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- La ecuación:
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Ecuación x+9=0
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Ecuación 2x^2+7x-9=0
-
Ecuación 5*x^2+9*x=0
-
Ecuación 3x^2=2x+x^3
- Expresar {x} en función de y en la ecuación:
- -6*x+11*y=-5
- -15*x+7*y=1
- -14*x+4*y=-16
- 11*x-9*y=-4
- Integral de d{x}:
- 0,25
- Suma de la serie:
-
0,25
-
Expresiones idénticas
- cos^2x+cos^2y= cero , veinticinco
- coseno de al cuadrado x más coseno de al cuadrado y es igual a 0,25
- coseno de al cuadrado x más coseno de al cuadrado y es igual a cero , veinticinco
- cos2x+cos2y=0,25
- cos²x+cos²y=0,25
- cos en el grado 2x+cos en el grado 2y=0,25
- cos^2x+cos^2y=O,25
-
Expresiones semejantes
- cos^2x-cos^2y=0,25
-
Expresiones con funciones
- Coseno cos
- cos(x-pi/3)=sqrt(3)*sin(x)
- cosπ(4x+72)\4=−2–√2.
- Cos(t)=sqrt(3)
- cos(2x)+sqrt2cos(pi/2+x)+1=0
- cos(x)=(23/50)
- Coseno cos
- cos(x-pi/3)=sqrt(3)*sin(x)
- cosπ(4x+72)\4=−2–√2.
- Cos(t)=sqrt(3)
- cos(2x)+sqrt2cos(pi/2+x)+1=0
- cos(x)=(23/50)
cos^2x+cos^2y=0,25 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 cos (x) + cos (y) = 1/4
$$\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} = \frac{1}{4}$$
Solución detallada
Tenemos la ecuación
$$\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} = \frac{1}{4}$$
cambiamos
$$\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} - \frac{1}{4} = 0$$
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} + \frac{3}{4} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = 0$$
$$c = \cos^{2}{\left(y \right)} + \frac{3}{4}$$
, entonces
La ecuación tiene dos raíces.
o
$$w_{1} = - \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2}$$
$$w_{2} = \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2}$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
$$\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} = \frac{1}{4}$$
cambiamos
$$\cos^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} - \frac{1}{4} = 0$$
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(y \right)} + \frac{3}{4} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = 0$$
$$c = \cos^{2}{\left(y \right)} + \frac{3}{4}$$
, 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
$$w_{1} = - \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2}$$
$$w_{2} = \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2}$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{4 \cos^{2}{\left(y \right)} + 3}}{2} \right)} + \pi$$
Gráfica
Respuesta rápida
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/ / _________________________________ \\ / / _________________________________ \\
| |-\/ -(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 //
$$x_{1} = - \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi$$
/ / _________________________________\\ / / _________________________________\\
| |\/ -(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 //
$$x_{2} = - \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi$$
/ / _________________________________ \\ / / _________________________________ \\
| |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) || | |-\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||
x3 = I*im|acos|-------------------------------------|| + re|acos|-------------------------------------||
\ \ 2 // \ \ 2 //
$$x_{3} = \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}$$
/ / _________________________________\\ / / _________________________________\\
| |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) || | |\/ -(1 + 2*cos(y))*(-1 + 2*cos(y)) ||
x4 = I*im|acos|-----------------------------------|| + re|acos|-----------------------------------||
\ \ 2 // \ \ 2 //
$$x_{4} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}$$
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 //
$$\left(\left(\left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right)$$
=
4*pi
$$4 \pi$$
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 ///
$$\left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + 2 \pi\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{- \left(2 \cos{\left(y \right)} - 1\right) \left(2 \cos{\left(y \right)} + 1\right)}}{2} \right)}\right)}\right)$$
=
/ / / _______________\\ / / _______________\\\ / / / _______________ \\ / / _______________ \\\ / / / _______________\\ / / _______________\\\ / / / _______________ \\ / / _______________ \\\ | | | / 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 ///
$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} - 2 \pi\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{\sqrt{1 - 4 \cos^{2}{\left(y \right)}}}{2} \right)}\right)} - 2 \pi\right)$$
(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)))