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-2
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cos^2z-sin^2z=-2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 cos (z) - sin (z) = -2
$$- \sin^{2}{\left(z \right)} + \cos^{2}{\left(z \right)} = -2$$
Solución detallada
Tenemos la ecuación
$$- \sin^{2}{\left(z \right)} + \cos^{2}{\left(z \right)} = -2$$
cambiamos
$$\cos{\left(2 z \right)} + 2 = 0$$
$$3 - 2 \sin^{2}{\left(z \right)} = 0$$
Sustituimos
$$w = \sin{\left(z \right)}$$
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
$$a = -2$$
$$b = 0$$
$$c = 3$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{\sqrt{6}}{2}$$
$$w_{2} = \frac{\sqrt{6}}{2}$$
hacemos cambio inverso
$$\sin{\left(z \right)} = w$$
Tenemos la ecuación
$$\sin{\left(z \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$z = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$z = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$z = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$z = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$z_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$z_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{6}}{2} \right)}$$
$$z_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$z_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$z_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{6}}{2} \right)}$$
$$z_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$z_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$- \sin^{2}{\left(z \right)} + \cos^{2}{\left(z \right)} = -2$$
cambiamos
$$\cos{\left(2 z \right)} + 2 = 0$$
$$3 - 2 \sin^{2}{\left(z \right)} = 0$$
Sustituimos
$$w = \sin{\left(z \right)}$$
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
$$a = -2$$
$$b = 0$$
$$c = 3$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-2) * (3) = 24
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{\sqrt{6}}{2}$$
$$w_{2} = \frac{\sqrt{6}}{2}$$
hacemos cambio inverso
$$\sin{\left(z \right)} = w$$
Tenemos la ecuación
$$\sin{\left(z \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$z = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$z = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$z = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$z = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$z_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$z_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{6}}{2} \right)}$$
$$z_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$z_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$z_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{6}}{2} \right)}$$
$$z_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$z_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$z_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}$$
Suma y producto de raíces
[src]
suma
/ / / ___\ \\ / / / ___\ \\ / / / ___\ \\ / / / ___\ \\ / / / ___\\\ / / / ___\\\ / / / ___\\\ / / / ___\\\
| | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / || | | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /||
| | -----------------|| | | -----------------|| | | -----------------|| | | -----------------|| | | ---------------|| | | ---------------|| | | ---------------|| | | ---------------||
| | 2 || | | 2 || | | 2 || | | 2 || | | 2 || | | 2 || | | 2 || | | 2 ||
- 2*re\atan\e // - 2*I*im\atan\e // + 2*re\atan\e // + 2*I*im\atan\e // + - 2*re\atan\e // - 2*I*im\atan\e // + 2*re\atan\e // + 2*I*im\atan\e //
$$\left(\left(\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right) + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right)\right) + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right)$$
=
0
$$0$$
producto
/ / / / ___\ \\ / / / ___\ \\\ / / / / ___\ \\ / / / ___\ \\\ / / / / ___\\\ / / / ___\\\\ / / / / ___\\\ / / / ___\\\\ | | | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / ||| | | | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / ||| | | | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /||| | | | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /||| | | | -----------------|| | | -----------------||| | | | -----------------|| | | -----------------||| | | | ---------------|| | | ---------------||| | | | ---------------|| | | ---------------||| | | | 2 || | | 2 ||| | | | 2 || | | 2 ||| | | | 2 || | | 2 ||| | | | 2 || | | 2 ||| \- 2*re\atan\e // - 2*I*im\atan\e ///*\2*re\atan\e // + 2*I*im\atan\e ///*\- 2*re\atan\e // - 2*I*im\atan\e ///*\2*re\atan\e // + 2*I*im\atan\e ///
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right)$$
=
2 2 / / / / ___\\\ / / / ___\\\\ / / / / ___\ \\ / / / ___\ \\\ | | | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /||| | | | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / ||| | | | ---------------|| | | ---------------||| | | | -----------------|| | | -----------------||| | | | 2 || | | 2 ||| | | | 2 || | | 2 ||| 16*\I*im\atan\e // + re\atan\e /// *\I*im\atan\e // + re\atan\e ///
$$16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right)^{2} \left(\operatorname{re}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}\right)^{2}$$
16*(i*im(atan(exp(i*atan(2*sqrt(2))/2))) + re(atan(exp(i*atan(2*sqrt(2))/2))))^2*(i*im(atan(exp(-i*atan(2*sqrt(2))/2))) + re(atan(exp(-i*atan(2*sqrt(2))/2))))^2
Respuesta rápida
[src]
/ / / ___\ \\ / / / ___\ \\
| | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / ||
| | -----------------|| | | -----------------||
| | 2 || | | 2 ||
z1 = - 2*re\atan\e // - 2*I*im\atan\e //
$$z_{1} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}$$
/ / / ___\ \\ / / / ___\ \\
| | -I*atan\2*\/ 2 / || | | -I*atan\2*\/ 2 / ||
| | -----------------|| | | -----------------||
| | 2 || | | 2 ||
z2 = 2*re\atan\e // + 2*I*im\atan\e //
$$z_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{- \frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}$$
/ / / ___\\\ / / / ___\\\
| | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /||
| | ---------------|| | | ---------------||
| | 2 || | | 2 ||
z3 = - 2*re\atan\e // - 2*I*im\atan\e //
$$z_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}$$
/ / / ___\\\ / / / ___\\\
| | I*atan\2*\/ 2 /|| | | I*atan\2*\/ 2 /||
| | ---------------|| | | ---------------||
| | 2 || | | 2 ||
z4 = 2*re\atan\e // + 2*I*im\atan\e //
$$z_{4} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(e^{\frac{i \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}} \right)}\right)}$$
z4 = 2*re(atan(exp(i*atan(2*sqrt(2))/2))) + 2*i*im(atan(exp(i*atan(2*sqrt(2))/2)))
Respuesta numérica
[src]
z1 = -1.5707963267949 + 0.658478948462408*i
z2 = 1.5707963267949 - 0.658478948462408*i
z3 = -1.5707963267949 - 0.658478948462408*i
z4 = 1.5707963267949 + 0.658478948462408*i
z4 = 1.5707963267949 + 0.658478948462408*i