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cos^2(x)+cos^3(x)+cos^4(x)+cos^5(x)=0

Solución

cos^{2} (x) + cos^{3} (x) + cos^{4} (x) + cos^{5} (x) = 0

Solución

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

Grados

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

cos^{2} (x) + cos^{3} (x) + cos^{4} (x) + cos^{5} (x) = 0

Usando el método de sustitución

cos^{2} (x) + cos^{3} (x) + cos^{4} (x) + cos^{5} (x) = 0

Sea:

cos (x) = u

u^{2} + u^{3} + u^{4} + u^{5} = 0

u^{2} + u^{3} + u^{4} + u^{5} = 0 : u = 0, u = - 1, u = i, u = - i

u^{2} + u^{3} + u^{4} + u^{5} = 0

Factorizar

u^{2} + u^{3} + u^{4} + u^{5} : u^{2} (u + 1) (u^{2} + 1)

u^{2} + u^{3} + u^{4} + u^{5}

Factorizar el termino común

u^{2} : u^{2} (u^{3} + u^{2} + u + 1)

u^{5} + u^{4} + u^{3} + u^{2}

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

u^{3} = u u^{2}

= u^{3} u^{2} + u^{2} u^{2} + u u^{2} + u^{2}

Factorizar el termino común

u^{2}

= u^{2} (u^{3} + u^{2} + u + 1)

= u^{2} (u^{3} + u^{2} + u + 1)

Factorizar

u^{3} + u^{2} + u + 1 : (u + 1) (u^{2} + 1)

u^{3} + u^{2} + u + 1

= (u^{3} + u^{2}) + (u + 1)

Factorizar

u^{2}

u^{3} + u^{2}

u^{2} (u + 1)

u^{3} + u^{2}

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

u^{3} = u u^{2}

= u u^{2} + u^{2}

Factorizar el termino común

u^{2}

= u^{2} (u + 1)

= (u + 1) + u^{2} (u + 1)

Factorizar el termino común

u + 1

= (u + 1) (u^{2} + 1)

= u^{2} (u + 1) (u^{2} + 1)

u^{2} (u + 1) (u^{2} + 1) = 0

Utilizando el teorema de factor cero: si

ab = 0

entonces

a = 0

b = 0

u = 0 or u + 1 = 0 or u^{2} + 1 = 0

Resolver

u + 1 = 0 : u = - 1

u + 1 = 0

Mover

1

al lado derecho

u + 1 = 0

Restar

1

de ambos lados

u + 1 - 1 = 0 - 1

Simplificar

u = - 1

u = - 1

Resolver

u^{2} + 1 = 0 : u = i, u = - i

u^{2} + 1 = 0

Mover

1

al lado derecho

u^{2} + 1 = 0

Restar

1

de ambos lados

u^{2} + 1 - 1 = 0 - 1

Simplificar

u^{2} = - 1

u^{2} = - 1

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = - 1, u = - - 1

Simplificar

- 1 : i

- 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= i

Simplificar

- - 1 : - i

- - 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= - i

u = i, u = - i

Las soluciones son

u = 0, u = - 1, u = i, u = - i

Sustituir en la ecuación

u = cos (x)

cos (x) = 0, cos (x) = - 1, cos (x) = i, cos (x) = - i

cos (x) = 0, cos (x) = - 1, cos (x) = i, cos (x) = - i

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

Soluciones generales para

cos (x) = 0

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

Soluciones generales para

cos (x) = - 1

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

cos (x) = i :

Sin solución

cos (x) = i

Sin soluci \overset{o}{ˊ} n

cos (x) = - i :

Sin solución

cos (x) = - i

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

Ejemplos populares

sin^2(x)-4sin(x)=-3

sin^{2} (x) - 4 sin (x) = - 3

1+7sinh(x)=4cosh^2(x)

1 + 7 sinh (x) = 4 cosh^{2} (x)

sin^2(x)-1+2cos(2x)-cos^2(x)=0

sin^{2} (x) - 1 + 2 cos (2 x) - cos^{2} (x) = 0

cos^2(x)+6cos(x)+5=0

cos^{2} (x) + 6 cos (x) + 5 = 0

sin(t)=-0.9397

sin (t) = - 0.9397