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tan(x)+4sin(x)=0

Solución

tan (x) + 4 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = - 104.47751 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

tan (x) + 4 sin (x) = 0

Expresar con seno, coseno

tan (x) + 4 sin (x)

Utilizar la identidad trigonométrica básica:

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

= \frac{sin ( x )}{cos ( x )} + 4 sin (x)

Simplificar

\frac{sin ( x )}{cos ( x )} + 4 sin (x) : \frac{sin ( x ) + 4 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} + 4 sin (x)

Convertir a fracción:

4 sin (x) = \frac{4 sin ( x ) cos ( x )}{cos ( x )}

= \frac{sin ( x )}{cos ( x )} + \frac{4 sin ( x ) cos ( x )}{cos ( 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 ) + 4 sin ( x ) cos ( x )}{cos ( x )}

= \frac{sin ( x ) + 4 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x ) + 4 cos ( x ) sin ( x )}{cos ( x )} = 0

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

sin (x) + 4 cos (x) sin (x) = 0

Factorizar

sin (x) + 4 cos (x) sin (x) : sin (x) (4 cos (x) + 1)

sin (x) + 4 cos (x) sin (x)

Factorizar el termino común

sin (x)

= sin (x) (1 + 4 cos (x))

sin (x) (4 cos (x) + 1) = 0

Resolver cada parte por separado

sin (x) = 0 or 4 cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

tabla de valores periódicos con

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

4 cos (x) + 1 = 0 : x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

4 cos (x) + 1 = 0

Mover

1

al lado derecho

4 cos (x) + 1 = 0

Restar

1

de ambos lados

4 cos (x) + 1 - 1 = 0 - 1

Simplificar

4 cos (x) = - 1

4 cos (x) = - 1

Dividir ambos lados entre

4

4 cos (x) = - 1

Dividir ambos lados entre

4

\frac{4 cos ( x )}{4} = \frac{- 1}{4}

Simplificar

cos (x) = - \frac{1}{4}

cos (x) = - \frac{1}{4}

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{1}{4}

Soluciones generales para

cos (x) = - \frac{1}{4}

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

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

Mostrar soluciones en forma decimal

x = 2 πn, x = π + 2 πn, x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

1 + 2 \cdot sin (x) = 2

1 + 2 \cdot sin (x) = 3

sin (α) = \frac{120 sin ( 6 0 ^{\circ} )}{1200}

15.8 \cdot sin (30 x - 90) + 5 = 17

2 sin^{(2)} (x) - sin (x) - 1 = 0