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6*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 6*sin (x) - 5*sin(x) - 2 = 0
$$\left(6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
cambiamos
$$6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2 = 0$$
$$\left(6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
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
$$a = 6$$
$$b = -5$$
$$c = -2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{5}{12} + \frac{\sqrt{73}}{12}$$
$$w_{2} = \frac{5}{12} - \frac{\sqrt{73}}{12}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)} + \pi$$
$$\left(6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
cambiamos
$$6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2 = 0$$
$$\left(6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 2 = 0$$
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
$$a = 6$$
$$b = -5$$
$$c = -2$$
, entonces
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (6) * (-2) = 73
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{5}{12} + \frac{\sqrt{73}}{12}$$
$$w_{2} = \frac{5}{12} - \frac{\sqrt{73}}{12}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)} + \pi$$
Respuesta rápida
[src]
/ ____\
|5 \/ 73 |
x1 = pi - asin|-- - ------|
\12 12 /
$$x_{1} = \pi - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}$$
/ ____\
|5 \/ 73 |
x2 = asin|-- - ------|
\12 12 /
$$x_{2} = \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}$$
/ / ____\\ / / ____\\
| |5 \/ 73 || | |5 \/ 73 ||
x3 = pi - re|asin|-- + ------|| - I*im|asin|-- + ------||
\ \12 12 // \ \12 12 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}$$
/ / ____\\ / / ____\\
| |5 \/ 73 || | |5 \/ 73 ||
x4 = I*im|asin|-- + ------|| + re|asin|-- + ------||
\ \12 12 // \ \12 12 //
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}$$
x4 = re(asin(5/12 + sqrt(73)/12)) + i*im(asin(5/12 + sqrt(73)/12))
Suma y producto de raíces
[src]
suma
/ ____\ / ____\ / / ____\\ / / ____\\ / / ____\\ / / ____\\
|5 \/ 73 | |5 \/ 73 | | |5 \/ 73 || | |5 \/ 73 || | |5 \/ 73 || | |5 \/ 73 ||
pi - asin|-- - ------| + asin|-- - ------| + pi - re|asin|-- + ------|| - I*im|asin|-- + ------|| + I*im|asin|-- + ------|| + re|asin|-- + ------||
\12 12 / \12 12 / \ \12 12 // \ \12 12 // \ \12 12 // \ \12 12 //
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}\right) + \left(\left(\operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)} + \left(\pi - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}\right)\right)$$
=
2*pi
$$2 \pi$$
producto
/ / ____\\ / ____\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ | |5 \/ 73 || |5 \/ 73 | | | |5 \/ 73 || | |5 \/ 73 ||| | | |5 \/ 73 || | |5 \/ 73 ||| |pi - asin|-- - ------||*asin|-- - ------|*|pi - re|asin|-- + ------|| - I*im|asin|-- + ------|||*|I*im|asin|-- + ------|| + re|asin|-- + ------||| \ \12 12 // \12 12 / \ \ \12 12 // \ \12 12 /// \ \ \12 12 // \ \12 12 ///
$$\left(\pi - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}\right) \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}\right)$$
=
/ / ____\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / ____\ | |5 \/ 73 || | | |5 \/ 73 || | |5 \/ 73 ||| | | |5 \/ 73 || | |5 \/ 73 ||| |5 \/ 73 | -|pi - asin|-- - ------||*|I*im|asin|-- + ------|| + re|asin|-- + ------|||*|-pi + I*im|asin|-- + ------|| + re|asin|-- + ------|||*asin|-- - ------| \ \12 12 // \ \ \12 12 // \ \12 12 /// \ \ \12 12 // \ \12 12 /// \12 12 /
$$- \left(\pi - \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5}{12} + \frac{\sqrt{73}}{12} \right)}\right)}\right) \operatorname{asin}{\left(\frac{5}{12} - \frac{\sqrt{73}}{12} \right)}$$
-(pi - asin(5/12 - sqrt(73)/12))*(i*im(asin(5/12 + sqrt(73)/12)) + re(asin(5/12 + sqrt(73)/12)))*(-pi + i*im(asin(5/12 + sqrt(73)/12)) + re(asin(5/12 + sqrt(73)/12)))*asin(5/12 - sqrt(73)/12)
Respuesta numérica
[src]
x1 = -75.6980284087978
x2 = -2.84178793094698
x3 = 31.1161218132551
x4 = -63.1316577944387
x5 = 68.8152336563326
x6 = -84.5231969242816
x7 = 62.5320483491531
x8 = -40.5408997740245
x9 = -88.264399023157
x10 = 37.3993071204347
x11 = 16.0077679905918
x12 = -71.9568263099224
x13 = -19.1493606441816
x14 = -31.7157312585407
x15 = 81.3816042706918
x16 = -69.4148431016183
x17 = -147.355049996077
x18 = 72.5564357552081
x19 = 97.6891769839264
x20 = -386.116091668902
x21 = -34.2577144668449
x22 = 28.5741386049509
x23 = -65.6736410027429
x24 = 16258.0417870498
x25 = 22.2909532977714
x26 = 66.2732504480285
x27 = 43.6824924276143
x28 = -25.4325459513612
x29 = 78.8396210623876
x30 = -78.240011617102
x31 = -59.3904556955633
x32 = -50.5652871800795
x33 = 3.4413973762326
x34 = 9.72458268341219
x35 = -46.8240850812041
x36 = -90.8063822314612
x37 = -9.12497323812657
x38 = 53.7068798336693
x39 = 41.1405092193101
x40 = 56.2488630419735
x41 = 18.549751198896
x42 = -44.2821018728999
x43 = 93.947974885051
x44 = 75.0984189635122
x45 = -6.58299002982239
x46 = -37.9989165657203
x47 = -94.5475843303366
x48 = 47.4236945264897
x49 = 24.8329365060755
x50 = 5.98338058453678
x51 = 91.4059916767468
x52 = 85.1228063695672
x53 = -107.113954944696
x54 = -0.299804722642808
x55 = -27.9745291596653
x56 = 87.6647895778714
x57 = -15.4081585453062
x58 = 12.2665658917164
x59 = 59.9900651408489
x60 = 49.9656777347939
x61 = 112.79753080659
x62 = 100.231160192231
x63 = 8492.02474737585
x64 = -21.6913438524857
x65 = -81.9812137159774
x65 = -81.9812137159774
Gráfico