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sin(x)= 1/(sqrt(5))\land cos(x)<0

Solución

sin (x) = \frac{1}{5} and cos (x) < 0

x = π - 0.46364 \dots + 2 πn

Decimal

x = 2.67794 \dots + 2 πn

Pasos de solución

sin (x) = \frac{1}{5} and cos (x) < 0

sin (x) = \frac{1}{5} : x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

sin (x) = \frac{1}{5}

Simplificar

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

Multiplicar por el conjugado

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Aplicar las leyes de los exponentes:

a a = a

55 = 5

= 5

= \frac{5}{5}

sin (x) = \frac{5}{5}

Aplicar propiedades trigonométricas inversas

sin (x) = \frac{5}{5}

Soluciones generales para

sin (x) = \frac{5}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{5}{5}) + 2 πn, x = π - arcsin (\frac{5}{5}) + 2 πn

x = arcsin (\frac{5}{5}) + 2 πn, x = π - arcsin (\frac{5}{5}) + 2 πn

Mostrar soluciones en forma decimal

x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

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

cos (x) < 0

Para

cos (x) < a

, si

- 1 < a \leq 1

entonces

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

arccos (0) + 2 πn < x < 2 π - arccos (0) + 2 πn

Simplificar

arccos (0) : \frac{π}{2}

arccos (0)

Utilizar la siguiente identidad trivial:

arccos (0) = \frac{π}{2}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{2}

Simplificar

2 π - arccos (0) : \frac{3 π}{2}

2 π - arccos (0)

Utilizar la siguiente identidad trivial:

arccos (0) = \frac{π}{2}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{2}

Simplificar

2 π - \frac{π}{2}

Convertir a fracción:

2 π = \frac{2 π 2}{2}

= \frac{2 π 2}{2} - \frac{π}{2}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 π 2 - π}{2}

2 π 2 - π = 3 π

2 π 2 - π

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 π - π

Sumar elementos similares:

4 π - π = 3 π

= 3 π

= \frac{3 π}{2}

= \frac{3 π}{2}

\frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn

Combinar los rangos

(x = 0.46364 \dots + 2 πn or x = π - 0.46364 \dots + 2 πn) and \frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn

Mezclar intervalos sobrepuestos

x = π - 0.46364 \dots + 2 πn