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-4*(sin(x))^2+11*sin(x)-3=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 - 4*sin (x) + 11*sin(x) - 3 = 0
$$\left(- 4 \sin^{2}{\left(x \right)} + 11 \sin{\left(x \right)}\right) - 3 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(- 4 \sin^{2}{\left(x \right)} + 11 \sin{\left(x \right)}\right) - 3 = 0$$
cambiamos
$$- 4 \sin^{2}{\left(x \right)} + 11 \sin{\left(x \right)} - 3 = 0$$
$$\left(- 4 \sin^{2}{\left(x \right)} + 11 \sin{\left(x \right)}\right) - 3 = 0$$
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
$$a = -4$$
$$b = 11$$
$$c = -3$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{11}{8} - \frac{\sqrt{73}}{8}$$
$$w_{2} = \frac{\sqrt{73}}{8} + \frac{11}{8}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
$$\left(- 4 \sin^{2}{\left(x \right)} + 11 \sin{\left(x \right)}\right) - 3 = 0$$
cambiamos
$$- 4 \sin^{2}{\left(x \right)} + 11 \sin{\left(x \right)} - 3 = 0$$
$$\left(- 4 \sin^{2}{\left(x \right)} + 11 \sin{\left(x \right)}\right) - 3 = 0$$
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
$$a = -4$$
$$b = 11$$
$$c = -3$$
, entonces
D = b^2 - 4 * a * c =
(11)^2 - 4 * (-4) * (-3) = 73
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{11}{8} - \frac{\sqrt{73}}{8}$$
$$w_{2} = \frac{\sqrt{73}}{8} + \frac{11}{8}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}$$
Respuesta rápida
[src]
/ ____\
|11 \/ 73 |
x1 = pi - asin|-- - ------|
\8 8 /
$$x_{1} = \pi - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}$$
/ ____\
|11 \/ 73 |
x2 = asin|-- - ------|
\8 8 /
$$x_{2} = \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}$$
/ / ____\\ / / ____\\
| |11 \/ 73 || | |11 \/ 73 ||
x3 = pi - re|asin|-- + ------|| - I*im|asin|-- + ------||
\ \8 8 // \ \8 8 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}$$
/ / ____\\ / / ____\\
| |11 \/ 73 || | |11 \/ 73 ||
x4 = I*im|asin|-- + ------|| + re|asin|-- + ------||
\ \8 8 // \ \8 8 //
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}$$
x4 = re(asin(sqrt(73)/8 + 11/8)) + i*im(asin(sqrt(73)/8 + 11/8))
Suma y producto de raíces
[src]
suma
/ ____\ / ____\ / / ____\\ / / ____\\ / / ____\\ / / ____\\
|11 \/ 73 | |11 \/ 73 | | |11 \/ 73 || | |11 \/ 73 || | |11 \/ 73 || | |11 \/ 73 ||
pi - asin|-- - ------| + asin|-- - ------| + pi - re|asin|-- + ------|| - I*im|asin|-- + ------|| + I*im|asin|-- + ------|| + re|asin|-- + ------||
\8 8 / \8 8 / \ \8 8 // \ \8 8 // \ \8 8 // \ \8 8 //
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}\right) + \left(\left(\operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)} + \left(\pi - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}\right)\right)$$
=
2*pi
$$2 \pi$$
producto
/ / ____\\ / ____\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ | |11 \/ 73 || |11 \/ 73 | | | |11 \/ 73 || | |11 \/ 73 ||| | | |11 \/ 73 || | |11 \/ 73 ||| |pi - asin|-- - ------||*asin|-- - ------|*|pi - re|asin|-- + ------|| - I*im|asin|-- + ------|||*|I*im|asin|-- + ------|| + re|asin|-- + ------||| \ \8 8 // \8 8 / \ \ \8 8 // \ \8 8 /// \ \ \8 8 // \ \8 8 ///
$$\left(\pi - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}\right) \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}\right)$$
=
/ / ____\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / ____\ | |11 \/ 73 || | | |11 \/ 73 || | |11 \/ 73 ||| | | |11 \/ 73 || | |11 \/ 73 ||| |11 \/ 73 | -|pi - asin|-- - ------||*|I*im|asin|-- + ------|| + re|asin|-- + ------|||*|-pi + I*im|asin|-- + ------|| + re|asin|-- + ------|||*asin|-- - ------| \ \8 8 // \ \ \8 8 // \ \8 8 /// \ \ \8 8 // \ \8 8 /// \8 8 /
$$- \left(\pi - \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{73}}{8} + \frac{11}{8} \right)}\right)}\right) \operatorname{asin}{\left(\frac{11}{8} - \frac{\sqrt{73}}{8} \right)}$$
-(pi - asin(11/8 - sqrt(73)/8))*(i*im(asin(11/8 + sqrt(73)/8)) + re(asin(11/8 + sqrt(73)/8)))*(-pi + i*im(asin(11/8 + sqrt(73)/8)) + re(asin(11/8 + sqrt(73)/8)))*asin(11/8 - sqrt(73)/8)
Respuesta numérica
[src]
x1 = -97.7014109699468
x2 = 38.0111505517408
x3 = 75.7102623948183
x4 = -112.785296820569
x5 = 65.6614070167224
x6 = 25.4447799373816
x7 = 1074.73672623637
x8 = 78.2277776310816
x9 = -22.3031872837918
x10 = -47.4359285125101
x11 = 9.11273925210613
x12 = -49.9534437487734
x13 = 6.59522401584283
x14 = 27.9622951736449
x15 = -18.5375172128755
x16 = 21.6791098664653
x17 = -93.9357408990305
x18 = -75.0861849774918
x19 = 34.2454804808245
x20 = 94.559818316357
x21 = -16.0200019766122
x22 = -62.5198143631326
x23 = 71.944592323902
x24 = -66.2854844340489
x25 = -78.8518550484081
x26 = -87.652555591851
x27 = 84.5109629382612
x28 = 53.0950364023632
x29 = 2.82955394492655
x30 = -56.236629055953
x31 = 100.843003623537
x32 = -60.0022991268693
x33 = -100.21892620621
x34 = 0.312038708663248
x35 = -5.97114659851634
x36 = 15.3959245592857
x37 = 97.0773335526203
x38 = 59.3782217095428
x39 = -43.6702584415939
x40 = 81.9934477019979
x41 = -72.5686697412285
x42 = 44.2943358589204
x43 = -31.1038878272347
x44 = -37.3870731344143
x45 = 69.4270770876387
x46 = 31.7279652445612
x47 = 63.1438917804591
x48 = -3.45363136225304
x49 = -53.7191138196897
x50 = -12.2543319056959
x51 = 90.7941482454408
x52 = -91.4182256627673
x53 = 50.5775211660999
x54 = -9.73681666943263
x55 = 19.161594630202
x56 = -24.8207025200551
x57 = 1319.78095321638
x58 = -81.3693702846714
x59 = -28.5863725909714
x60 = 88.2766330091775
x61 = -34.869557898151
x62 = -85.1350403555877
x63 = 12.8784093230224
x64 = -68.8029996703122
x65 = 56.8607064732795
x66 = 46.8118510951836
x67 = 40.5286657880041
x68 = -41.1527432053306
x68 = -41.1527432053306