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Popular >

tan(2x)=5sin(2x)

Solución

tan (2 x) = 5 sin (2 x)

Solución

x = πn, x = \frac{π}{2} + πn, x = \frac{1.36943 \dots}{2} + πn, x = π - \frac{1.36943 \dots}{2} + πn

Grados

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 39.23152 \dots^{\circ} + 18 0^{\circ} n, x = 140.76847 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

tan (2 x) = 5 sin (2 x)

Restar

5 sin (2 x)

de ambos lados

tan (2 x) - 5 sin (2 x) = 0

Expresar con seno, coseno

tan (2 x) - 5 sin (2 x)

Utilizar la identidad trigonométrica básica:

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

= \frac{sin ( 2 x )}{cos ( 2 x )} - 5 sin (2 x)

Simplificar

\frac{sin ( 2 x )}{cos ( 2 x )} - 5 sin (2 x) : \frac{sin ( 2 x ) - 5 sin ( 2 x ) cos ( 2 x )}{cos ( 2 x )}

\frac{sin ( 2 x )}{cos ( 2 x )} - 5 sin (2 x)

Convertir a fracción:

5 sin (2 x) = \frac{5 sin ( 2 x ) cos ( 2 x )}{cos ( 2 x )}

= \frac{sin ( 2 x )}{cos ( 2 x )} - \frac{5 sin ( 2 x ) cos ( 2 x )}{cos ( 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 ( 2 x ) - 5 sin ( 2 x ) cos ( 2 x )}{cos ( 2 x )}

= \frac{sin ( 2 x ) - 5 sin ( 2 x ) cos ( 2 x )}{cos ( 2 x )}

\frac{sin ( 2 x ) - 5 cos ( 2 x ) sin ( 2 x )}{cos ( 2 x )} = 0

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

sin (2 x) - 5 cos (2 x) sin (2 x) = 0

Factorizar

sin (2 x) - 5 cos (2 x) sin (2 x) : - sin (2 x) (5 cos (2 x) - 1)

sin (2 x) - 5 cos (2 x) sin (2 x)

Factorizar el termino común

- sin (2 x)

= - sin (2 x) (- 1 + 5 cos (2 x))

- sin (2 x) (5 cos (2 x) - 1) = 0

Resolver cada parte por separado

sin (2 x) = 0 or 5 cos (2 x) - 1 = 0

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

sin (2 x) = 0

Soluciones generales para

sin (2 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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Resolver

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Dividir ambos lados entre

2

2 x = 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplificar

x = πn

x = πn

Resolver

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Dividir ambos lados entre

2

2 x = π + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

5 cos (2 x) - 1 = 0 : x = \frac{arccos ( \frac{1}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

5 cos (2 x) - 1 = 0

Mover

1

al lado derecho

5 cos (2 x) - 1 = 0

Sumar

1

a ambos lados

5 cos (2 x) - 1 + 1 = 0 + 1

Simplificar

5 cos (2 x) = 1

5 cos (2 x) = 1

Dividir ambos lados entre

5

5 cos (2 x) = 1

Dividir ambos lados entre

5

\frac{5 cos ( 2 x )}{5} = \frac{1}{5}

Simplificar

cos (2 x) = \frac{1}{5}

cos (2 x) = \frac{1}{5}

Aplicar propiedades trigonométricas inversas

cos (2 x) = \frac{1}{5}

Soluciones generales para

cos (2 x) = \frac{1}{5}

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

2 x = arccos (\frac{1}{5}) + 2 πn, 2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

2 x = arccos (\frac{1}{5}) + 2 πn, 2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

Resolver

2 x = arccos (\frac{1}{5}) + 2 πn : x = \frac{arccos ( \frac{1}{5} )}{2} + πn

2 x = arccos (\frac{1}{5}) + 2 πn

Dividir ambos lados entre

2

2 x = arccos (\frac{1}{5}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{arccos ( \frac{1}{5} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{arccos ( \frac{1}{5} )}{2} + πn

x = \frac{arccos ( \frac{1}{5} )}{2} + πn

Resolver

2 x = 2 π - arccos (\frac{1}{5}) + 2 πn : x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

Dividir ambos lados entre

2

2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{1}{5} )}{2} + \frac{2 πn}{2}

Simplificar

x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

x = \frac{arccos ( \frac{1}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

Combinar toda las soluciones

x = πn, x = \frac{π}{2} + πn, x = \frac{arccos ( \frac{1}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

Mostrar soluciones en forma decimal

x = πn, x = \frac{π}{2} + πn, x = \frac{1.36943 \dots}{2} + πn, x = π - \frac{1.36943 \dots}{2} + πn

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