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Popular Trigonometría >

3sin(2x+30)=tan(2x+30)

Solución

3 sin (2 x + 30) = tan (2 x + 30)

Solución

x = πn - 15, x = \frac{π}{2} + πn - 15, x = \frac{1.23095 \dots}{2} + πn - 15, x = π - \frac{1.23095 \dots}{2} + πn - 15

Grados

x = - 859.43669 \dots^{\circ} + 18 0^{\circ} n, x = - 769.43669 \dots^{\circ} + 18 0^{\circ} n, x = - 824.17230 \dots^{\circ} + 18 0^{\circ} n, x = - 714.70108 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

3 sin (2 x + 30) = tan (2 x + 30)

Restar

tan (2 x + 30)

de ambos lados

3 sin (2 x + 30) - tan (2 x + 30) = 0

Expresar con seno, coseno

- tan (30 + 2 x) + 3 sin (30 + 2 x)

Utilizar la identidad trigonométrica básica:

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

= - \frac{sin ( 30 + 2 x )}{cos ( 30 + 2 x )} + 3 sin (30 + 2 x)

Simplificar

- \frac{sin ( 30 + 2 x )}{cos ( 30 + 2 x )} + 3 sin (30 + 2 x) : \frac{- sin ( 30 + 2 x ) + 3 sin ( 30 + 2 x ) cos ( 30 + 2 x )}{cos ( 30 + 2 x )}

- \frac{sin ( 30 + 2 x )}{cos ( 30 + 2 x )} + 3 sin (30 + 2 x)

Convertir a fracción:

3 sin (2 x + 30) = \frac{3 sin ( 30 + 2 x ) cos ( 30 + 2 x )}{cos ( 30 + 2 x )}

= - \frac{sin ( 30 + 2 x )}{cos ( 30 + 2 x )} + \frac{3 sin ( 30 + 2 x ) cos ( 30 + 2 x )}{cos ( 30 + 2 x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- sin ( 30 + 2 x ) + 3 sin ( 30 + 2 x ) cos ( 30 + 2 x )}{cos ( 30 + 2 x )}

= \frac{- sin ( 30 + 2 x ) + 3 sin ( 30 + 2 x ) cos ( 30 + 2 x )}{cos ( 30 + 2 x )}

\frac{- sin ( 30 + 2 x ) + 3 cos ( 30 + 2 x ) sin ( 30 + 2 x )}{cos ( 30 + 2 x )} = 0

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

- sin (30 + 2 x) + 3 cos (30 + 2 x) sin (30 + 2 x) = 0

Factorizar

- sin (30 + 2 x) + 3 cos (30 + 2 x) sin (30 + 2 x) : sin (2 (x + 15)) (3 cos (2 (x + 15)) - 1)

- sin (30 + 2 x) + 3 cos (30 + 2 x) sin (30 + 2 x)

Factorizar el termino común

sin (30 + 2 x)

= sin (30 + 2 x) (- 1 + 3 cos (30 + 2 x))

Factorizar

2 x + 30 : 2 (x + 15)

2 x + 30

Factorizar el termino común

2 : 2 (x + 15)

2 x + 30

Reescribir

30

como

2 \cdot 15

= 2 x + 2 \cdot 15

Factorizar el termino común

2

= 2 (x + 15)

= 2 (x + 15)

= sin (2 x + 30) (3 cos (2 (x + 15)) - 1)

Factorizar

2 x + 30 : 2 (x + 15)

2 x + 30

Factorizar el termino común

2 : 2 (x + 15)

2 x + 30

Reescribir

30

como

2 \cdot 15

= 2 x + 2 \cdot 15

Factorizar el termino común

2

= 2 (x + 15)

= 2 (x + 15)

= sin (2 (x + 15)) (3 cos (2 (x + 15)) - 1)

