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csc(A)*tan(A)=sin(A)

Solución

csc (A) \cdot tan (A) = sin (A)

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

Pasos de solución

csc (A) tan (A) = sin (A)

Restar

sin (A)

de ambos lados

csc (A) tan (A) - sin (A) = 0

Expresar con seno, coseno

- sin (A) + csc (A) tan (A)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

= - sin (A) + \frac{1}{sin ( A )} tan (A)

Utilizar la identidad trigonométrica básica:

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

= - sin (A) + \frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )}

Simplificar

- sin (A) + \frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )} : \frac{- sin ( A ) cos ( A ) + 1}{cos ( A )}

- sin (A) + \frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )}

\frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )} = \frac{1}{cos ( A )}

\frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )}

Multiplicar fracciones:

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

= \frac{1 \cdot sin ( A )}{sin ( A ) cos ( A )}

Eliminar los terminos comunes:

sin (A)

= \frac{1}{cos ( A )}

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

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

Re-escribir usando identidades trigonométricas

1 - cos (A) sin (A)

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 A )}{2}

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

Mover

1

al lado derecho

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

Restar

1

de ambos lados

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

Simplificar

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

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

Multiplicar ambos lados por

2

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

Multiplicar ambos lados por

2

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

Simplificar

- sin (2 A) = - 2

- sin (2 A) = - 2

Dividir ambos lados entre

- 1

- sin (2 A) = - 2

Dividir ambos lados entre

- 1

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

Simplificar

sin (2 A) = 2

sin (2 A) = 2

- 1 \leq sin (x) \leq 1

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

cos (x) = \frac{2}{3}

(sec (3 x) - 2) (tan (2 x) - 1) = 0

cos (2 x) \cdot 2 - 2 cos (x) = 0

- 2 sin (2 x) = \frac{π}{12}

cos (x) = \frac{sec ( x )}{4}