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+8)(x-5)>0
-
(x+8)*(x-5)>0
- (x-7)^2<sqrt11(x-7)
-
x^2-4*x+3>0
- Derivada de:
-
cos^23x
- Integral de d{x}:
- cos^23x
-
Expresiones idénticas
- cos^23x>= uno
- coseno de al cuadrado 3x más o igual a 1
- coseno de al cuadrado 3x más o igual a uno
- cos23x>=1
- cos²3x>=1
- cos en el grado 23x>=1
-
Expresiones con funciones
- Coseno cos
- cos^2(x)<sin(x)
- cos(2*x)-2cos(x)>0
- cos(x-(pi/6))>=-1/2
- cos(x+pi/3)>=1/2
- cos(2*pi*x)>1
cos^23x>=1 desigualdades
Solución
Solución detallada
Se da la desigualdad:
$$\cos^{23}{\left(x \right)} \geq 1$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\cos^{23}{\left(x \right)} = 1$$
Resolvemos:
Tenemos la ecuación
$$\cos^{23}{\left(x \right)} = 1$$
cambiamos
$$\cos^{23}{\left(x \right)} - 1 = 0$$
$$\cos^{23}{\left(x \right)} - 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
Tenemos la ecuación
$$w^{23} - 1 = 0$$
Ya que la potencia en la ecuación es igual a = 23 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 23 de las dos partes de la ecuación:
Obtenemos:
$$\sqrt[23]{w^{23}} = \sqrt[23]{1}$$
o
$$w = 1$$
Obtenemos la respuesta: w = 1
Las demás 22 raíces son complejas.
hacemos el cambio:
$$z = w$$
entonces la ecuación será así:
$$z^{23} = 1$$
Cualquier número complejo se puede presentar que:
$$z = r e^{i p}$$
sustituimos en la ecuación
$$r^{23} e^{23 i p} = 1$$
donde
$$r = 1$$
- módulo del número complejo
Sustituyamos r:
$$e^{23 i p} = 1$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(23 p \right)} + \cos{\left(23 p \right)} = 1$$
es decir
$$\cos{\left(23 p \right)} = 1$$
y
$$\sin{\left(23 p \right)} = 0$$
entonces
$$p = \frac{2 \pi N}{23}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
$$z_{1} = 1$$
$$z_{2} = - \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)}$$
$$z_{3} = - \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)}$$
$$z_{4} = \cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$z_{5} = \cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$z_{6} = - \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$z_{7} = - \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$z_{8} = \cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$z_{9} = \cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$z_{10} = - \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$z_{11} = - \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$z_{12} = \cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$z_{13} = \cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$z_{14} = - \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$z_{15} = - \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$z_{16} = \cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$z_{17} = \cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$z_{18} = - \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$z_{19} = - \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$z_{20} = \cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$z_{21} = \cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$z_{22} = - \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)}$$
$$z_{23} = - \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)}$$
hacemos cambio inverso
$$z = w$$
$$w = z$$
Entonces la respuesta definitiva es:
$$w_{1} = 1$$
$$w_{2} = - \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)}$$
$$w_{3} = - \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)}$$
$$w_{4} = \cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$w_{5} = \cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$w_{6} = - \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$w_{7} = - \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$w_{8} = \cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$w_{9} = \cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$w_{10} = - \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$w_{11} = - \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$w_{12} = \cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$w_{13} = \cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$w_{14} = - \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$w_{15} = - \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$w_{16} = \cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$w_{17} = \cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$w_{18} = - \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$w_{19} = - \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$w_{20} = \cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$w_{21} = \cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$w_{22} = - \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)}$$
$$w_{23} = - \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n - \pi$$
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
$$x_{3} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{4} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{5} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{6} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{7} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{8} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{9} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{10} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{11} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{12} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{13} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{14} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{15} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{16} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{17} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{18} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{19} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{20} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{21} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{22} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{23} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
$$x_{24} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
$$x_{25} = \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{26} = \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{27} = \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{28} = \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{29} = \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{30} = \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{31} = \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{32} = \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{33} = \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{34} = \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{35} = \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{36} = \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{37} = \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{38} = \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{39} = \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{40} = \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{41} = \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{42} = \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{43} = \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{44} = \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{45} = \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
$$x_{46} = \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
Descartamos las soluciones complejas:
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
Las raíces dadas
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
lo sustituimos en la expresión
$$\cos^{23}{\left(x \right)} \geq 1$$
$$\cos^{23}{\left(- \frac{1}{10} \right)} \geq 1$$
pero
Entonces
$$x \leq 0$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq 0 \wedge x \leq 2 \pi$$
$$\cos^{23}{\left(x \right)} \geq 1$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\cos^{23}{\left(x \right)} = 1$$
Resolvemos:
Tenemos la ecuación
$$\cos^{23}{\left(x \right)} = 1$$
cambiamos
$$\cos^{23}{\left(x \right)} - 1 = 0$$
$$\cos^{23}{\left(x \right)} - 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
Tenemos la ecuación
$$w^{23} - 1 = 0$$
Ya que la potencia en la ecuación es igual a = 23 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 23 de las dos partes de la ecuación:
Obtenemos:
$$\sqrt[23]{w^{23}} = \sqrt[23]{1}$$
o
$$w = 1$$
Obtenemos la respuesta: w = 1
Las demás 22 raíces son complejas.
