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2*sin(x)^2+sqrt(2*sin(x))=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2        __________    
2*sin (x) + \/ 2*sin(x)  = 0
2sin⁡(x)+2sin⁡2(x)=0\sqrt{2 \sin{\left(x \right)}} + 2 \sin^{2}{\left(x \right)} = 0
Simplificar
Solución detallada
Tenemos la ecuación
2sin⁡(x)+2sin⁡2(x)=0\sqrt{2 \sin{\left(x \right)}} + 2 \sin^{2}{\left(x \right)} = 0
cambiamos
2sin⁡(x)+2sin⁡2(x)=0\sqrt{2} \sqrt{\sin{\left(x \right)}} + 2 \sin^{2}{\left(x \right)} = 0
2sin⁡(x)+2sin⁡2(x)=0\sqrt{2 \sin{\left(x \right)}} + 2 \sin^{2}{\left(x \right)} = 0
Sustituimos
w=sin⁡(x)w = \sin{\left(x \right)}
Tenemos la ecuación
2w+2w2=0\sqrt{2} \sqrt{w} + 2 w^{2} = 0
Evidentemente:
w0 = 0

luego,
cambiamos
w32=−22w^{\frac{3}{2}} = - \frac{\sqrt{2}}{2}
Ya que la potencia en la ecuación es igual a = 3/2 y miembro libre = -sqrt(2)/2 < 0,
significa que la ecuación correspondiente no tiene soluciones reales

Las demás 2 raíces son complejas.
hacemos el cambio:
z=wz = w
entonces la ecuación será así:
z32=−22z^{\frac{3}{2}} = - \frac{\sqrt{2}}{2}
Cualquier número complejo se puede presentar que:
z=reipz = r e^{i p}
sustituimos en la ecuación
(reip)32=−22\left(r e^{i p}\right)^{\frac{3}{2}} = - \frac{\sqrt{2}}{2}
donde
r=2232r = \frac{2^{\frac{2}{3}}}{2}
- módulo del número complejo
Sustituyamos r:
e3ip2=−1e^{\frac{3 i p}{2}} = -1
Usando la fórmula de Euler hallemos las raíces para p
isin⁡(3p2)+cos⁡(3p2)=−1i \sin{\left(\frac{3 p}{2} \right)} + \cos{\left(\frac{3 p}{2} \right)} = -1
es decir
cos⁡(3p2)=−1\cos{\left(\frac{3 p}{2} \right)} = -1
y
sin⁡(3p2)=0\sin{\left(\frac{3 p}{2} \right)} = 0
entonces
p=4πN3+2π3p = \frac{4 \pi N}{3} + \frac{2 \pi}{3}
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
z1=(2564−2563i4)2z_{1} = \left(\frac{2^{\frac{5}{6}}}{4} - \frac{2^{\frac{5}{6}} \sqrt{3} i}{4}\right)^{2}
z2=(2564+2563i4)2z_{2} = \left(\frac{2^{\frac{5}{6}}}{4} + \frac{2^{\frac{5}{6}} \sqrt{3} i}{4}\right)^{2}
hacemos cambio inverso
z=wz = w
w=zw = z

