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Popular >

-cos(2x)-4sin^2(x)+2=0

Solución

- cos (2 x) - 4 sin^{2} (x) + 2 = 0

Solución

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Grados

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Pasos de solución

- cos (2 x) - 4 sin^{2} (x) + 2 = 0

Re-escribir usando identidades trigonométricas

2 - cos (2 x) - 4 sin^{2} (x)

Utilizar la identidad trigonométrica del ángulo doble:

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

= 2 - (1 - 2 sin^{2} (x)) - 4 sin^{2} (x)

Simplificar

2 - (1 - 2 sin^{2} (x)) - 4 sin^{2} (x) : - 2 sin^{2} (x) + 1

2 - (1 - 2 sin^{2} (x)) - 4 sin^{2} (x)

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

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

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

= 2 - 1 + 2 sin^{2} (x) - 4 sin^{2} (x)

Simplificar

2 - 1 + 2 sin^{2} (x) - 4 sin^{2} (x) : - 2 sin^{2} (x) + 1

2 - 1 + 2 sin^{2} (x) - 4 sin^{2} (x)

Sumar elementos similares:

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

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

Restar:

2 - 1 = 1

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

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

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

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

Usando el método de sustitución

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

Sea:

sin (x) = u

1 - 2 u^{2} = 0

1 - 2 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

1 - 2 u^{2} = 0

Mover

1

al lado derecho

1 - 2 u^{2} = 0

Restar

1

de ambos lados

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

Simplificar

- 2 u^{2} = - 1

- 2 u^{2} = - 1

Dividir ambos lados entre

- 2

- 2 u^{2} = - 1

Dividir ambos lados entre

- 2

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

Simplificar

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

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

Para

x^{2} = f (a)

las soluciones son

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

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

Sustituir en la ecuación

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

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

sin (x) = \frac{1}{2}

Soluciones generales para

sin (x) = \frac{1}{2}

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

Soluciones generales para

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

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combinar toda las soluciones

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Ejemplos populares

sec(θ)-2cos(θ)=0

sec (θ) - 2 cos (θ) = 0

sin(θ)= 1/(sqrt(2)),0<= θ<= 4pi

sin (θ) = \frac{1}{2}, 0 \leq θ \leq 4 π

8sin(θ)cos(θ)tan(θ)=csc(θ)

8 sin (θ) cos (θ) tan (θ) = csc (θ)

(cot(θ))/(sec(θ))=sin(θ)

\frac{cot ( θ )}{sec ( θ )} = sin (θ)

16sin(2x)=0

16 sin (2 x) = 0