Sr Examen
Otras calculadoras
- Sistemas de desigualdades paso a paso
- Ecuaciones básicas paso a paso
- Sistemas de ecuaciones paso a paso
- Solución de desigualdades trigonométricas en línea
- Resolución de desigualdades exponenciales en línea
- Resolución de desigualdades con un módulo en línea
-
¿Cómo usar?
- Desigualdades:
-
(x+3)/(5-2x)<0
-
x^2>9
-
x^2-10x<0
- 5(y-1,4)-6<4y-1,5
- Derivada de:
-
cos(x)^(2)-sin(x)^(2)
-
Expresiones idénticas
- cos(x)^(dos)-sin(x)^(dos)<sqrt(dos)/ dos
- coseno de (x) en el grado (2) menos seno de (x) en el grado (2) menos raíz cuadrada de (2) dividir por 2
- coseno de (x) en el grado (dos) menos seno de (x) en el grado (dos) menos raíz cuadrada de (dos) dividir por dos
- cos(x)^(2)-sin(x)^(2)<√(2)/2
- cos(x)(2)-sin(x)(2)<sqrt(2)/2
- cosx2-sinx2<sqrt2/2
- cosx^2-sinx^2<sqrt2/2
- cos(x)^(2)-sin(x)^(2)<sqrt(2) dividir por 2
-
Expresiones semejantes
- cos(x)^(2)+sin(x)^(2)<sqrt(2)/2
- (sqrt2/2)^x*(1/16)^(-x+1)>4^x/64
- sin2x>sqrt2/2
- sin(8*x+7*pi/22)<=(-sqrt(2))/2
- cosx^(2)-sinx^(2)<sqrt(2)/2
-
Expresiones con funciones
- Coseno cos
- cos(x)>3/5
- cos(2x)+5cos(x)+3>=0
- cos>1/2
- cos(x\2)<=-0,5
- cos2x+cos3x>0
- Seno sin
- sin(x)/(x^2+3*x)>0
- sin(x+pi/4)>0
- sin(x)>=(-sqrt(3)/2)
- sin(x)>scrt3/2
- sin(x/7)>-1,5
- Raíz cuadrada sqrt
- sqrt(x+4)<=2-x
- sqrt2*sin(pi/4+x/2)>=1
- sqrt(8+2x-x^2)>6-3x
- sqrt(1-3x)+sqrt(6+2x)+sqrt(5+x)<=6
- sqrt(1+x)<=1
cos(x)^(2)-sin(x)^(2)
Solución
Ha introducido
[src]
___
2 2 \/ 2
cos (x) - sin (x) < -----
2
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
-sin(x)^2 + cos(x)^2 < sqrt(2)/2
Solución detallada
Se da la desigualdad:
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{\sqrt{2}}{2}$$
cambiamos
$$\cos{\left(2 x \right)} - \frac{\sqrt{2}}{2} = 0$$
$$- 2 \sin^{2}{\left(x \right)} - \frac{\sqrt{2}}{2} + 1 = 0$$
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
$$a = -2$$
$$b = 0$$
$$c = 1 - \frac{\sqrt{2}}{2}$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-2) * (1 - sqrt(2)/2) = 8 - 4*sqrt(2)
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{\sqrt{8 - 4 \sqrt{2}}}{4}$$
$$w_{2} = \frac{\sqrt{8 - 4 \sqrt{2}}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
Las raíces dadas
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10}$$
lo sustituimos en la expresión
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
$$- \sin^{2}{\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10} \right)} + \cos^{2}{\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10} \right)} < \frac{\sqrt{2}}{2}$$
/ / _________________________________________________________\\ / / _________________________________________________________\\
| | / ___________ ___________ || | | / ___________ ___________ ||
| | / ___ / ___ ___ / ___ || | | / ___ / ___ ___ / ___ || ___
2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || 2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || \/ 2
cos |-- + 2*atan|-------------------------------------------------------------|| - sin |-- + 2*atan|-------------------------------------------------------------|| < -----
|10 | _____________ || |10 | _____________ || 2
| | / ___ || | | / ___ ||
\ \ \/ 3 - 2*\/ 2 // \ \ \/ 3 - 2*\/ 2 //
pero
/ / _________________________________________________________\\ / / _________________________________________________________\\
| | / ___________ ___________ || | | / ___________ ___________ ||
| | / ___ / ___ ___ / ___ || | | / ___ / ___ ___ / ___ || ___
2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || 2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || \/ 2
cos |-- + 2*atan|-------------------------------------------------------------|| - sin |-- + 2*atan|-------------------------------------------------------------|| > -----
|10 | _____________ || |10 | _____________ || 2
| | / ___ || | | / ___ ||
\ \ \/ 3 - 2*\/ 2 // \ \ \/ 3 - 2*\/ 2 //
Entonces
$$x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
_____ _____
/ \ / \
-------ο-------ο-------ο-------ο-------
x1 x3 x4 x2
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x > 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
Solución de la desigualdad en el gráfico
Respuesta rápida 2
[src]
/ ___________\ / ___________\
| / ___ | | / ___ |
|\/ 2 - \/ 2 | |\/ 2 - \/ 2 |
(atan|--------------|, pi - atan|--------------|)
| ___________| | ___________|
| / ___ | | / ___ |
\\/ 2 + \/ 2 / \\/ 2 + \/ 2 /
$$x\ in\ \left(\operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}, \pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}\right)$$
x in Interval.open(atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2)), pi - atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2)))
Respuesta rápida
[src]
/ / ___________\ / ___________\ \
| | / ___ | | / ___ | |
| |\/ 2 - \/ 2 | |\/ 2 - \/ 2 | |
And|x < pi - atan|--------------|, atan|--------------| < x|
| | ___________| | ___________| |
| | / ___ | | / ___ | |
\ \\/ 2 + \/ 2 / \\/ 2 + \/ 2 / /
$$x < \pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)} \wedge \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)} < x$$
(atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))) < x)∧(x < pi - atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))))
