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4sin^2x/3=3 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2       
4*sin (x)    
--------- = 3
    3
4sin⁡2(x)3=3\frac{4 \sin^{2}{\left(x \right)}}{3} = 3
Simplificar
Solución detallada
Tenemos la ecuación
4sin⁡2(x)3=3\frac{4 \sin^{2}{\left(x \right)}}{3} = 3
cambiamos
4sin⁡2(x)3−3=0\frac{4 \sin^{2}{\left(x \right)}}{3} - 3 = 0
4sin⁡2(x)3−3=0\frac{4 \sin^{2}{\left(x \right)}}{3} - 3 = 0
Sustituimos
w=sin⁡(x)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:
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=43a = \frac{4}{3}
b=0b = 0
c=−3c = -3
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (4/3) * (-3) = 16

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

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

o
w1=32w_{1} = \frac{3}{2}
w2=−32w_{2} = - \frac{3}{2}
hacemos cambio inverso
sin⁡(x)=w\sin{\left(x \right)} = w
Tenemos la ecuación
sin⁡(x)=w\sin{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(32)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}
x1=2πn+asin⁡(32)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(−32)x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{3}{2} \right)}
x2=2πn−asin⁡(32)x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{2} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn+π−asin⁡(32)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}
x3=2πn+π−asin⁡(32)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(−32)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{3}{2} \right)}
x4=2πn+π+asin⁡(32)x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{3}{2} \right)}
Gráfica
0-80-60-40-2020406080-10010005
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = pi - re(asin(3/2)) - I*im(asin(3/2))
x1=−re⁡(asin⁡(32))+π−iim⁡(asin⁡(32))x_{1} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} 
x2 = pi + I*im(asin(3/2)) + re(asin(3/2))
x2=re⁡(asin⁡(32))+π+iim⁡(asin⁡(32))x_{2} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} 
x3 = -re(asin(3/2)) - I*im(asin(3/2))
x3=−re⁡(asin⁡(32))−iim⁡(asin⁡(32))x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} 
x4 = I*im(asin(3/2)) + re(asin(3/2))
x4=re⁡(asin⁡(32))+iim⁡(asin⁡(32))x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} 
x4 = re(asin(3/2)) + i*im(asin(3/2))
Suma y producto de raíces [src] 
suma
pi - re(asin(3/2)) - I*im(asin(3/2)) + pi + I*im(asin(3/2)) + re(asin(3/2)) + -re(asin(3/2)) - I*im(asin(3/2)) + I*im(asin(3/2)) + re(asin(3/2))
(re⁡(asin⁡(32))+iim⁡(asin⁡(32)))+(((re⁡(asin⁡(32))+π+iim⁡(asin⁡(32)))+(−re⁡(asin⁡(32))+π−iim⁡(asin⁡(32))))+(−re⁡(asin⁡(32))−iim⁡(asin⁡(32))))\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) + \left(\left(\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right)\right) 
=
2*pi
2π2 \pi 
producto
(pi - re(asin(3/2)) - I*im(asin(3/2)))*(pi + I*im(asin(3/2)) + re(asin(3/2)))*(-re(asin(3/2)) - I*im(asin(3/2)))*(I*im(asin(3/2)) + re(asin(3/2)))
(−re⁡(asin⁡(32))+π−iim⁡(asin⁡(32)))(re⁡(asin⁡(32))+π+iim⁡(asin⁡(32)))(−re⁡(asin⁡(32))−iim⁡(asin⁡(32)))(re⁡(asin⁡(32))+iim⁡(asin⁡(32)))\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) 
=
                                 2                                                                               
(I*im(asin(3/2)) + re(asin(3/2))) *(pi + I*im(asin(3/2)) + re(asin(3/2)))*(-pi + I*im(asin(3/2)) + re(asin(3/2)))
(re⁡(asin⁡(32))+iim⁡(asin⁡(32)))2(−π+re⁡(asin⁡(32))+iim⁡(asin⁡(32)))(re⁡(asin⁡(32))+π+iim⁡(asin⁡(32)))\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right)^{2} \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) 
(i*im(asin(3/2)) + re(asin(3/2)))^2*(pi + i*im(asin(3/2)) + re(asin(3/2)))*(-pi + i*im(asin(3/2)) + re(asin(3/2)))
Respuesta numérica [src] 
x1 = 1.5707963267949 + 0.962423650119207*i
x2 = 4.71238898038469 - 0.962423650119207*i
x3 = -1.5707963267949 + 0.962423650119207*i
x4 = 1.5707963267949 - 0.962423650119207*i
x4 = 1.5707963267949 - 0.962423650119207*i
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