Find the value of cos - - i sin -n where - R- n - I- i - -1

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De-Moivre's Theorem

Find the valu...

Question

Find the value of $(c o s θ + i s i n θ)^{n}$ where $θ ϵ R, n ϵ I, i = \sqrt{- 1}$

n (c o s θ + i s i n θ)

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n c o s θ + i {s i n}^{n} θ

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c o s n θ + i s i n n θ

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$c o s n θ + i n s i n θ$

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Solution

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(c o s θ + i s i n θ)^{n}

θ ϵ R, n ϵ I, i = \sqrt{- 1}

(1 + cos θ + i sin θ)^{n} + (1 + cos θ - i sin θ)^{n} =

(c o s θ + i s i n θ)^{n}

θ ϵ R, n ϵ I, i = \sqrt{- 1}

{(\frac{1 + cos θ + i sin θ}{1 + cos θ - i sin θ})}^{n} = cos n θ + i sin n θ,

i = \sqrt{- 1}

(1 + cos θ + i sin θ)^{n} + (1 + cos θ - i sin θ)^{n}

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