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cos(2*x)+5*sin(x)-3=0

cos(2*x)+5*sin(x)-3=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
cos(2*x) + 5*sin(x) - 3 = 0
$$\left(5 \sin{\left(x \right)} + \cos{\left(2 x \right)}\right) - 3 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(5 \sin{\left(x \right)} + \cos{\left(2 x \right)}\right) - 3 = 0$$
cambiamos
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} - 3 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} - 2 = 0$$
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
$$a = -2$$
$$b = 5$$
$$c = -2$$
, entonces
D = b^2 - 4 * a * c = 

(5)^2 - 4 * (-2) * (-2) = 9

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

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

o
$$w_{1} = \frac{1}{2}$$
$$w_{2} = 2$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
Gráfica
Suma y producto de raíces [src]
suma
pi   5*pi   pi        /      ___\   pi        /      ___\
-- + ---- + -- - I*log\2 - \/ 3 / + -- - I*log\2 + \/ 3 /
6     6     2                       2                    
$$\left(\frac{\pi}{2} - i \log{\left(\sqrt{3} + 2 \right)}\right) + \left(\left(\frac{\pi}{6} + \frac{5 \pi}{6}\right) + \left(\frac{\pi}{2} - i \log{\left(2 - \sqrt{3} \right)}\right)\right)$$
=
            /      ___\        /      ___\
2*pi - I*log\2 + \/ 3 / - I*log\2 - \/ 3 /
$$2 \pi - i \log{\left(\sqrt{3} + 2 \right)} - i \log{\left(2 - \sqrt{3} \right)}$$
producto
pi 5*pi /pi        /      ___\\ /pi        /      ___\\
--*----*|-- - I*log\2 - \/ 3 /|*|-- - I*log\2 + \/ 3 /|
6   6   \2                    / \2                    /
$$\frac{\pi}{6} \frac{5 \pi}{6} \left(\frac{\pi}{2} - i \log{\left(2 - \sqrt{3} \right)}\right) \left(\frac{\pi}{2} - i \log{\left(\sqrt{3} + 2 \right)}\right)$$
=
    2 /            /      ___\\ /            /      ___\\
5*pi *\pi - 2*I*log\2 + \/ 3 //*\pi - 2*I*log\2 - \/ 3 //
---------------------------------------------------------
                           144                           
$$\frac{5 \pi^{2} \left(\pi - 2 i \log{\left(2 - \sqrt{3} \right)}\right) \left(\pi - 2 i \log{\left(\sqrt{3} + 2 \right)}\right)}{144}$$
5*pi^2*(pi - 2*i*log(2 + sqrt(3)))*(pi - 2*i*log(2 - sqrt(3)))/144
Respuesta rápida [src]
     pi
x1 = --
     6 
$$x_{1} = \frac{\pi}{6}$$
     5*pi
x2 = ----
      6  
$$x_{2} = \frac{5 \pi}{6}$$
     pi        /      ___\
x3 = -- - I*log\2 - \/ 3 /
     2                    
$$x_{3} = \frac{\pi}{2} - i \log{\left(2 - \sqrt{3} \right)}$$
     pi        /      ___\
x4 = -- - I*log\2 + \/ 3 /
     2                    
$$x_{4} = \frac{\pi}{2} - i \log{\left(\sqrt{3} + 2 \right)}$$
x4 = pi/2 - i*log(sqrt(3) + 2)
Respuesta numérica [src]
x1 = 534.594349885863
x2 = -3.66519142918809
x3 = 90.5825881785057
x4 = -49.7418836818384
x5 = -81.1578102177363
x6 = -79.0634151153431
x7 = 804.771318094585
x8 = 38.2227106186758
x9 = -74.8746249105567
x10 = 88.4881930761125
x11 = 31.9395253114962
x12 = 44.5058959258554
x13 = 27.7507351067098
x14 = -5.75958653158129
x15 = 15.1843644923507
x16 = 13.0899693899575
x17 = -93.7241808320955
x18 = 101.054563690472
x19 = -35.081117965086
x20 = -28.7979326579064
x21 = -91.6297857297023
x22 = -72.7802298081635
x23 = -68.5914396033772
x24 = 75.9218224617533
x25 = 96.8657734856853
x26 = -9.94837673636768
x27 = 34.0339204138894
x28 = -56.025068989018
x29 = 65.4498469497874
x30 = -24.60914245312
x31 = -66.497044500984
x32 = 94.7713783832921
x33 = -30.8923277602996
x34 = -16.2315620435473
x35 = 69.6386371545737
x36 = -43.4586983746588
x37 = 8.90117918517108
x38 = 52.8834763354282
x39 = 63.3554518473942
x40 = 19.3731546971371
x41 = -18.3259571459405
x42 = 21.4675497995303
x43 = 2.61799387799149
x44 = -62.3082542961976
x45 = -41.3643032722656
x46 = 25.6563400043166
x47 = -85.3466004225227
x48 = 6.80678408277789
x49 = -97.9129710368819
x50 = 50.789081233035
x51 = -87.4409955249159
x52 = 84.2994028713261
x53 = -53.9306738866248
x54 = -204.727121258935
x55 = -22.5147473507269
x56 = -60.2138591938044
x57 = 46.6002910282486
x58 = -100.007366139275
x59 = 40.317105721069
x60 = 2113.76825709033
x61 = -12.0427718387609
x62 = -2190.21367832768
x63 = 82.2050077689329
x64 = -47.6474885794452
x65 = 57.0722665402146
x66 = 59.1666616426078
x67 = 78.0162175641465
x68 = 71.733032256967
x69 = -37.1755130674792
x70 = 0.523598775598299
x71 = 7056.54069873827
x71 = 7056.54069873827
Gráfico
cos(2*x)+5*sin(x)-3=0 la ecuación