Trigonometría Ejemplos

\sin (5 x)

$\sin (5 x)$

Paso 1

Un buen método para expandir

\sin (5 x)

$\sin (5 x)$ es usar el teorema de DeMoivre

(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))

$(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . Cuando

r = 1

$r = 1$ ,

\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$

Paso 2

Expande el lado derecho de

\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ con el teorema del binomio.

Expandir

{(\cos (x) + i \cdot \sin (x))}^{5}

${(\cos (x) + i \cdot \sin (x))}^{5}$

Paso 3

Usa el teorema del binomio.

\cos^{5} (x) + 5 \cos^{4} (x) (i \sin (x)) + 10 \cos^{3} (x) {(i \sin (x))}^{2} + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) (i \sin (x)) + 10 \cos^{3} (x) {(i \sin (x))}^{2} + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4

Simplifica los términos.

Toca para ver más pasos...

Paso 4.1

Simplifica cada término.

Toca para ver más pasos...

Paso 4.1.1

Aplica la regla del producto a

i \sin (x)

$i \sin (x)$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cos^{3} (x) (i^{2} \sin^{2} (x)) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cos^{3} (x) (i^{2} \sin^{2} (x)) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.2

Reescribe con la propiedad conmutativa de la multiplicación.

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot i^{2} \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot i^{2} \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.3

Reescribe

i^{2}

$i^{2}$ como

- 1

$- 1$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot - 1 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot - 1 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.4

Multiplica

10

$10$ por

- 1

$- 1$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.5

Aplica la regla del producto a

i \sin (x)

$i \sin (x)$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) (i^{3} \sin^{3} (x)) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) (i^{3} \sin^{3} (x)) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.6

Reescribe con la propiedad conmutativa de la multiplicación.

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot i^{3} \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot i^{3} \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.7

Factoriza

i^{2}

$i^{2}$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (i^{2} \cdot i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (i^{2} \cdot i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.8

Reescribe

i^{2}

$i^{2}$ como

- 1

$- 1$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (- 1 \cdot i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (- 1 \cdot i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.9

Reescribe

- 1 i

$- 1 i$ como

- i

$- i$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (- i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (- i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.10

Multiplica

- 1

$- 1$ por

10

$10$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Paso 4.1.11

Aplica la regla del producto a

i \sin (x)

$i \sin (x)$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (i^{4} \sin^{4} (x)) + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (i^{4} \sin^{4} (x)) + {(i \sin (x))}^{5}$

Paso 4.1.12

Reescribe

i^{4}

$i^{4}$ como

1

$1$ .

Toca para ver más pasos...

Paso 4.1.12.1

Reescribe

i^{4}

$i^{4}$ como

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) ({(i^{2})}^{2} \sin^{4} (x)) + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) ({(i^{2})}^{2} \sin^{4} (x)) + {(i \sin (x))}^{5}$

Paso 4.1.12.2

Reescribe

i^{2}

$i^{2}$ como

- 1

$- 1$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) ({(- 1)}^{2} \sin^{4} (x)) + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) ({(- 1)}^{2} \sin^{4} (x)) + {(i \sin (x))}^{5}$

Paso 4.1.12.3

Eleva

- 1

$- 1$ a la potencia de

2

$2$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (1 \sin^{4} (x)) + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (1 \sin^{4} (x)) + {(i \sin (x))}^{5}$

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (1 \sin^{4} (x)) + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (1 \sin^{4} (x)) + {(i \sin (x))}^{5}$

Paso 4.1.13

Multiplica

\sin^{4} (x)

$\sin^{4} (x)$ por

1

$1$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(i \sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(i \sin (x))}^{5}$

Paso 4.1.14

Aplica la regla del producto a

i \sin (x)

$i \sin (x)$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i^{5} \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i^{5} \sin^{5} (x)$

Paso 4.1.15

Factoriza

i^{4}

$i^{4}$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i^{4} i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i^{4} i \sin^{5} (x)$

Paso 4.1.16

Reescribe

i^{4}

$i^{4}$ como

1

$1$ .

Toca para ver más pasos...

Paso 4.1.16.1

Reescribe

i^{4}

$i^{4}$ como

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(i^{2})}^{2} i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(i^{2})}^{2} i \sin^{5} (x)$

Paso 4.1.16.2

Reescribe

i^{2}

$i^{2}$ como

- 1

$- 1$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(- 1)}^{2} i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(- 1)}^{2} i \sin^{5} (x)$

Paso 4.1.16.3

Eleva

- 1

$- 1$ a la potencia de

2

$2$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + 1 i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + 1 i \sin^{5} (x)$

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + 1 i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + 1 i \sin^{5} (x)$

Paso 4.1.17

Multiplica

i

$i$ por

1

$1$ .

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$

Paso 4.2

Reordena los factores en

\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$ .

\cos^{5} (x) + 5 i \cos^{4} (x) \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)

$\cos^{5} (x) + 5 i \cos^{4} (x) \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$

\cos^{5} (x) + 5 i \cos^{4} (x) \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)

$\cos^{5} (x) + 5 i \cos^{4} (x) \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$

Paso 5

Saca las expresiones con la parte imaginaria, que son equivalentes a

\sin (5 x)

$\sin (5 x)$ . Elimina el número imaginario

i

$i$ .

\sin (5 x) = 5 \cos^{4} (x) \sin (x) - 10 \cos^{2} (x) \sin^{3} (x) + \sin^{5} (x)

$\sin (5 x) = 5 \cos^{4} (x) \sin (x) - 10 \cos^{2} (x) \sin^{3} (x) + \sin^{5} (x)$

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