285361359
x33 = 22.7509686141212
x34 = -32.1757465748906
x35 = -101.290784953866
x36 = -132.706711489764
x37 = 35.3173392284804
x38 = -33.797699150495
x39 = 47.8837098428396
x40 = -27.5145138433155
x41 = 60.4500804571988
x42 = -95.0075996466865
x43 = -69.8748584179681
x44 = 93.4879595687011
x45 = -52.6472550720338
x46 = -88.7244143395069
x47 = 11.8065505753665
x48 = -57.308487803609
x49 = 87.2047742615215
x50 = -0.759820038992677
x51 = 23245.4038639499
x52 = -40.0808844576746
x53 = 68.3552183399828
x54 = -1445.8924406903
x55 = 43.2224771112644
x56 = 66.7332657643783
x57 = 3.90141269258247
x58 = -8.6649579217767
x59 = 91.8660069930967
x60 = -25.892561267711
x61 = -77.7799963007522
x62 = 3145.49406628238
x63 = 99.7711448758807
x64 = -96.6295522222909
x65 = 16.4677833069416
x66 = 112.33751549024
x67 = -7.04300534617226
x68 = 49.505662418444
x69 = -65.213625686393
x70 = 98.1491923002763
x71 = 80.9215889543419
x72 = 30.6561064969053
x72 = 30.6561064969053
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