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3cos^2(x)-cos(2x)=1

Solución

3 cos^{2} (x) - cos (2 x) = 1

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

Grados

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

Pasos de solución

3 cos^{2} (x) - cos (2 x) = 1

Restar

1

de ambos lados

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

Re-escribir usando identidades trigonométricas

- 1 - cos (2 x) + 3 cos^{2} (x)

Utilizar la identidad trigonométrica del ángulo doble:

cos (2 x) = 2 cos^{2} (x) - 1

= - 1 - (2 cos^{2} (x) - 1) + 3 cos^{2} (x)

Simplificar

- 1 - (2 cos^{2} (x) - 1) + 3 cos^{2} (x) : cos^{2} (x)

- 1 - (2 cos^{2} (x) - 1) + 3 cos^{2} (x)

- (2 cos^{2} (x) - 1) : - 2 cos^{2} (x) + 1

- (2 cos^{2} (x) - 1)

Poner los parentesis

= - (2 cos^{2} (x)) - (- 1)

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

= - 1 - 2 cos^{2} (x) + 1 + 3 cos^{2} (x)

Simplificar

- 1 - 2 cos^{2} (x) + 1 + 3 cos^{2} (x) : cos^{2} (x)

- 1 - 2 cos^{2} (x) + 1 + 3 cos^{2} (x)

Agrupar términos semejantes

= - 2 cos^{2} (x) + 3 cos^{2} (x) - 1 + 1

Sumar elementos similares:

- 2 cos^{2} (x) + 3 cos^{2} (x) = cos^{2} (x)

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

- 1 + 1 = 0

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

cos^{2} (x) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

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