Precálculo Ejemplos

$\frac{\sin (x + y)}{\sin (x - y)}$

Paso 1

Reescribe $\frac{\sin (x + y)}{\sin (x - y)}$ como un producto.

$\sin (x + y) \frac{1}{\sin (x - y)}$

Paso 2

Escribe $\sin (x + y)$ como una fracción con el denominador $1$ .

$\frac{\sin (x + y)}{1} \cdot \frac{1}{\sin (x - y)}$

Paso 3

Simplifica.

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Paso 3.1

Divide $\sin (x + y)$ por $1$ .

$\sin (x + y) \frac{1}{\sin (x - y)}$

Paso 3.2

Convierte de $\frac{1}{\sin (x - y)}$ a $\csc (x - y)$ .

$\sin (x + y) \csc (x - y)$

Paso 4

Aplica la suma de la razón de los ángulos.

$(\sin (x) \cos (y) + \cos (x) \sin (y)) \csc (x - y)$

Paso 5

Aplica la diferencia de la razón de los ángulos.

$(\sin (x) \cos (y) + \cos (x) \sin (y)) \frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 6

Aplica la propiedad distributiva.

$\sin (x) \cos (y) \frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)} + \cos (x) \sin (y) \frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 7

Multiplica $\sin (x) \cos (y) \frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$ .

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Paso 7.1

Combina $\frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$ y $\sin (x)$ .

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) \sin (x)}{\sec (x) \csc (y) - \csc (x) \sec (y)} \cos (y) + \cos (x) \sin (y) \frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 7.2

Combina $\frac{\sec (x) \sec (y) \csc (x) \csc (y) \sin (x)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$ y $\cos (y)$ .

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) \sin (x) \cos (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)} + \cos (x) \sin (y) \frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 8

Multiplica $\cos (x) \sin (y) \frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$ .

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Paso 8.1

Combina $\frac{\sec (x) \sec (y) \csc (x) \csc (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$ y $\cos (x)$ .

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) \sin (x) \cos (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)} + \frac{\sec (x) \sec (y) \csc (x) \csc (y) \cos (x)}{\sec (x) \csc (y) - \csc (x) \sec (y)} \sin (y)$

Paso 8.2

Combina $\frac{\sec (x) \sec (y) \csc (x) \csc (y) \cos (x)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$ y $\sin (y)$ .

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) \sin (x) \cos (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)} + \frac{\sec (x) \sec (y) \csc (x) \csc (y) \cos (x) \sin (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 9

Combina los numeradores sobre el denominador común.

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) \sin (x) \cos (y) + \sec (x) \sec (y) \csc (x) \csc (y) \cos (x) \sin (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 10

Factoriza $\sec (x) \sec (y) \csc (x) \csc (y)$ de $\sec (x) \sec (y) \csc (x) \csc (y) \sin (x) \cos (y) + \sec (x) \sec (y) \csc (x) \csc (y) \cos (x) \sin (y)$ .

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Paso 10.1

Factoriza $\sec (x) \sec (y) \csc (x) \csc (y)$ de $\sec (x) \sec (y) \csc (x) \csc (y) \sin (x) \cos (y)$ .

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) (\sin (x) \cos (y)) + \sec (x) \sec (y) \csc (x) \csc (y) \cos (x) \sin (y)}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 10.2

Factoriza $\sec (x) \sec (y) \csc (x) \csc (y)$ de $\sec (x) \sec (y) \csc (x) \csc (y) \cos (x) \sin (y)$ .

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) (\sin (x) \cos (y)) + \sec (x) \sec (y) \csc (x) \csc (y) (\cos (x) \sin (y))}{\sec (x) \csc (y) - \csc (x) \sec (y)}$

Paso 10.3

Factoriza $\sec (x) \sec (y) \csc (x) \csc (y)$ de $\sec (x) \sec (y) \csc (x) \csc (y) (\sin (x) \cos (y)) + \sec (x) \sec (y) \csc (x) \csc (y) (\cos (x) \sin (y))$ .

$\frac{\sec (x) \sec (y) \csc (x) \csc (y) (\sin (x) \cos (y) + \cos (x) \sin (y))}{\sec (x) \csc (y) - \csc (x) \sec (y)}$