hacemos el cambio:
$$z = w$$
entonces la ecuación será así:
$$z^{23} = 1$$
Cualquier número complejo se puede presentar que:
$$z = r e^{i p}$$
sustituimos en la ecuación
$$r^{23} e^{23 i p} = 1$$
donde
$$r = 1$$
- módulo del número complejo
Sustituyamos r:
$$e^{23 i p} = 1$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(23 p \right)} + \cos{\left(23 p \right)} = 1$$
es decir
$$\cos{\left(23 p \right)} = 1$$
y
$$\sin{\left(23 p \right)} = 0$$
entonces
$$p = \frac{2 \pi N}{23}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
$$z_{1} = 1$$
$$z_{2} = - \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)}$$
$$z_{3} = - \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)}$$
$$z_{4} = \cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$z_{5} = \cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$z_{6} = - \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$z_{7} = - \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$z_{8} = \cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$z_{9} = \cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$z_{10} = - \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$z_{11} = - \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$z_{12} = \cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$z_{13} = \cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$z_{14} = - \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$z_{15} = - \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$z_{16} = \cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$z_{17} = \cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$z_{18} = - \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$z_{19} = - \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$z_{20} = \cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$z_{21} = \cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$z_{22} = - \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)}$$
$$z_{23} = - \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)}$$
hacemos cambio inverso
$$z = w$$
$$w = z$$
Entonces la respuesta definitiva es:
$$w_{1} = 1$$
$$w_{2} = - \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)}$$
$$w_{3} = - \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)}$$
$$w_{4} = \cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$w_{5} = \cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)}$$
$$w_{6} = - \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$w_{7} = - \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)}$$
$$w_{8} = \cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$w_{9} = \cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)}$$
$$w_{10} = - \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$w_{11} = - \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)}$$
$$w_{12} = \cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$w_{13} = \cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)}$$
$$w_{14} = - \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$w_{15} = - \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)}$$
$$w_{16} = \cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$w_{17} = \cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)}$$
$$w_{18} = - \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$w_{19} = - \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)}$$
$$w_{20} = \cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$w_{21} = \cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)}$$
$$w_{22} = - \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)}$$
$$w_{23} = - \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n - \pi$$
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
$$x_{3} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{4} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{5} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{6} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{7} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{8} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{9} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{10} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{11} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{12} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{13} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{14} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{15} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{16} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{17} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{18} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{19} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{20} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{21} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{22} = 2 \pi - \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{23} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
$$x_{24} = 2 \pi - \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
$$x_{25} = \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} - i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{26} = \operatorname{acos}{\left(- \cos{\left(\frac{\pi}{23} \right)} + i \sin{\left(\frac{\pi}{23} \right)} \right)}$$
$$x_{27} = \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} - i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{28} = \operatorname{acos}{\left(\cos{\left(\frac{2 \pi}{23} \right)} + i \sin{\left(\frac{2 \pi}{23} \right)} \right)}$$
$$x_{29} = \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} - i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{30} = \operatorname{acos}{\left(- \cos{\left(\frac{3 \pi}{23} \right)} + i \sin{\left(\frac{3 \pi}{23} \right)} \right)}$$
$$x_{31} = \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} - i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{32} = \operatorname{acos}{\left(\cos{\left(\frac{4 \pi}{23} \right)} + i \sin{\left(\frac{4 \pi}{23} \right)} \right)}$$
$$x_{33} = \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} - i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{34} = \operatorname{acos}{\left(- \cos{\left(\frac{5 \pi}{23} \right)} + i \sin{\left(\frac{5 \pi}{23} \right)} \right)}$$
$$x_{35} = \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} - i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{36} = \operatorname{acos}{\left(\cos{\left(\frac{6 \pi}{23} \right)} + i \sin{\left(\frac{6 \pi}{23} \right)} \right)}$$
$$x_{37} = \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} - i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{38} = \operatorname{acos}{\left(- \cos{\left(\frac{7 \pi}{23} \right)} + i \sin{\left(\frac{7 \pi}{23} \right)} \right)}$$
$$x_{39} = \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} - i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{40} = \operatorname{acos}{\left(\cos{\left(\frac{8 \pi}{23} \right)} + i \sin{\left(\frac{8 \pi}{23} \right)} \right)}$$
$$x_{41} = \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} - i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{42} = \operatorname{acos}{\left(- \cos{\left(\frac{9 \pi}{23} \right)} + i \sin{\left(\frac{9 \pi}{23} \right)} \right)}$$
$$x_{43} = \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} - i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{44} = \operatorname{acos}{\left(\cos{\left(\frac{10 \pi}{23} \right)} + i \sin{\left(\frac{10 \pi}{23} \right)} \right)}$$
$$x_{45} = \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} - i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
$$x_{46} = \operatorname{acos}{\left(- \cos{\left(\frac{11 \pi}{23} \right)} + i \sin{\left(\frac{11 \pi}{23} \right)} \right)}$$
Descartamos las soluciones complejas:
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
Las raíces dadas
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
lo sustituimos en la expresión
$$\cos^{23}{\left(x \right)} \geq 1$$
$$\cos^{23}{\left(- \frac{1}{10} \right)} \geq 1$$
23
cos (1/10) >= 1
pero
23
cos (1/10) < 1
Entonces
$$x \leq 0$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq 0 \wedge x \leq 2 \pi$$
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x1 x2
Solución de la desigualdad en el gráfico