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3*sin^2(x)-2*cos(x)+2=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2                      
3*sin (x) - 2*cos(x) + 2 = 0
(3sin⁡2(x)−2cos⁡(x))+2=0\left(3 \sin^{2}{\left(x \right)} - 2 \cos{\left(x \right)}\right) + 2 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(3sin⁡2(x)−2cos⁡(x))+2=0\left(3 \sin^{2}{\left(x \right)} - 2 \cos{\left(x \right)}\right) + 2 = 0
cambiamos
−3cos⁡2(x)−2cos⁡(x)+5=0- 3 \cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 5 = 0
−3cos⁡2(x)−2cos⁡(x)+5=0- 3 \cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 5 = 0
Sustituimos
w=cos⁡(x)w = \cos{\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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−3a = -3
b=−2b = -2
c=5c = 5
, entonces
D = b^2 - 4 * a * c = 

(-2)^2 - 4 * (-3) * (5) = 64

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

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

o
w1=−53w_{1} = - \frac{5}{3}
w2=1w_{2} = 1
hacemos cambio inverso
cos⁡(x)=w\cos{\left(x \right)} = w
Tenemos la ecuación
cos⁡(x)=w\cos{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
sustituimos w:
x1=πn+acos⁡(w1)x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}
x1=πn+acos⁡(−53)x_{1} = \pi n + \operatorname{acos}{\left(- \frac{5}{3} \right)}
x1=πn+acos⁡(−53)x_{1} = \pi n + \operatorname{acos}{\left(- \frac{5}{3} \right)}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(1)x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}
x2=πnx_{2} = \pi n
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(−53)x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{5}{3} \right)}
x3=πn−π+acos⁡(−53)x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{5}{3} \right)}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(1)x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}
x4=πn−πx_{4} = \pi n - \pi
Gráfica
0-80-60-40-2020406080-100100010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = 0
x1=0x_{1} = 0 
x2 = 2*im(atanh(2)) - 2*I*re(atanh(2))
x2=2im⁡(atanh⁡(2))−2ire⁡(atanh⁡(2))x_{2} = 2 \operatorname{im}{\left(\operatorname{atanh}{\left(2 \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(2 \right)}\right)} 
x3 = -2*im(atanh(2)) + 2*I*re(atanh(2))
x3=−2im⁡(atanh⁡(2))+2ire⁡(atanh⁡(2))x_{3} = - 2 \operatorname{im}{\left(\operatorname{atanh}{\left(2 \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(2 \right)}\right)} 
x3 = -2*im(atanh(2)) + 2*i*re(atanh(2))
Suma y producto de raíces [src] 
suma
2*im(atanh(2)) - 2*I*re(atanh(2)) + -2*im(atanh(2)) + 2*I*re(atanh(2))
(2im⁡(atanh⁡(2))−2ire⁡(atanh⁡(2)))+(−2im⁡(atanh⁡(2))+2ire⁡(atanh⁡(2)))\left(2 \operatorname{im}{\left(\operatorname{atanh}{\left(2 \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(2 \right)}\right)}\right) + \left(- 2 \operatorname{im}{\left(\operatorname{atanh}{\left(2 \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(2 \right)}\right)}\right) 
=
0
00 
producto
0*(2*im(atanh(2)) - 2*I*re(atanh(2)))*(-2*im(atanh(2)) + 2*I*re(atanh(2)))
(−2im⁡(atanh⁡(2))+2ire⁡(atanh⁡(2)))0(2im⁡(atanh⁡(2))−2ire⁡(atanh⁡(2)))\left(- 2 \operatorname{im}{\left(\operatorname{atanh}{\left(2 \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(2 \right)}\right)}\right) 0 \left(2 \operatorname{im}{\left(\operatorname{atanh}{\left(2 \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(2 \right)}\right)}\right) 
=
0
00 
0
Respuesta numérica [src] 
x1 = -31.4159266196534
x2 = -251.327412273949
x3 = -69.115038643136
x4 = 12.566370114504
x5 = 94.2477796093524
x6 = 75.3982235560588
x7 = 81.6814094553991
x8 = -75.3982235939661
x9 = -12.5663703510437
x10 = -62.831853345356
x11 = 18.849555664849
x12 = 0.0
x13 = -25.1327410787185
x14 = 81.6814091796372
x15 = -100.530964659879
x16 = 75.3982239567878
x17 = -18.8495561981144
x18 = 12.5663704494723
x19 = -25.1327414892758
x20 = 188.495559563197
x21 = 50.2654824463453
x22 = -50.265482293023
x23 = -62.831852783388
x24 = 100.530964764843
x25 = 31.415926803001
x26 = -87.9645943587335
x27 = -94.2477794510094
x28 = -31.4159267079561
x29 = 69.1150386618644
x30 = 25.1327409497728
x31 = 6.28318528424044
x32 = -6.28318513525716
x33 = 43.9822971694365
x34 = 69.1150379182055
x35 = -31.4159268178552
x36 = -37.6991118771797
x37 = -56.5486678045987
x38 = 62.8318528192291
x39 = 18.8495560486233
x40 = 25.132741513973
x41 = 69.1150380982552
x42 = 31.4159264316657
x43 = -87.9645946197325
x44 = -75.3982238652115
x45 = 94.2477791537892
x46 = -56.5486675053392
x47 = 56.5486678237887
x48 = -12.5663706887756
x49 = 37.6991120222947
x50 = -43.9822971745689
x51 = -69.1150382090074
x52 = -81.6814090380623
x53 = -18.8495556344567
x54 = 87.9645943357949
x55 = 56.5486676070721
x56 = 62.83185317253
x56 = 62.83185317253
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