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Popular Trigonometría >

(tan(x))/(sin(x))=cos(x)

Solución

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

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

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

Restar

cos (x)

de ambos lados

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

Simplificar

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

Expresar con seno, coseno

tan (x) - cos (x) sin (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Convertir a fracción:

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

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

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

sin (x) - cos (x) sin (x) cos (x)

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

cos (x) sin (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)

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

Sumar:

1 + 1 = 2

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

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

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

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

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

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

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

Factorizar

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

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

Factorizar el termino común

- sin (x)

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

Factorizar

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

cos^{2} (x) - 1

Reescribir

1

como

1^{2}

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

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

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

Resolver cada parte por separado

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

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Desplace

1

a la derecha

cos (x) + 1 = 0

Restar

1

de ambos lados

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

Simplificar

cos (x) = - 1

cos (x) = - 1

Soluciones generales para

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Desplace

1

a la derecha

cos (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

cos (x) = 1

cos (x) = 1

Soluciones generales para

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 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

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Siendo que la ecuación esta indefinida para:

2 πn, π + 2 πn

Sin soluci \overset{o}{ˊ} n para x \in R

Gráfica

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