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cos^2x+cos^2y=0,25 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2         2         
cos (x) + cos (y) = 1/4
cos⁡2(x)+cos⁡2(y)=14\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} = \frac{1}{4}
Simplificar
Solución detallada
Tenemos la ecuación
cos⁡2(x)+cos⁡2(y)=14\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} = \frac{1}{4}
cambiamos
cos⁡2(x)+cos⁡2(y)−14=0\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} - \frac{1}{4} = 0
−sin⁡2(x)+cos⁡2(y)+34=0- \sin^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} + \frac{3}{4} = 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=cos⁡2(y)+34c = \cos^{2}{\left(y \right)} + \frac{3}{4}
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (-1) * (3/4 + cos(y)^2) = 3 + 4*cos(y)^2

La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

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

o
w1=−4cos⁡2(y)+32w_{1} = - \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2}
w2=4cos⁡2(y)+32w_{2} = \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 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⁡(−4cos⁡2(y)+32)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}
x1=2πn−asin⁡(4cos⁡2(y)+32)x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(4cos⁡2(y)+32)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}
x2=2πn+asin⁡(4cos⁡2(y)+32)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(−4cos⁡2(y)+32)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi
x3=2πn+asin⁡(4cos⁡2(y)+32)+πx_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(4cos⁡2(y)+32)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi
x4=2πn−asin⁡(4cos⁡2(y)+32)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi
Gráfica
Respuesta rápida [src] 
         /    /   _________________________________ \\              /    /   _________________________________ \\
         |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||              |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||
x1 = - re|acos|-------------------------------------|| + 2*pi - I*im|acos|-------------------------------------||
         \    \                  2                  //              \    \                  2                  //
x1=−re⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))−iim⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+2πx_{1} = - \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi 
         /    /  _________________________________\\              /    /  _________________________________\\
         |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||              |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||
x2 = - re|acos|-----------------------------------|| + 2*pi - I*im|acos|-----------------------------------||
         \    \                 2                 //              \    \                 2                 //
x2=−re⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))−iim⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+2πx_{2} = - \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi 
         /    /   _________________________________ \\     /    /   _________________________________ \\
         |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||     |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||
x3 = I*im|acos|-------------------------------------|| + re|acos|-------------------------------------||
         \    \                  2                  //     \    \                  2                  //
x3=re⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+iim⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))x_{3} = \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} 
         /    /  _________________________________\\     /    /  _________________________________\\
         |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||     |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||
x4 = I*im|acos|-----------------------------------|| + re|acos|-----------------------------------||
         \    \                 2                 //     \    \                 2                 //
x4=re⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+iim⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))x_{4} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} 
x4 = re(acos(sqrt(-(2*cos(y) - 1)*(2*cos(y) + 1))/2)) + i*im(acos(sqrt(-(2*cos(y) - 1)*(2*cos(y) + 1))/2))
Suma y producto de raíces [src] 
suma
    /    /   _________________________________ \\              /    /   _________________________________ \\       /    /  _________________________________\\              /    /  _________________________________\\       /    /   _________________________________ \\     /    /   _________________________________ \\       /    /  _________________________________\\     /    /  _________________________________\\
    |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||              |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||       |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||              |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||       |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||     |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||       |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||     |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||
- re|acos|-------------------------------------|| + 2*pi - I*im|acos|-------------------------------------|| + - re|acos|-----------------------------------|| + 2*pi - I*im|acos|-----------------------------------|| + I*im|acos|-------------------------------------|| + re|acos|-------------------------------------|| + I*im|acos|-----------------------------------|| + re|acos|-----------------------------------||
