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3cos2x-5sin(x)+1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
3*cos(2*x) - 5*sin(x) + 1 = 0
$$\left(- 5 \sin{\left(x \right)} + 3 \cos{\left(2 x \right)}\right) + 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(- 5 \sin{\left(x \right)} + 3 \cos{\left(2 x \right)}\right) + 1 = 0$$
cambiamos
$$- 6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} + 4 = 0$$
$$- 6 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} + 4 = 0$$
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
$$a = -6$$
$$b = -5$$
$$c = 4$$
, entonces
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (-6) * (4) = 121

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

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

o
$$w_{1} = - \frac{4}{3}$$
$$w_{2} = \frac{1}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{4}{3} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{4}{3} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{6}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{4}{3} \right)}$$
$$x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{4}{3} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{5 \pi}{6}$$
Gráfica
Respuesta rápida [src]
     pi
x1 = --
     6 
$$x_{1} = \frac{\pi}{6}$$
     5*pi
x2 = ----
      6  
$$x_{2} = \frac{5 \pi}{6}$$
                 /      ___\
       pi        |4   \/ 7 |
x3 = - -- + I*log|- - -----|
       2         \3     3  /
$$x_{3} = - \frac{\pi}{2} + i \log{\left(\frac{4}{3} - \frac{\sqrt{7}}{3} \right)}$$
                 /      ___\
       pi        |4   \/ 7 |
x4 = - -- + I*log|- + -----|
       2         \3     3  /
$$x_{4} = - \frac{\pi}{2} + i \log{\left(\frac{\sqrt{7}}{3} + \frac{4}{3} \right)}$$
x4 = -pi/2 + i*log(sqrt(7)/3 + 4/3)
Suma y producto de raíces [src]
suma
                        /      ___\               /      ___\
pi   5*pi     pi        |4   \/ 7 |     pi        |4   \/ 7 |
-- + ---- + - -- + I*log|- - -----| + - -- + I*log|- + -----|
6     6       2         \3     3  /     2         \3     3  /
$$\left(\left(\frac{\pi}{6} + \frac{5 \pi}{6}\right) + \left(- \frac{\pi}{2} + i \log{\left(\frac{4}{3} - \frac{\sqrt{7}}{3} \right)}\right)\right) + \left(- \frac{\pi}{2} + i \log{\left(\frac{\sqrt{7}}{3} + \frac{4}{3} \right)}\right)$$
=
     /      ___\        /      ___\
     |4   \/ 7 |        |4   \/ 7 |
I*log|- - -----| + I*log|- + -----|
     \3     3  /        \3     3  /
$$i \log{\left(\frac{4}{3} - \frac{\sqrt{7}}{3} \right)} + i \log{\left(\frac{\sqrt{7}}{3} + \frac{4}{3} \right)}$$
producto
        /            /      ___\\ /            /      ___\\
pi 5*pi |  pi        |4   \/ 7 || |  pi        |4   \/ 7 ||
--*----*|- -- + I*log|- - -----||*|- -- + I*log|- + -----||
6   6   \  2         \3     3  // \  2         \3     3  //
$$\frac{\pi}{6} \frac{5 \pi}{6} \left(- \frac{\pi}{2} + i \log{\left(\frac{4}{3} - \frac{\sqrt{7}}{3} \right)}\right) \left(- \frac{\pi}{2} + i \log{\left(\frac{\sqrt{7}}{3} + \frac{4}{3} \right)}\right)$$
=
      /          /           2\\ /          /           2\\
      |          |/      ___\ || |          |/      ___\ ||
    2 |          |\4 + \/ 7 / || |          |\4 - \/ 7 / ||
5*pi *|pi - I*log|------------||*|pi - I*log|------------||
      \          \     9      // \          \     9      //
-----------------------------------------------------------
                            144                            
$$\frac{5 \pi^{2} \left(\pi - i \log{\left(\frac{\left(4 - \sqrt{7}\right)^{2}}{9} \right)}\right) \left(\pi - i \log{\left(\frac{\left(\sqrt{7} + 4\right)^{2}}{9} \right)}\right)}{144}$$
5*pi^2*(pi - i*log((4 + sqrt(7))^2/9))*(pi - i*log((4 - sqrt(7))^2/9))/144
Respuesta numérica [src]
x1 = -47.6474885794452
x2 = 21.4675497995303
x3 = -93.7241808320955
x4 = 69.6386371545737
x5 = 52.8834763354282
x6 = -79.0634151153431
x7 = -66.497044500984
x8 = -72.7802298081635
x9 = -24.60914245312
x10 = 84.2994028713261
x11 = 63.3554518473942
x12 = 90.5825881785057
x13 = 115.715329407224
x14 = 78.0162175641465
x15 = 2.61799387799149
x16 = -41.3643032722656
x17 = -43.4586983746588
x18 = -37.1755130674792
x19 = -1203.75358510049
x20 = -22.5147473507269
x21 = 96.8657734856853
x22 = 0.523598775598299
x23 = 38.2227106186758
x24 = -68.5914396033772
x25 = 44.5058959258554
x26 = -60.2138591938044
x27 = 94.7713783832921
x28 = -53.9306738866248
x29 = 50.789081233035
x30 = 27.7507351067098
x31 = -5.75958653158129
x32 = -87.4409955249159
x33 = 65.4498469497874
x34 = -16.2315620435473
x35 = -56.025068989018
x36 = 19.3731546971371
x37 = -9.94837673636768
x38 = -12.0427718387609
x39 = -3.66519142918809
x40 = 6.80678408277789
x41 = -81.1578102177363
x42 = -97.9129710368819
x43 = 88.4881930761125
x44 = -18.3259571459405
x45 = 31.9395253114962
x46 = 46.6002910282486
x47 = 40.317105721069
x48 = 82.2050077689329
x49 = 8.90117918517108
x50 = 75.9218224617533
x51 = -91.6297857297023
x52 = 34.0339204138894
x53 = 71.733032256967
x54 = -85.3466004225227
x55 = -100.007366139275
x56 = -62.3082542961976
x57 = -49.7418836818384
x58 = -110.479341651241
x59 = 25.6563400043166
x60 = -35.081117965086
x60 = -35.081117965086