Trigonometría Ejemplos

$\cos (6 x)$

Un buen método para expandir $\cos (6 x)$ es utilizar el teorema de DeMoivre $(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . Si $r = 1$ , $\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

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

Expande el lado derecho de $\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ usando el teorema binomial.

Expandir: ${(\cos (x) + i \cdot \sin (x))}^{6}$

Utilizar el teorema del binomio.

$\cos^{6} (x) + 6 \cos^{5} (x) (i \sin (x)) + 15 \cos^{4} (x) {(i \sin (x))}^{2} + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Simplifica los términos

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Simplifique cada término.

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Aplicar la regla del producto a $i \sin (x)$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cos^{4} (x) (i^{2} \sin^{2} (x)) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribir utilizando la propiedad conmutativa de la multiplicación.

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cdot i^{2} \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribe $i^{2}$ como $- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cdot - 1 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Multiplicar $15$ por $- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Aplicar la regla del producto a $i \sin (x)$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) (i^{3} \sin^{3} (x)) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribir utilizando la propiedad conmutativa de la multiplicación.

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot i^{3} \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Factoriza $i^{2}$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (i^{2} \cdot i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribe $i^{2}$ como $- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (- 1 \cdot i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribe $- 1 i$ como $- i$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (- i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Multiplicar $- 1$ por $20$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Aplicar la regla del producto a $i \sin (x)$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) (i^{4} \sin^{4} (x)) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribir utilizando la propiedad conmutativa de la multiplicación.

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot i^{4} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribe $i^{4}$ como $1$ .

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Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot {(i^{2})}^{2} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Reescribe $i^{2}$ como $- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot {(- 1)}^{2} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Elevar $- 1$ a la potencia de $2$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot 1 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Multiplicar $15$ por $1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Aplicar la regla del producto a $i \sin (x)$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i^{5} \sin^{5} (x)) + {(i \sin (x))}^{6}$

Factoriza $i^{4}$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i^{4} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Reescribe $i^{4}$ como $1$ .

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Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) ({(i^{2})}^{2} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Reescribe $i^{2}$ como $- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) ({(- 1)}^{2} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Elevar $- 1$ a la potencia de $2$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (1 i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Multiplicar $i$ por $1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Aplicar la regla del producto a $i \sin (x)$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{6} \sin^{6} (x)$

Factoriza $i^{4}$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{4} i^{2} \sin^{6} (x)$

Reescribe $i^{4}$ como $1$ .

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Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + {(i^{2})}^{2} i^{2} \sin^{6} (x)$

Reescribe $i^{2}$ como $- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + {(- 1)}^{2} i^{2} \sin^{6} (x)$

Elevar $- 1$ a la potencia de $2$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + 1 i^{2} \sin^{6} (x)$

Multiplicar $i^{2}$ por $1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{2} \sin^{6} (x)$

Reescribe $i^{2}$ como $- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - 1 \sin^{6} (x)$

Reescribe $- 1 \sin^{6} (x)$ como $- \sin^{6} (x)$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - \sin^{6} (x)$

Reordena los factores en $\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - \sin^{6} (x)$ .

$\cos^{6} (x) + 6 i \cos^{5} (x) \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 i \cos (x) \sin^{5} (x) - \sin^{6} (x)$

Quite las expresiones con la parte imaginaria, las cuales son iguales a $\cos (6 x)$ . Remueva el número imaginario $i$ .

$\cos (6 x) = \cos^{6} (x) - 15 \cos^{4} (x) \sin^{2} (x) + 15 \cos^{2} (x) \sin^{4} (x) - \sin^{6} (x)$