Solución
Ha introducido
[src]
___
2 2 \/ 2
cos (x) - sin (x) < -----
2
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
-sin(x)^2 + cos(x)^2 < sqrt(2)/2
Solución detallada
Se da la desigualdad:
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{\sqrt{2}}{2}$$
cambiamos
$$\cos{\left(2 x \right)} - \frac{\sqrt{2}}{2} = 0$$
$$- 2 \sin^{2}{\left(x \right)} - \frac{\sqrt{2}}{2} + 1 = 0$$
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
$$a = -2$$
$$b = 0$$
$$c = 1 - \frac{\sqrt{2}}{2}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{\sqrt{8 - 4 \sqrt{2}}}{4}$$
$$w_{2} = \frac{\sqrt{8 - 4 \sqrt{2}}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
Las raíces dadas
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10}$$
lo sustituimos en la expresión
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
$$- \sin^{2}{\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10} \right)} + \cos^{2}{\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10} \right)} < \frac{\sqrt{2}}{2}$$
pero
Entonces
$$x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x > 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{\sqrt{2}}{2}$$
cambiamos
$$\cos{\left(2 x \right)} - \frac{\sqrt{2}}{2} = 0$$
$$- 2 \sin^{2}{\left(x \right)} - \frac{\sqrt{2}}{2} + 1 = 0$$
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
$$a = -2$$
$$b = 0$$
$$c = 1 - \frac{\sqrt{2}}{2}$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-2) * (1 - sqrt(2)/2) = 8 - 4*sqrt(2)
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{\sqrt{8 - 4 \sqrt{2}}}{4}$$
$$w_{2} = \frac{\sqrt{8 - 4 \sqrt{2}}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{8 - 4 \sqrt{2}}}{4} \right)} + \pi$$
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
Las raíces dadas
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10}$$
lo sustituimos en la expresión
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} < \frac{\sqrt{2}}{2}$$
$$- \sin^{2}{\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10} \right)} + \cos^{2}{\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} - \frac{1}{10} \right)} < \frac{\sqrt{2}}{2}$$
/ / _________________________________________________________\\ / / _________________________________________________________\\
| | / ___________ ___________ || | | / ___________ ___________ ||
| | / ___ / ___ ___ / ___ || | | / ___ / ___ ___ / ___ || ___
2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || 2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || \/ 2
cos |-- + 2*atan|-------------------------------------------------------------|| - sin |-- + 2*atan|-------------------------------------------------------------|| < -----
|10 | _____________ || |10 | _____________ || 2
| | / ___ || | | / ___ ||
\ \ \/ 3 - 2*\/ 2 // \ \ \/ 3 - 2*\/ 2 //
pero
/ / _________________________________________________________\\ / / _________________________________________________________\\
| | / ___________ ___________ || | | / ___________ ___________ ||
| | / ___ / ___ ___ / ___ || | | / ___ / ___ ___ / ___ || ___
2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || 2|1 |\/ 5 - 2*\/ 2 + 4*\/ 2 + \/ 2 - 2*\/ 2 *\/ 2 + \/ 2 || \/ 2
cos |-- + 2*atan|-------------------------------------------------------------|| - sin |-- + 2*atan|-------------------------------------------------------------|| > -----
|10 | _____________ || |10 | _____________ || 2
| | / ___ || | | / ___ ||
\ \ \/ 3 - 2*\/ 2 // \ \ \/ 3 - 2*\/ 2 //
Entonces
$$x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
_____ _____
/ \ / \
-------ο-------ο-------ο-------ο-------
x1 x3 x4 x2Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > - 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < - 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
$$x > 2 \operatorname{atan}{\left(\frac{\sqrt{- 4 \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 2 \sqrt{2} \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)} \wedge x < 2 \operatorname{atan}{\left(\frac{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{2} + 2} - 2 \sqrt{2} + 5 + 4 \sqrt{\sqrt{2} + 2}}}{\sqrt{3 - 2 \sqrt{2}}} \right)}$$
Solución de la desigualdad en el gráfico
Respuesta rápida 2
[src]
/ ___________\ / ___________\
| / ___ | | / ___ |
|\/ 2 - \/ 2 | |\/ 2 - \/ 2 |
(atan|--------------|, pi - atan|--------------|)
| ___________| | ___________|
| / ___ | | / ___ |
\\/ 2 + \/ 2 / \\/ 2 + \/ 2 /
$$x\ in\ \left(\operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}, \pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}\right)$$
x in Interval.open(atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2)), pi - atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2)))
Respuesta rápida
[src]
/ / ___________\ / ___________\ \ | | / ___ | | / ___ | | | |\/ 2 - \/ 2 | |\/ 2 - \/ 2 | | And|x < pi - atan|--------------|, atan|--------------| < x| | | ___________| | ___________| | | | / ___ | | / ___ | | \ \\/ 2 + \/ 2 / \\/ 2 + \/ 2 / /
$$x < \pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)} \wedge \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)} < x$$
(atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))) < x)∧(x < pi - atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))))