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Popular Trigonometría >

sqrt(3)tan(x)-2sin(x)=0

Solución

3 tan (x) - 2 sin (x) = 0

Solución

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Pasos de solución

3 tan (x) - 2 sin (x) = 0

Expresar con seno, coseno

- 2 sin (x) + 3 tan (x)

Utilizar la identidad trigonométrica básica:

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

= - 2 sin (x) + 3 \frac{sin ( x )}{cos ( x )}

Simplificar

- 2 sin (x) + 3 \frac{sin ( x )}{cos ( x )} : \frac{- 2 sin ( x ) cos ( x ) + 3 sin ( x )}{cos ( x )}

- 2 sin (x) + 3 \frac{sin ( x )}{cos ( x )}

Multiplicar

3 \frac{sin ( x )}{cos ( x )} : \frac{3 sin ( x )}{cos ( x )}

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

Multiplicar fracciones:

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

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

= - 2 sin (x) + \frac{3 sin ( x )}{cos ( x )}

Convertir a fracción:

2 sin (x) = \frac{2 sin ( x ) cos ( x )}{cos ( x )}

= - \frac{2 sin ( x ) cos ( x )}{cos ( x )} + \frac{sin ( x ) 3}{cos ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 2 sin ( x ) cos ( x ) + sin ( x ) 3}{cos ( x )}

= \frac{- 2 sin ( x ) cos ( x ) + 3 sin ( x )}{cos ( x )}

\frac{sin ( x ) 3 - 2 cos ( x ) sin ( x )}{cos ( x )} = 0

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

sin (x) 3 - 2 cos (x) sin (x) = 0

Factorizar

sin (x) 3 - 2 cos (x) sin (x) : sin (x) (3 - 2 cos (x))

sin (x) 3 - 2 cos (x) sin (x)

Factorizar el termino común

sin (x)

= sin (x) (3 - 2 cos (x))

sin (x) (3 - 2 cos (x)) = 0

Resolver cada parte por separado

sin (x) = 0 or 3 - 2 cos (x) = 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

3 - 2 cos (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

3 - 2 cos (x) = 0

Desplace

3

a la derecha

3 - 2 cos (x) = 0

Restar

3

de ambos lados

3 - 2 cos (x) - 3 = 0 - 3

Simplificar

- 2 cos (x) = - 3

- 2 cos (x) = - 3

Dividir ambos lados entre

- 2

- 2 cos (x) = - 3

Dividir ambos lados entre

- 2

\frac{- 2 cos ( x )}{- 2} = \frac{- 3}{- 2}

Simplificar

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

Soluciones generales para

cos (x) = \frac{3}{2}

cos (x)

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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Gráfica

Ejemplos populares

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