English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

verificar (sin(3x)-sin(5x))/(2cos(4x))=-sin(x)

Solución

verificar

\frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )} = - sin (x)

Verdadero

Pasos de solución

\frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )} = - sin (x)

Manipular el lado derecho

\frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )}

Re-escribir usando identidades trigonométricas

\frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )}

Utilizar la identidad suma-producto:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= \frac{2 sin ( \frac{3 x - 5 x}{2} ) cos ( \frac{3 x + 5 x}{2} )}{2 cos ( 4 x )}

Simplificar

\frac{2 sin ( \frac{3 x - 5 x}{2} ) cos ( \frac{3 x + 5 x}{2} )}{2 cos ( 4 x )} : - sin (x)

\frac{2 sin ( \frac{3 x - 5 x}{2} ) cos ( \frac{3 x + 5 x}{2} )}{2 cos ( 4 x )}

2 sin (\frac{3 x - 5 x}{2}) cos (\frac{3 x + 5 x}{2}) = 2 sin (- \frac{2 x}{2}) cos (\frac{8 x}{2})

2 sin (\frac{3 x - 5 x}{2}) cos (\frac{3 x + 5 x}{2})

\frac{3 x - 5 x}{2} = - \frac{2 x}{2}

\frac{3 x - 5 x}{2}

Sumar elementos similares:

3 x - 5 x = - 2 x

= \frac{- 2 x}{2}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 x}{2}

= 2 sin (- \frac{2 x}{2}) cos (\frac{3 x + 5 x}{2})

Sumar elementos similares:

3 x + 5 x = 8 x

= 2 sin (- \frac{2 x}{2}) cos (\frac{8 x}{2})

= \frac{2 sin ( - \frac{2 x}{2} ) cos ( \frac{8 x}{2} )}{2 cos ( 4 x )}

Dividir:

\frac{2}{2} = 1

= \frac{sin ( - \frac{2 x}{2} ) cos ( \frac{8 x}{2} )}{cos ( 4 x )}

Utilizar la razón trigonométrica de ángulo negativo:

sin (- x) = - sin (x)

= \frac{cos ( \frac{8 x}{2} ) ( - sin ( \frac{2 x}{2} ) )}{cos ( 4 x )}

Quitar los parentesis:

(- a) = - a

= \frac{- cos ( \frac{8 x}{2} ) sin ( \frac{2 x}{2} )}{cos ( 4 x )}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{cos ( \frac{8 x}{2} ) sin ( \frac{2 x}{2} )}{cos ( 4 x )}

Dividir:

\frac{8}{2} = 4

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

Dividir:

\frac{2}{2} = 1

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

Eliminar los terminos comunes:

cos (4 x)

= - sin (x)

= - sin (x)

= - sin (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

2 tan (θ) (1 + cos (2 θ)) = 2 sin (2 θ)

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

tan (a) + cot (a) = 2 csc (2 a)

\frac{sec ^{2} ( t )}{csc ( t )} = sec (t) tan (t)

\frac{sec ( t ) - 1}{tan ( t )} = \frac{tan ( t )}{sec ( t ) + 1}