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العربية

tan(x)= 3/(10cos(x)),0<x<90

Solución

tan (x) = \frac{3}{10 cos ( x )}, 0^{\circ} < x < 9 0^{\circ}

x = 0.30469 \dots

Pasos de solución

tan (x) = \frac{3}{10 cos ( x )}, 0^{\circ} < x < 9 0^{\circ}

Restar

\frac{3}{10 cos ( x )}

de ambos lados

tan (x) - \frac{3}{10 cos ( x )} = 0

Simplificar

tan (x) - \frac{3}{10 cos ( x )} : \frac{10 tan ( x ) cos ( x ) - 3}{10 cos ( x )}

tan (x) - \frac{3}{10 cos ( x )}

Convertir a fracción:

tan (x) = \frac{tan ( x ) 10 cos ( x )}{10 cos ( x )}

= \frac{tan ( x ) \cdot 10 cos ( x )}{10 cos ( x )} - \frac{3}{10 cos ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{tan ( x ) \cdot 10 cos ( x ) - 3}{10 cos ( x )}

\frac{10 tan ( x ) cos ( x ) - 3}{10 cos ( x )} = 0

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

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

Re-escribir usando identidades trigonométricas

10 tan (x) cos (x) - 3

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)

= - 3 + 10 sin (x)

- 3 + 10 sin (x) = 0

Desplace

3

a la derecha

- 3 + 10 sin (x) = 0

Sumar

3

a ambos lados

- 3 + 10 sin (x) + 3 = 0 + 3

Simplificar

10 sin (x) = 3

10 sin (x) = 3

Dividir ambos lados entre

10

10 sin (x) = 3

Dividir ambos lados entre

10

\frac{10 sin ( x )}{10} = \frac{3}{10}

Simplificar

sin (x) = \frac{3}{10}

sin (x) = \frac{3}{10}

Aplicar propiedades trigonométricas inversas

sin (x) = \frac{3}{10}

Soluciones generales para

sin (x) = \frac{3}{10}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{3}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{10}) + 36 0^{\circ} n

x = arcsin (\frac{3}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{10}) + 36 0^{\circ} n

Soluciones para el rango

0 < x < 9 0^{\circ}

x = arcsin (\frac{3}{10})

Mostrar soluciones en forma decimal

x = 0.30469 \dots