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cos(θ)= 2/3 \land sin(θ)<0

Solución

cos (θ) = \frac{2}{3} and sin (θ) < 0

θ = 5.44211 \dots + 2 πn

Notación de intervalos

θ = 5.44211 \dots + 2 πn

Pasos de solución

cos (θ) = \frac{2}{3} and sin (θ) < 0

cos (θ) = \frac{2}{3} : θ = 0.84106 \dots + 2 πn, θ = 2 π - 0.84106 \dots + 2 πn

cos (θ) = \frac{2}{3}

Aplicar propiedades trigonométricas inversas

cos (θ) = \frac{2}{3}

Soluciones generales para

cos (θ) = \frac{2}{3}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

θ = arccos (\frac{2}{3}) + 2 πn, θ = 2 π - arccos (\frac{2}{3}) + 2 πn

θ = arccos (\frac{2}{3}) + 2 πn, θ = 2 π - arccos (\frac{2}{3}) + 2 πn

Mostrar soluciones en forma decimal

θ = 0.84106 \dots + 2 πn, θ = 2 π - 0.84106 \dots + 2 πn

sin (θ) < 0 : - π + 2 πn < θ < 2 πn

sin (θ) < 0

Para

sin (x) < a

, si

- 1 < a \leq 1

entonces

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (0) + 2 πn < θ < arcsin (0) + 2 πn

Simplificar

- π - arcsin (0) : - π

- π - arcsin (0)

Utilizar la siguiente identidad trivial:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= - π - 0

- π - 0 = - π

= - π

Simplificar

arcsin (0) : 0

arcsin (0)

Utilizar la siguiente identidad trivial:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0

- π + 2 πn < θ < 0 + 2 πn

Simplificar

- π + 2 πn < θ < 2 πn

Combinar los rangos

(θ = 0.84106 \dots + 2 πn or θ = 2 π - 0.84106 \dots + 2 πn) and - π + 2 πn < θ < 2 πn

Mezclar intervalos sobrepuestos

θ = 5.44211 \dots + 2 πn