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(1+2sin^2(x)+2sin(x)+2sin(x)*cos(x))/(2sin(x)*cos(x)-1)=1 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
         2                                    
1 + 2*sin (x) + 2*sin(x) + 2*sin(x)*cos(x)    
------------------------------------------ = 1
           2*sin(x)*cos(x) - 1
((2sin⁡2(x)+1)+2sin⁡(x))+2sin⁡(x)cos⁡(x)2sin⁡(x)cos⁡(x)−1=1\frac{\left(\left(2 \sin^{2}{\left(x \right)} + 1\right) + 2 \sin{\left(x \right)}\right) + 2 \sin{\left(x \right)} \cos{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)} - 1} = 1
Simplificar
Solución detallada
Tenemos la ecuación
((2sin⁡2(x)+1)+2sin⁡(x))+2sin⁡(x)cos⁡(x)2sin⁡(x)cos⁡(x)−1=1\frac{\left(\left(2 \sin^{2}{\left(x \right)} + 1\right) + 2 \sin{\left(x \right)}\right) + 2 \sin{\left(x \right)} \cos{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)} - 1} = 1
cambiamos
2(sin⁡2(x)+sin⁡(x)+1)sin⁡(2x)−1=0\frac{2 \left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 1\right)}{\sin{\left(2 x \right)} - 1} = 0
−1+((2sin⁡2(x)+1)+2sin⁡(x))+2sin⁡(x)cos⁡(x)2sin⁡(x)cos⁡(x)−1=0-1 + \frac{\left(\left(2 \sin^{2}{\left(x \right)} + 1\right) + 2 \sin{\left(x \right)}\right) + 2 \sin{\left(x \right)} \cos{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)} - 1} = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
Tenemos la ecuación:
−1+2wsin⁡(x)+2sin⁡2(x)+2sin⁡(x)+12wsin⁡(x)−1=0-1 + \frac{2 w \sin{\left(x \right)} + 2 \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 1}{2 w \sin{\left(x \right)} - 1} = 0
Multipliquemos las dos partes de la ecuación por el denominador -1 + 2*w*sin(x)
obtendremos:
2sin⁡2(x)+2sin⁡(x)+2=02 \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 2 = 0
Abrimos los paréntesis en el miembro izquierdo de la ecuación
2 + 2*sinx^2 + 2*sinx = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
2sin⁡2(x)+2sin⁡(x)=−22 \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} = -2
Esta ecuación no tiene soluciones
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:
Gráfica
0-80-60-40-2020406080-100100-100000005000000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
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         |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||            |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||
pi + I*im|asin|- - -------|| + re|asin|- - -------|| + pi + I*im|asin|- + -------|| + re|asin|- + -------|| + - re|asin|- - -------|| - I*im|asin|- - -------|| + - re|asin|- + -------|| - I*im|asin|- + -------||
         \    \2      2   //     \    \2      2   //            \    \2      2   //     \    \2      2   //       \    \2      2   //       \    \2      2   //       \    \2      2   //       \    \2      2   //
(−re⁡(asin⁡(12+3i2))−iim⁡(asin⁡(12+3i2)))+(((re⁡(asin⁡(12−3i2))+π+iim⁡(asin⁡(12−3i2)))+(re⁡(asin⁡(12+3i2))+π+iim⁡(asin⁡(12+3i2))))+(−re⁡(asin⁡(12−3i2))−iim⁡(asin⁡(12−3i2))))\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) + \left(\left(\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right)\right) 
=
2*pi
2π2 \pi 
producto
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|         |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||| |         |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||| |    |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||| |    |    |1   I*\/ 3 ||       |    |1   I*\/ 3 |||