sin (2 (x + 15)) (3 cos (2 (x + 15)) - 1) = 0

Resolver cada parte por separado

sin (2 (x + 15)) = 0 or 3 cos (2 (x + 15)) - 1 = 0

sin (2 (x + 15)) = 0 : x = πn - 15, x = \frac{π}{2} + πn - 15

sin (2 (x + 15)) = 0

Soluciones generales para

sin (2 (x + 15)) = 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}

2 (x + 15) = 0 + 2 πn, 2 (x + 15) = π + 2 πn

2 (x + 15) = 0 + 2 πn, 2 (x + 15) = π + 2 πn

Resolver

2 (x + 15) = 0 + 2 πn : x = πn - 15

2 (x + 15) = 0 + 2 πn

0 + 2 πn = 2 πn

2 (x + 15) = 2 πn

Dividir ambos lados entre

2

2 (x + 15) = 2 πn

Dividir ambos lados entre

2

\frac{2 ( x + 15 )}{2} = \frac{2 πn}{2}

Simplificar

x + 15 = πn

x + 15 = πn

Desplace

15

a la derecha

x + 15 = πn

Restar

15

de ambos lados

x + 15 - 15 = πn - 15

Simplificar

x = πn - 15

x = πn - 15

Resolver

2 (x + 15) = π + 2 πn : x = \frac{π}{2} + πn - 15

2 (x + 15) = π + 2 πn

Dividir ambos lados entre

2

2 (x + 15) = π + 2 πn

Dividir ambos lados entre

2

\frac{2 ( x + 15 )}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplificar

x + 15 = \frac{π}{2} + πn

x + 15 = \frac{π}{2} + πn

Desplace

15

a la derecha

x + 15 = \frac{π}{2} + πn

Restar

15

de ambos lados

x + 15 - 15 = \frac{π}{2} + πn - 15

Simplificar

x = \frac{π}{2} + πn - 15

x = \frac{π}{2} + πn - 15

x = πn - 15, x = \frac{π}{2} + πn - 15

3 cos (2 (x + 15)) - 1 = 0 : x = \frac{arccos ( \frac{1}{3} )}{2} + πn - 15, x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

3 cos (2 (x + 15)) - 1 = 0

Desplace

1

a la derecha

3 cos (2 (x + 15)) - 1 = 0

Sumar

1

a ambos lados

3 cos (2 (x + 15)) - 1 + 1 = 0 + 1

Simplificar

3 cos (2 (x + 15)) = 1

3 cos (2 (x + 15)) = 1

Dividir ambos lados entre

3

3 cos (2 (x + 15)) = 1

Dividir ambos lados entre

3

\frac{3 cos ( 2 ( x + 15 ) )}{3} = \frac{1}{3}

Simplificar

cos (2 (x + 15)) = \frac{1}{3}

cos (2 (x + 15)) = \frac{1}{3}

Aplicar propiedades trigonométricas inversas

cos (2 (x + 15)) = \frac{1}{3}

Soluciones generales para

cos (2 (x + 15)) = \frac{1}{3}

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

2 (x + 15) = arccos (\frac{1}{3}) + 2 πn, 2 (x + 15) = 2 π - arccos (\frac{1}{3}) + 2 πn

2 (x + 15) = arccos (\frac{1}{3}) + 2 πn, 2 (x + 15) = 2 π - arccos (\frac{1}{3}) + 2 πn

Resolver

2 (x + 15) = arccos (\frac{1}{3}) + 2 πn : x = \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

2 (x + 15) = arccos (\frac{1}{3}) + 2 πn

Dividir ambos lados entre

2

2 (x + 15) = arccos (\frac{1}{3}) + 2 πn

Dividir ambos lados entre

2

\frac{2 ( x + 15 )}{2} = \frac{arccos ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplificar

x + 15 = \frac{arccos ( \frac{1}{3} )}{2} + πn

x + 15 = \frac{arccos ( \frac{1}{3} )}{2} + πn

Desplace

15

a la derecha

x + 15 = \frac{arccos ( \frac{1}{3} )}{2} + πn

Restar

15

de ambos lados

x + 15 - 15 = \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

Simplificar

x = \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

x = \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

Resolver

2 (x + 15) = 2 π - arccos (\frac{1}{3}) + 2 πn : x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

2 (x + 15) = 2 π - arccos (\frac{1}{3}) + 2 πn

Dividir ambos lados entre

2

2 (x + 15) = 2 π - arccos (\frac{1}{3}) + 2 πn

Dividir ambos lados entre

2

\frac{2 ( x + 15 )}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplificar

x + 15 = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

x + 15 = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

Desplace

15

a la derecha

x + 15 = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

Restar

15

de ambos lados

x + 15 - 15 = π - \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

Simplificar

x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

x = \frac{arccos ( \frac{1}{3} )}{2} + πn - 15, x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

Combinar toda las soluciones

x = πn - 15, x = \frac{π}{2} + πn - 15, x = \frac{arccos ( \frac{1}{3} )}{2} + πn - 15, x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn - 15

Mostrar soluciones en forma decimal

x = πn - 15, x = \frac{π}{2} + πn - 15, x = \frac{1.23095 \dots}{2} + πn - 15, x = π - \frac{1.23095 \dots}{2} + πn - 15

Gráfica

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