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If n is any integer, then (cos Θ + i sin Θ)n = cos nΘ + i sin nΘ. This is known as De Movre’s Theorem.

Remarks:

  • Writing the binomial expansion of (cos Θ + i sin Θ)n and equating the real part to cos nΘ and the imaginary part to sin nΘ, we get

cos nΘ = cosn Θ – nc2 cosn–2Θ sin2Θ + nc4 cosn–4Θ sin4Θ + ………
sin nΘ = nc1 cosn–1Θ sinΘ – nc3 cosn–3Θ sin3Θ + nc5 cosn–5Θ sin5Θ +

DE MOIVER’S Theorem

 

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