|pi + I*im|asin|- - -------|| + re|asin|- - -------|||*|pi + I*im|asin|- + -------|| + re|asin|- + -------|||*|- re|asin|- - -------|| - I*im|asin|- - -------|||*|- re|asin|- + -------|| - I*im|asin|- + -------|||
\         \    \2      2   //     \    \2      2   /// \         \    \2      2   //     \    \2      2   /// \    \    \2      2   //       \    \2      2   /// \    \    \2      2   //       \    \2      2   ///
(re⁡(asin⁡(12−3i2))+π+iim⁡(asin⁡(12−3i2)))(re⁡(asin⁡(12+3i2))+π+iim⁡(asin⁡(12+3i2)))(−re⁡(asin⁡(12−3i2))−iim⁡(asin⁡(12−3i2)))(−re⁡(asin⁡(12+3i2))−iim⁡(asin⁡(12+3i2)))\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) 
=
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|    |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||| |    |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||| |         |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||| |         |    |1   I*\/ 3 ||     |    |1   I*\/ 3 |||
|I*im|asin|- + -------|| + re|asin|- + -------|||*|I*im|asin|- - -------|| + re|asin|- - -------|||*|pi + I*im|asin|- + -------|| + re|asin|- + -------|||*|pi + I*im|asin|- - -------|| + re|asin|- - -------|||
\    \    \2      2   //     \    \2      2   /// \    \    \2      2   //     \    \2      2   /// \         \    \2      2   //     \    \2      2   /// \         \    \2      2   //     \    \2      2   ///
(re⁡(asin⁡(12−3i2))+iim⁡(asin⁡(12−3i2)))(re⁡(asin⁡(12+3i2))+iim⁡(asin⁡(12+3i2)))(re⁡(asin⁡(12−3i2))+π+iim⁡(asin⁡(12−3i2)))(re⁡(asin⁡(12+3i2))+π+iim⁡(asin⁡(12+3i2)))\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) 
(i*im(asin(1/2 + i*sqrt(3)/2)) + re(asin(1/2 + i*sqrt(3)/2)))*(i*im(asin(1/2 - i*sqrt(3)/2)) + re(asin(1/2 - i*sqrt(3)/2)))*(pi + i*im(asin(1/2 + i*sqrt(3)/2)) + re(asin(1/2 + i*sqrt(3)/2)))*(pi + i*im(asin(1/2 - i*sqrt(3)/2)) + re(asin(1/2 - i*sqrt(3)/2)))
Respuesta rápida [src] 
              /    /        ___\\     /    /        ___\\
              |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||
x1 = pi + I*im|asin|- - -------|| + re|asin|- - -------||
              \    \2      2   //     \    \2      2   //
x1=re⁡(asin⁡(12−3i2))+π+iim⁡(asin⁡(12−3i2))x_{1} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} 
              /    /        ___\\     /    /        ___\\
              |    |1   I*\/ 3 ||     |    |1   I*\/ 3 ||
x2 = pi + I*im|asin|- + -------|| + re|asin|- + -------||
              \    \2      2   //     \    \2      2   //
x2=re⁡(asin⁡(12+3i2))+π+iim⁡(asin⁡(12+3i2))x_{2} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} 
         /    /        ___\\       /    /        ___\\
         |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||
x3 = - re|asin|- - -------|| - I*im|asin|- - -------||
         \    \2      2   //       \    \2      2   //
x3=−re⁡(asin⁡(12−3i2))−iim⁡(asin⁡(12−3i2))x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} 
         /    /        ___\\       /    /        ___\\
         |    |1   I*\/ 3 ||       |    |1   I*\/ 3 ||
x4 = - re|asin|- + -------|| - I*im|asin|- + -------||
         \    \2      2   //       \    \2      2   //
x4=−re⁡(asin⁡(12+3i2))−iim⁡(asin⁡(12+3i2))x_{4} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} 
x4 = -re(asin(1/2 + sqrt(3)*i/2)) - i*im(asin(1/2 + sqrt(3)*i/2))
Respuesta numérica [src] 
x1 = 3.51632708629853 - 0.831442945529311*i
x2 = 3.51632708629853 + 0.831442945529311*i
x3 = -0.37473443270874 + 0.831442945529311*i
x4 = -0.37473443270874 - 0.831442945529311*i
x4 = -0.37473443270874 - 0.831442945529311*i
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