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sin(x)= 1/(sec(x))

Solución

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

x = \frac{π}{4} + πn

Grados

x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

\frac{1}{sec ( x )}

de ambos lados

sin (x) - \frac{1}{sec ( x )} = 0

Simplificar

sin (x) - \frac{1}{sec ( x )} : \frac{sin ( x ) sec ( x ) - 1}{sec ( x )}

sin (x) - \frac{1}{sec ( x )}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

\frac{sin ( x ) sec ( x ) - 1}{sec ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (x) sec (x) - 1 = 0

Re-escribir usando identidades trigonométricas

sin (x) sec (x) - 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Expresar con seno, coseno

sec (x) sin (x)

Utilizar la identidad trigonométrica básica:

sec (x) = \frac{1}{cos ( x )}

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

Simplificar

\frac{1}{cos ( x )} sin (x) : \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} sin (x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

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

Utilizar la identidad trigonométrica básica:

\frac{sin ( x )}{cos ( x )} = tan (x)

= tan (x)

= - 1 + tan (x)

- 1 + tan (x) = 0

Mover

1

al lado derecho

- 1 + tan (x) = 0

Sumar

1

a ambos lados

- 1 + tan (x) + 1 = 0 + 1

Simplificar

tan (x) = 1

tan (x) = 1

Soluciones generales para

tan (x) = 1

tan (x)

tabla de valores periódicos con

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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

1 = tan (\frac{x}{3})

cos (12 x) = 0

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

2 ∣ sin (4 x) ∣ = 0