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Popular Trigonometría >

2sin(x)*tan(x)+5sin(x)=-2cos(x)

Solución

2 sin (x) \cdot tan (x) + 5 sin (x) = - 2 cos (x)

Solución

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

Grados

x = - 26.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

- 2 cos (x)

de ambos lados

2 sin (x) tan (x) + 5 sin (x) + 2 cos (x) = 0

Expresar con seno, coseno

2 cos (x) + 5 sin (x) + 2 sin (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Multiplicar fracciones:

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

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

sin (x) \cdot 2 sin (x) = 2 sin^{2} (x)

sin (x) \cdot 2 sin (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= 2 sin^{1 + 1} (x)

Sumar:

1 + 1 = 2

= 2 sin^{2} (x)

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

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

Convertir a fracción:

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

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

2 cos (x) cos (x) + 5 sin (x) cos (x) + 2 sin^{2} (x) = 2 cos^{2} (x) + 5 sin (x) cos (x) + 2 sin^{2} (x)

2 cos (x) cos (x) + 5 sin (x) cos (x) + 2 sin^{2} (x)

2 cos (x) cos (x) = 2 cos^{2} (x)

2 cos (x) cos (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= 2 cos^{1 + 1} (x)

Sumar:

1 + 1 = 2

= 2 cos^{2} (x)

= 2 cos^{2} (x) + 5 sin (x) cos (x) + 2 sin^{2} (x)

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

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

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

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

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x) = 0

Factorizar

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x) : (2 sin (x) + cos (x)) (sin (x) + 2 cos (x))

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x)

Factorizar la expresión

2 sin^{2} (x) + 5 sin (x) cos (x) + 2 cos^{2} (x)

Definición

Factores de

4 : 1, 2, 4

4

Divisores (factores)

Encontrar los factores primos de

4 : 2, 2

4

4

divida por

24 = 2 \cdot 2

= 2 \cdot 2

2

es un número primo, por lo tanto, no es posible factorizar mas

= 2 \cdot 2

Agregar factores primos:

2

Agregar 1 y su propio número

4

1, 4

Divisores de

4

1, 2, 4

Por cada dos factores tales que

u * v = 4,

revisar si

u + v = 5

Revisar

u = 1, v = 4 : u * v = 4, u + v = 5 \Rightarrow

VerdaderoRevisar

u = 2, v = 2 : u * v = 4, u + v = 4 \Rightarrow

Falso

u = 1, v = 4

Agrupar en

(a x^{2} + ux y) + (vx y + c y^{2})

(2 sin^{2} (x) + sin (x) cos (x)) + (4 sin (x) cos (x) + 2 cos^{2} (x))

= (2 sin^{2} (x) + sin (x) cos (x)) + (4 sin (x) cos (x) + 2 cos^{2} (x))

Factorizar

sin (x)

2 sin^{2} (x) + sin (x) cos (x)

sin (x) (2 sin (x) + cos (x))

2 sin^{2} (x) + sin (x) cos (x)

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

= 2 sin (x) sin (x) + sin (x) cos (x)

Factorizar el termino común

sin (x)

= sin (x) (2 sin (x) + cos (x))

Factorizar

2 cos (x)

4 sin (x) cos (x) + 2 cos^{2} (x)

2 cos (x) (2 sin (x) + cos (x))

4 sin (x) cos (x) + 2 cos^{2} (x)

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

cos^{2} (x) = cos (x) cos (x)

= 4 sin (x) cos (x) + 2 cos (x) cos (x)

Reescribir

4

como

2 \cdot 2

= 2 \cdot 2 sin (x) cos (x) + 2 cos (x) cos (x)

Factorizar el termino común

2 cos (x)

= 2 cos (x) (2 sin (x) + cos (x))

= sin (x) (2 sin (x) + cos (x)) + 2 cos (x) (2 sin (x) + cos (x))

Factorizar el termino común

2 sin (x) + cos (x)

= (2 sin (x) + cos (x)) (sin (x) + 2 cos (x))

(2 sin (x) + cos (x)) (sin (x) + 2 cos (x)) = 0

Resolver cada parte por separado

2 sin (x) + cos (x) = 0 or sin (x) + 2 cos (x) = 0

2 sin (x) + cos (x) = 0 : x = arctan (- \frac{1}{2}) + πn

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

Re-escribir usando identidades trigonométricas

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

Dividir ambos lados entre

cos (x), cos (x) \neq = 0

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

Simplificar

\frac{2 sin ( x )}{cos ( x )} + 1 = 0

Utilizar la identidad trigonométrica básica:

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

2 tan (x) + 1 = 0

2 tan (x) + 1 = 0

Desplace

1

a la derecha

2 tan (x) + 1 = 0

Restar

1

de ambos lados

2 tan (x) + 1 - 1 = 0 - 1

Simplificar

2 tan (x) = - 1

2 tan (x) = - 1

Dividir ambos lados entre

2

2 tan (x) = - 1

Dividir ambos lados entre

2

\frac{2 tan ( x )}{2} = \frac{- 1}{2}

Simplificar

tan (x) = - \frac{1}{2}

tan (x) = - \frac{1}{2}

Aplicar propiedades trigonométricas inversas

tan (x) = - \frac{1}{2}

Soluciones generales para

tan (x) = - \frac{1}{2}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

sin (x) + 2 cos (x) = 0 : x = arctan (- 2) + πn

sin (x) + 2 cos (x) = 0

Re-escribir usando identidades trigonométricas

sin (x) + 2 cos (x) = 0

Dividir ambos lados entre

cos (x), cos (x) \neq = 0

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

Simplificar

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

Utilizar la identidad trigonométrica básica:

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

tan (x) + 2 = 0

tan (x) + 2 = 0

Desplace

2

a la derecha

tan (x) + 2 = 0

Restar

2

de ambos lados

tan (x) + 2 - 2 = 0 - 2

Simplificar

tan (x) = - 2

tan (x) = - 2

Aplicar propiedades trigonométricas inversas

tan (x) = - 2

Soluciones generales para

tan (x) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 2) + πn

x = arctan (- 2) + πn

Combinar toda las soluciones

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

Mostrar soluciones en forma decimal

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

Gráfica

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