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(sin(x))/(cos(x))+(cos(x))/(sec(x))=1

Solución

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

Solución

x = 2 πn, x = π + 2 πn

Grados

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

Pasos de solución

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

Restar

1

de ambos lados

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

Simplificar

\frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sec ( x )} - 1 : \frac{sin ( x ) sec ( x ) + cos ^{2} ( x ) - cos ( x ) sec ( x )}{cos ( x ) sec ( x )}

\frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sec ( x )} - 1

Convertir a fracción:

1 = \frac{1}{1}

= \frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sec ( x )} - \frac{1}{1}

Mínimo común múltiplo de

cos (x), sec (x), 1 : cos (x) sec (x)

cos (x), sec (x), 1

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan en al menos una de las expresiones factorizadas

= cos (x) sec (x)

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

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

multiplicar el denominador y el numerador por

sec (x)

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

Para

\frac{cos ( x )}{sec ( x )} :

multiplicar el denominador y el numerador por

cos (x)

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

Para

\frac{1}{1} :

multiplicar el denominador y el numerador por

cos (x) sec (x)

\frac{1}{1} = \frac{1 \cdot cos ( x ) sec ( x )}{1 \cdot cos ( x ) sec ( x )} = \frac{cos ( x ) sec ( x )}{cos ( x ) sec ( x )}

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

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

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

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

Re-escribir usando identidades trigonométricas

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

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

cos^{2} (x)

Utilizar la identidad pitagórica:

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

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

= 1 - sin^{2} (x)

Factorizar

1 - sin^{2} (x) : (1 + sin (x)) (1 - sin (x))

1 - sin^{2} (x)

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

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

= (1 + sin (x)) (1 - sin (x))

= (1 + sin (x)) (1 - sin (x))

= (1 + sin (x)) (1 - sin (x)) - cos (x) sec (x) + sec (x) sin (x)

Utilizar la identidad trigonométrica básica:

cos (x) = \frac{1}{sec ( x )}

= (1 + sin (x)) (1 - sin (x)) - \frac{1}{sec ( x )} sec (x) + sec (x) sin (x)

Simplificar

(1 + sin (x)) (1 - sin (x)) - \frac{1}{sec ( x )} sec (x) + sec (x) sin (x) : sec (x) sin (x) - sin^{2} (x)

(1 + sin (x)) (1 - sin (x)) - \frac{1}{sec ( x )} sec (x) + sec (x) sin (x)

\frac{1}{sec ( x )} sec (x) = 1

\frac{1}{sec ( x )} sec (x)

Multiplicar fracciones:

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

= \frac{1 \cdot sec ( x )}{sec ( x )}

Eliminar los terminos comunes:

sec (x)

= 1

= (sin (x) + 1) (- sin (x) + 1) - 1 + sec (x) sin (x)

Expandir

(1 + sin (x)) (1 - sin (x)) : 1 - sin^{2} (x)

(1 + sin (x)) (1 - sin (x))

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = sin (x)

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (x)

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

Simplificar

1 - sin^{2} (x) - 1 + sec (x) sin (x) : sec (x) sin (x) - sin^{2} (x)

1 - sin^{2} (x) - 1 + sec (x) sin (x)

Agrupar términos semejantes

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

1 - 1 = 0

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

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

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

- sin^{2} (x) + sec (x) sin (x) = 0

Factorizar

- sin^{2} (x) + sec (x) sin (x) : sin (x) (- sin (x) + sec (x))

- sin^{2} (x) + sec (x) sin (x)

Aplicar las leyes de los exponentes:

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

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

= - sin (x) sin (x) + sec (x) sin (x)

Factorizar el termino común

sin (x)

= sin (x) (- sin (x) + sec (x))

sin (x) (- sin (x) + sec (x)) = 0

Resolver cada parte por separado

sin (x) = 0 or - sin (x) + sec (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

- sin (x) + sec (x) = 0 :

Sin solución

- sin (x) + sec (x) = 0

Expresar con seno, coseno

sec (x) - sin (x)

Utilizar la identidad trigonométrica básica:

sec (x) = \frac{1}{cos ( x )}

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

Simplificar

\frac{1}{cos ( x )} - sin (x) : \frac{1 - sin ( x ) cos ( x )}{cos ( x )}

\frac{1}{cos ( x )} - sin (x)

Convertir a fracción:

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

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

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

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

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

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

Re-escribir usando identidades trigonométricas

1 - cos (x) sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

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

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

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

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

Mover

1

al lado derecho

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

Restar

1

de ambos lados

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

Simplificar

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

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

Multiplicar ambos lados por

2

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

Multiplicar ambos lados por

2

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

Simplificar

- sin (2 x) = - 2

- sin (2 x) = - 2

Dividir ambos lados entre

- 1

- sin (2 x) = - 2

Dividir ambos lados entre

- 1

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

Simplificar

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Ejemplos populares

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tan (θ) = 1770

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2 sin^{2} (x) - \frac{5}{4} = - cos^{2} (x)

2sqrt(3)sin(2x)=sqrt(3)

23 sin (2 x) = 3

(sin^2(x)-tan^2(x))/(csc(x))=sin(x)

\frac{sin ^{2} ( x ) - tan ^{2} ( x )}{csc ( x )} = sin (x)