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Popular >

1-cos(x)=2+cos(x)

Solución

1 - cos (x) = 2 + cos (x)

Solución

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

Grados

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Pasos de solución

1 - cos (x) = 2 + cos (x)

Usando el método de sustitución

1 - cos (x) = 2 + cos (x)

Sea:

cos (x) = u

1 - u = 2 + u

1 - u = 2 + u : u = - \frac{1}{2}

1 - u = 2 + u

Mover

1

al lado derecho

1 - u = 2 + u

Restar

1

de ambos lados

1 - u - 1 = 2 + u - 1

Simplificar

- u = u + 1

- u = u + 1

Mover

u

al lado izquierdo

- u = u + 1

Restar

u

de ambos lados

- u - u = u + 1 - u

Simplificar

- 2 u = 1

- 2 u = 1

Dividir ambos lados entre

- 2

- 2 u = 1

Dividir ambos lados entre

- 2

\frac{- 2 u}{- 2} = \frac{1}{- 2}

Simplificar

u = - \frac{1}{2}

u = - \frac{1}{2}

Sustituir en la ecuación

u = cos (x)

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

Soluciones generales para

cos (x) = - \frac{1}{2}

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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combinar toda las soluciones

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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