    \    \                  2                  //              \    \                  2                  //       \    \                 2                 //              \    \                 2                 //       \    \                  2                  //     \    \                  2                  //       \    \                 2                 //     \    \                 2                 //
(((−re⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))−iim⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+2π)+(−re⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))−iim⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+2π))+(re⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+iim⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))))+(re⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+iim⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2)))\left(\left(\left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right) 
=
4*pi
4π4 \pi 
producto
/    /    /   _________________________________ \\              /    /   _________________________________ \\\ /    /    /  _________________________________\\              /    /  _________________________________\\\ /    /    /   _________________________________ \\     /    /   _________________________________ \\\ /    /    /  _________________________________\\     /    /  _________________________________\\\
|    |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||              |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||| |    |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||              |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||| |    |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||     |    |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y))  ||| |    |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||     |    |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) |||
|- re|acos|-------------------------------------|| + 2*pi - I*im|acos|-------------------------------------|||*|- re|acos|-----------------------------------|| + 2*pi - I*im|acos|-----------------------------------|||*|I*im|acos|-------------------------------------|| + re|acos|-------------------------------------|||*|I*im|acos|-----------------------------------|| + re|acos|-----------------------------------|||
\    \    \                  2                  //              \    \                  2                  /// \    \    \                 2                 //              \    \                 2                 /// \    \    \                  2                  //     \    \                  2                  /// \    \    \                 2                 //     \    \                 2                 ///
(−re⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))−iim⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+2π)(−re⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))−iim⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+2π)(re⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+iim⁡(acos⁡(−−(2cos⁡(y)−1)(2cos⁡(y)+1)2)))(re⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2))+iim⁡(acos⁡(−(2cos⁡(y)−1)(2cos⁡(y)+1)2)))\left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right) 
=
/    /    /   _______________\\     /    /   _______________\\\ /    /    /    _______________ \\     /    /    _______________ \\\ /            /    /   _______________\\     /    /   _______________\\\ /            /    /    _______________ \\     /    /    _______________ \\\
|    |    |  /          2    ||     |    |  /          2    ||| |    |    |   /          2     ||     |    |   /          2     ||| |            |    |  /          2    ||     |    |  /          2    ||| |            |    |   /          2     ||     |    |   /          2     |||
|    |    |\/  1 - 4*cos (y) ||     |    |\/  1 - 4*cos (y) ||| |    |    |-\/  1 - 4*cos (y)  ||     |    |-\/  1 - 4*cos (y)  ||| |            |    |\/  1 - 4*cos (y) ||     |    |\/  1 - 4*cos (y) ||| |            |    |-\/  1 - 4*cos (y)  ||     |    |-\/  1 - 4*cos (y)  |||
|I*im|acos|------------------|| + re|acos|------------------|||*|I*im|acos|--------------------|| + re|acos|--------------------|||*|-2*pi + I*im|acos|------------------|| + re|acos|------------------|||*|-2*pi + I*im|acos|--------------------|| + re|acos|--------------------|||
\    \    \        2         //     \    \        2         /// \    \    \         2          //     \    \         2          /// \            \    \        2         //     \    \        2         /// \            \    \         2          //     \    \         2          ///
(re⁡(acos⁡(−1−4cos⁡2(y)2))+iim⁡(acos⁡(−1−4cos⁡2(y)2)))(re⁡(acos⁡(1−4cos⁡2(y)2))+iim⁡(acos⁡(1−4cos⁡2(y)2)))(re⁡(acos⁡(−1−4cos⁡2(y)2))+iim⁡(acos⁡(−1−4cos⁡2(y)2))−2π)(re⁡(acos⁡(1−4cos⁡2(y)2))+iim⁡(acos⁡(1−4cos⁡2(y)2))−2π)\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} - 2 \pi\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} - 2 \pi\right) 
(i*im(acos(sqrt(1 - 4*cos(y)^2)/2)) + re(acos(sqrt(1 - 4*cos(y)^2)/2)))*(i*im(acos(-sqrt(1 - 4*cos(y)^2)/2)) + re(acos(-sqrt(1 - 4*cos(y)^2)/2)))*(-2*pi + i*im(acos(sqrt(1 - 4*cos(y)^2)/2)) + re(acos(sqrt(1 - 4*cos(y)^2)/2)))*(-2*pi + i*im(acos(-sqrt(1 - 4*cos(y)^2)/2)) + re(acos(-sqrt(1 - 4*cos(y)^2)/2)))
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