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Expresiones con funciones
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6*sin(x)^2-sin(x)-1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 6*sin (x) - sin(x) - 1 = 0
$$\left(6 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(6 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$6 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} - 1 = 0$$
$$\left(6 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 1 = 0$$
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
$$a = 6$$
$$b = -1$$
$$c = -1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{1}{2}$$
$$w_{2} = - \frac{1}{3}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{3} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
$$\left(6 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$6 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} - 1 = 0$$
$$\left(6 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 1 = 0$$
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
$$a = 6$$
$$b = -1$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (6) * (-1) = 25
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{1}{2}$$
$$w_{2} = - \frac{1}{3}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{3} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
Suma y producto de raíces
[src]
suma
pi 5*pi -- + ---- + pi + asin(1/3) - asin(1/3) 6 6
$$- \operatorname{asin}{\left(\frac{1}{3} \right)} + \left(\left(\frac{\pi}{6} + \frac{5 \pi}{6}\right) + \left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right)\right)$$
=
2*pi
$$2 \pi$$
producto
pi 5*pi --*----*(pi + asin(1/3))*(-asin(1/3)) 6 6
$$\frac{\pi}{6} \frac{5 \pi}{6} \left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right) \left(- \operatorname{asin}{\left(\frac{1}{3} \right)}\right)$$
=
2
-5*pi *(pi + asin(1/3))*asin(1/3)
---------------------------------
36
$$- \frac{5 \pi^{2} \left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right) \operatorname{asin}{\left(\frac{1}{3} \right)}}{36}$$
-5*pi^2*(pi + asin(1/3))*asin(1/3)/36
Respuesta rápida
[src]
pi
x1 = --
6
$$x_{1} = \frac{\pi}{6}$$
5*pi
x2 = ----
6
$$x_{2} = \frac{5 \pi}{6}$$
x3 = pi + asin(1/3)
$$x_{3} = \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
x4 = -asin(1/3)
$$x_{4} = - \operatorname{asin}{\left(\frac{1}{3} \right)}$$
x4 = -asin(1/3)
Respuesta numérica
[src]
x1 = 9.7646148702235
x2 = -3.66519142918809
x3 = 68.7752014695213
x4 = 38.2227106186758
x5 = 88.4881930761125
x6 = -31.7557634453521
x7 = -25.4725781381725
x8 = -68.5914396033772
x9 = 96.8657734856853
x10 = -9.94837673636768
x11 = -84.4831647374703
x12 = -56.025068989018
x13 = -24.60914245312
x14 = 47.463726713301
x15 = -16.2315620435473
x16 = 62.4920161623417
x17 = 25.6563400043166
x18 = -103.332720659009
x19 = -97.9129710368819
x20 = -53.9306738866248
x21 = -100.007366139275
x22 = -82.0212459027887
x23 = 16.0478001774031
x24 = 90.5825881785057
x25 = -81.1578102177363
x26 = -90.7663500446499
x27 = -21.6513116656744
x28 = 56.2088308551622
x29 = -91.6297857297023
x30 = 487.286698215872
x31 = 34.0339204138894
x32 = 93.9079426982397
x33 = 69.6386371545737
x34 = 24.7929043192642
x35 = 2.61799387799149
x36 = -85.3466004225227
x37 = 49.9256455479826
x38 = 66.3132826348398
x39 = 22.3309854845827
x40 = -12.0427718387609
x41 = 82.2050077689329
x42 = -47.6474885794452
x43 = 97.7292091707377
x44 = 12.2265337049051
x45 = 75.9218224617533
x46 = -27.934496972854
x47 = -49.7418836818384
x48 = 44.5058959258554
x49 = 27.7507351067098
x50 = -93.7241808320955
x51 = -38.0389487525316
x52 = -34.2176822800336
x53 = -65.6336088159315
x54 = 5.94334839772546
x55 = 52.8834763354282
x56 = 63.3554518473942
x57 = 19.3731546971371
x58 = -19.1893928309929
x59 = -71.9167941231111
x60 = 46.6002910282486
x61 = -75.7380605956092
x62 = -78.1999794302907
x63 = -231.954257590046
x64 = 91.4460238635581
x65 = 78.0162175641465
x66 = 0.523598775598299
x67 = -69.4548752884296
x68 = 31.9395253114962
x69 = -46.7840528943928
x70 = -5.75958653158129
x71 = 53.7469120204806
x72 = -40.5008675872132
x73 = -12.9062075238133
x74 = -63.17168998125
x75 = 3.48142956304392
x76 = 18.5097190120846
x77 = 8.90117918517108
x78 = -62.3082542961976
x79 = -18.3259571459405
x80 = -41.3643032722656
x81 = 100.191128005419
x82 = 84.2994028713261
x83 = 60.0300973276602
x84 = -60.2138591938044
x85 = 40.317105721069
x86 = 8299.74795387478
x87 = 71.733032256967
x87 = 71.733032256967
Gráfico