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sin2(x)+1/2=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2      1    
sin (x) + - = 0
          2
sin⁡2(x)+12=0\sin^{2}{\left(x \right)} + \frac{1}{2} = 0
Simplificar
Solución detallada
Tenemos la ecuación
sin⁡2(x)+12=0\sin^{2}{\left(x \right)} + \frac{1}{2} = 0
cambiamos
sin⁡2(x)+12=0\sin^{2}{\left(x \right)} + \frac{1}{2} = 0
sin⁡2(x)+12=0\sin^{2}{\left(x \right)} + \frac{1}{2} = 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=1a = 1
b=0b = 0
c=12c = \frac{1}{2}
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (1/2) = -2

Como D < 0 la ecuación
no tiene raíces reales,
pero hay raíces complejas.
w1 = (-b + sqrt(D)) / (2*a)

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

o
w1=2i2w_{1} = \frac{\sqrt{2} i}{2}
w2=−2i2w_{2} = - \frac{\sqrt{2} i}{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⁡(2i2)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2} i}{2} \right)}
x1=2πn+iasinh⁡(22)x_{1} = 2 \pi n + i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(−2i2)x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2} i}{2} \right)}
x2=2πn−iasinh⁡(22)x_{2} = 2 \pi n - i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn+π−asin⁡(2i2)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{2} i}{2} \right)}
x3=2πn+π−iasinh⁡(22)x_{3} = 2 \pi n + \pi - i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(−2i2)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{2} i}{2} \right)}
x4=2πn+π+iasinh⁡(22)x_{4} = 2 \pi n + \pi + i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}
Gráfica
0-80-60-40-2020406080-10010002
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
             /  ___\
             |\/ 2 |
x1 = -I*asinh|-----|
             \  2  /
x1=−iasinh⁡(22)x_{1} = - i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)} 
            /  ___\
            |\/ 2 |
x2 = I*asinh|-----|
            \  2  /
x2=iasinh⁡(22)x_{2} = i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)} 
                 /  ___\
                 |\/ 2 |
x3 = pi - I*asinh|-----|
                 \  2  /
x3=π−iasinh⁡(22)x_{3} = \pi - i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)} 
                 /  ___\
                 |\/ 2 |
x4 = pi + I*asinh|-----|
                 \  2  /
x4=π+iasinh⁡(22)x_{4} = \pi + i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)} 
x4 = pi + i*asinh(sqrt(2)/2)
Suma y producto de raíces [src] 
suma
         /  ___\          /  ___\               /  ___\               /  ___\
         |\/ 2 |          |\/ 2 |               |\/ 2 |               |\/ 2 |
- I*asinh|-----| + I*asinh|-----| + pi - I*asinh|-----| + pi + I*asinh|-----|
         \  2  /          \  2  /               \  2  /               \  2  /
((π−iasinh⁡(22))+(−iasinh⁡(22)+iasinh⁡(22)))+(π+iasinh⁡(22))\left(\left(\pi - i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}\right) + \left(- i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)} + i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}\right)\right) + \left(\pi + i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}\right) 
=
2*pi
2π2 \pi 
producto
        /  ___\        /  ___\ /            /  ___\\ /            /  ___\\
        |\/ 2 |        |\/ 2 | |            |\/ 2 || |            |\/ 2 ||
-I*asinh|-----|*I*asinh|-----|*|pi - I*asinh|-----||*|pi + I*asinh|-----||
        \  2  /        \  2  / \            \  2  // \            \  2  //
−iasinh⁡(22)iasinh⁡(22)(π−iasinh⁡(22))(π+iasinh⁡(22))- i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)} i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)} \left(\pi - i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}\right) \left(\pi + i \operatorname{asinh}{\left(\frac{\sqrt{2}}{2} \right)}\right) 
=
      /  ___\ /            /  ___\\
     2|\/ 2 | |  2        2|\/ 2 ||
asinh |-----|*|pi  + asinh |-----||
      \  2  / \            \  2  //
(asinh⁡2(22)+π2)asinh⁡2(22)\left(\operatorname{asinh}^{2}{\left(\frac{\sqrt{2}}{2} \right)} + \pi^{2}\right) \operatorname{asinh}^{2}{\left(\frac{\sqrt{2}}{2} \right)} 
asinh(sqrt(2)/2)^2*(pi^2 + asinh(sqrt(2)/2)^2)
Respuesta numérica [src] 
x1 = -0.658478948462408*i
x2 = 0.658478948462408*i
x3 = 3.14159265358979 - 0.658478948462408*i
x4 = 3.14159265358979 + 0.658478948462408*i
x4 = 3.14159265358979 + 0.658478948462408*i
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