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3tan(x)= 1/(cos(x))

Solución

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

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

Grados

x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Restar

\frac{1}{cos ( x )}

de ambos lados

3 tan (x) - \frac{1}{cos ( x )} = 0

Simplificar

3 tan (x) - \frac{1}{cos ( x )} : \frac{3 tan ( x ) cos ( x ) - 1}{cos ( x )}

3 tan (x) - \frac{1}{cos ( x )}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

\frac{3 tan ( x ) cos ( x ) - 1}{cos ( x )} = 0

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

3 tan (x) cos (x) - 1 = 0

Re-escribir usando identidades trigonométricas

3 tan (x) cos (x) - 1

cos (x) tan (x) = sin (x)

cos (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= sin (x)

= sin (x)

= - 1 + 3 sin (x)

- 1 + 3 sin (x) = 0

Mover

1

al lado derecho

- 1 + 3 sin (x) = 0

Sumar

1

a ambos lados

- 1 + 3 sin (x) + 1 = 0 + 1

Simplificar

3 sin (x) = 1

3 sin (x) = 1

Dividir ambos lados entre

3

3 sin (x) = 1

Dividir ambos lados entre

3

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

Simplificar

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

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

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

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

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

Mostrar soluciones en forma decimal

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

cos (x) - sin (x) = 0, - π \leq x \leq π

5 - tan^{2} (x) = 4

2 sec^{2} (θ) - 3 = tan^{2} (θ)

sin (π - x) + cos (x) = 0

sin (x) = - sin (2 x)