Entonces la respuesta definitiva es:
w0 = 0

w1=(2564−2563i4)2w_{1} = \left(\frac{2^{\frac{5}{6}}}{4} - \frac{2^{\frac{5}{6}} \sqrt{3} i}{4}\right)^{2}
w2=(2564+2563i4)2w_{2} = \left(\frac{2^{\frac{5}{6}}}{4} + \frac{2^{\frac{5}{6}} \sqrt{3} i}{4}\right)^{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:
Gráfica
0-80-60-40-2020406080-10010005
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = 0
x1=0x_{1} = 0 
         /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\
         |    |2   *\1 - I*\/ 3 /||       |    |2   *\1 - I*\/ 3 /||
x2 = - re|asin|------------------|| - I*im|asin|------------------||
         \    \        4         //       \    \        4         //
x2=−re⁡(asin⁡(223(1−3i)4))−iim⁡(asin⁡(223(1−3i)4))x_{2} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)} 
         /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\
         |    |2   *\1 + I*\/ 3 /||       |    |2   *\1 + I*\/ 3 /||
x3 = - re|asin|------------------|| - I*im|asin|------------------||
         \    \        4         //       \    \        4         //
x3=−re⁡(asin⁡(223(1+3i)4))−iim⁡(asin⁡(223(1+3i)4))x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)} 
x3 = -re(asin(2^(2/3)*(1 + sqrt(3)*i)/4)) - i*im(asin(2^(2/3)*(1 + sqrt(3)*i)/4))
Suma y producto de raíces [src] 
suma
    /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\
    |    |2   *\1 - I*\/ 3 /||       |    |2   *\1 - I*\/ 3 /||       |    |2   *\1 + I*\/ 3 /||       |    |2   *\1 + I*\/ 3 /||
- re|asin|------------------|| - I*im|asin|------------------|| + - re|asin|------------------|| - I*im|asin|------------------||
    \    \        4         //       \    \        4         //       \    \        4         //       \    \        4         //
(−re⁡(asin⁡(223(1+3i)4))−iim⁡(asin⁡(223(1+3i)4)))+(−re⁡(asin⁡(223(1−3i)4))−iim⁡(asin⁡(223(1−3i)4)))\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)}\right) 
=
    /    / 2/3 /        ___\\\     /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\
    |    |2   *\1 + I*\/ 3 /||     |    |2   *\1 - I*\/ 3 /||       |    |2   *\1 + I*\/ 3 /||       |    |2   *\1 - I*\/ 3 /||
- re|asin|------------------|| - re|asin|------------------|| - I*im|asin|------------------|| - I*im|asin|------------------||
    \    \        4         //     \    \        4         //       \    \        4         //       \    \        4         //
−re⁡(asin⁡(223(1+3i)4))−re⁡(asin⁡(223(1−3i)4))−iim⁡(asin⁡(223(1+3i)4))−iim⁡(asin⁡(223(1−3i)4))- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)} - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)} 
producto
  /    /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\\ /    /    / 2/3 /        ___\\\       /    / 2/3 /        ___\\\\
  |    |    |2   *\1 - I*\/ 3 /||       |    |2   *\1 - I*\/ 3 /||| |    |    |2   *\1 + I*\/ 3 /||       |    |2   *\1 + I*\/ 3 /|||
0*|- re|asin|------------------|| - I*im|asin|------------------|||*|- re|asin|------------------|| - I*im|asin|------------------|||
  \    \    \        4         //       \    \        4         /// \    \    \        4         //       \    \        4         ///
0(−re⁡(asin⁡(223(1−3i)4))−iim⁡(asin⁡(223(1−3i)4)))(−re⁡(asin⁡(223(1+3i)4))−iim⁡(asin⁡(223(1+3i)4)))0 \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right)}{4} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right)}{4} \right)}\right)}\right) 
=
0
00 
0
Respuesta numérica [src] 
x1 = -34.2305552182376 - 0.673688674164692*i
x2 = 53.7340390822766 + 0.673688674164692*i
x3 = -31.742890507148 + 0.673688674164692*i
x4 = 31.0889625646478 - 0.673688674164692*i
x5 = 3.46855662483991 + 0.673688674164692*i
x6 = -75.7251876574051 + 0.673688674164692*i
x7 = -9.09781398951927 - 0.673688674164692*i
x8 = -25.4597051999685 - 0.673688674164692*i
x9 = 9.75174193201949 - 0.673688674164692*i
x10 = 16.0349272391991 - 0.673688674164692*i
x11 = 91.4331509253541 - 0.673688674164692*i
x12 = 47.450853775097 + 0.673688674164692*i
x13 = 68.7880744077253 - 0.673688674164692*i
x14 = 56.2217037933662 - 0.673688674164692*i
x15 = 68.7880744077253 + 0.673688674164692*i
x16 = 62.5048891005458 + 0.673688674164692*i
x17 = 97.7163362325337 + 0.673688674164692*i
x18 = 60.0172243894562 - 0.673688674164654*i
x19 = 18.5225919502886 + 0.673688674164692*i
x20 = 24.8057772574682 + 0.673688674164692*i
x21 = -63.158817043046 + 0.673688674164692*i
x22 = -25.4597051999685 + 0.673688674164692*i
x23 = -84.4960376756743 - 0.673688674164692*i
x24 = 0.0
x25 = 78.8667803109949 + 0.673688674164692*i
x26 = -69.4420023502256 + 0.673688674164692*i
x27 = -88.2915582717643 + 0.673688674164692*i
x28 = 91.4331509253541 + 0.673688674164692*i
x29 = 81.3544450220845 + 0.673688674164692*i
x30 = -78.2128523684947 - 0.673688674164692*i
x31 = -40.5137405254172 + 0.673688674164692*i
x32 = -53.0801111397764 + 0.673688674164692*i
x33 = -40.5137405254172 - 0.673688674164692*i
x34 = -46.7969258325968 - 0.673688674164692*i
x35 = 3.46855662483991 - 0.673688674164692*i
x36 = -27.947369911058 - 0.673688674164692*i
x37 = -59.363296446956 + 0.673688674164692*i
x38 = 87.6376303292641 - 0.673688674164692*i
x39 = 97.7163362325337 - 0.673688674164692*i
x40 = 53.7340390822766 - 0.673688674164692*i
x41 = -71.9296670613151 + 0.673688674164692*i
x42 = -53.0801111397764 - 0.673688674164692*i
x43 = 47.450853775097 - 0.673688674164692*i
x44 = -94.5747435789439 - 0.673688674164692*i
x45 = -90.7792229828539 - 0.673688674164692*i
x46 = -34.2305552182376 + 0.673688674164692*i
x47 = -15.3809992966989 + 0.673688674164692*i
x48 = 28.6012978535583 - 0.673688674164692*i
x49 = 24.8057772574682 - 0.673688674164692*i
x50 = -19.1765198927889 - 0.673688674164692*i
x51 = -84.4960376756743 + 0.673688674164692*i
x52 = -19.1765198927889 + 0.673688674164692*i
x53 = 34.8844831607378 + 0.673688674164692*i
x53 = 34.8844831607378 + 0.673688674164692*i
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