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- Coseno cos
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- sin(x)=1,5
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cos(2*x)+3*sin(x)-2=0 la ecuación
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Solución
Ha introducido
[src]
cos(2*x) + 3*sin(x) - 2 = 0
$$\left(3 \sin{\left(x \right)} + \cos{\left(2 x \right)}\right) - 2 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(3 \sin{\left(x \right)} + \cos{\left(2 x \right)}\right) - 2 = 0$$
cambiamos
$$3 \sin{\left(x \right)} + \cos{\left(2 x \right)} - 2 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} - 1 = 0$$
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
$$a = -2$$
$$b = 3$$
$$c = -1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{1}{2}$$
$$w_{2} = 1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
$$\left(3 \sin{\left(x \right)} + \cos{\left(2 x \right)}\right) - 2 = 0$$
cambiamos
$$3 \sin{\left(x \right)} + \cos{\left(2 x \right)} - 2 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} - 1 = 0$$
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
$$a = -2$$
$$b = 3$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(3)^2 - 4 * (-2) * (-1) = 1
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{1}{2}$$
$$w_{2} = 1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
Respuesta numérica
[src]
x1 = 20.4203521563718
x2 = -3.66519142918809
x3 = 38.2227106186758
x4 = -74.8746249105567
x5 = 88.4881930761125
x6 = 15.1843644923507
x7 = -111.526539747198
x8 = -68.5914396033772
x9 = 96.8657734856853
x10 = -9.94837673636768
x11 = -56.025068989018
x12 = -24.60914245312
x13 = -16.2315620435473
x14 = 39.2699080534577
x15 = 58.1194642838711
x16 = 25.6563400043166
x17 = -86.3937978734206
x18 = -97.9129710368819
x19 = -53.9306738866248
x20 = -23.5619450004191
x21 = -100.007366139275
x22 = 26.7035375608868
x23 = -79.0634151153431
x24 = 90.5825881785057
x25 = -61.2610565227993
x26 = -81.1578102177363
x27 = -4.71238891024587
x28 = -35.081117965086
x29 = -28.7979326579064
x30 = -91.6297857297023
x31 = 34.0339204138894
x32 = 70.6858346623125
x33 = 89.5353905764122
x34 = 76.9690200112793
x35 = 69.6386371545737
x36 = 2.61799387799149
x37 = 45.5530934927802
x38 = 139.800872765188
x39 = -85.3466004225227
x40 = 14.1371670807075
x41 = -29.8451300991955
x42 = -36.128315425571
x43 = -12.0427718387609
x44 = 82.2050077689329
x45 = -47.6474885794452
x46 = 57.0722665402146
x47 = -48.694685953454
x48 = -37.1755130674792
x49 = -54.9778712302784
x50 = 75.9218224617533
x51 = -49.7418836818384
x52 = 64.4026493137203
x53 = -80.1106125842073
x54 = 44.5058959258554
x55 = 27.7507351067098
x56 = 13.0899693899575
x57 = -93.7241808320955
x58 = -72.7802298081635
x59 = -67.5442421566185
x60 = -30.8923277602996
x61 = -66.497044500984
x62 = 94.7713783832921
x63 = -43.4586983746588
x64 = 52.8834763354282
x65 = 63.3554518473942
x66 = 19.3731546971371
x67 = 21.4675497995303
x68 = 6.80678408277789
x69 = 50.789081233035
x70 = -87.4409955249159
x71 = -10.9955739548254
x72 = 46.6002910282486
x73 = 78.0162175641465
x74 = 51.8362788893194
x75 = 7.85398173225698
x76 = 0.523598775598299
x77 = 95.8185760456344
x78 = 31.9395253114962
x79 = -5.75958653158129
x80 = 101.054563690472
x81 = -48.6946859020565
x82 = -42.4115007405972
x83 = 83.2522050326052
x84 = 65.4498469497874
x85 = 1.57079638342343
x86 = 32.9867230481337
x87 = 8.90117918517108
x88 = -62.3082542961976
x89 = -18.3259571459405
x90 = -41.3643032722656
x91 = -73.8274274914538
x92 = -73.8274272806479
x93 = 84.2994028713261
x94 = -17.278759508473
x95 = -22.5147473507269
x96 = -60.2138591938044
x97 = 40.317105721069
x98 = -92.6769829494681
x99 = 59.1666616426078
x100 = 71.733032256967
x100 = 71.733032256967
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