De moivre theorem:

If n is any integer, then (cosθ + i sinθ)ⁿ = cosⁿθ + i sinⁿθ. This is called as De Movre's Theorem.

Remarks:

• Writing the binomial expansion of (cosθ + i sinθ)ⁿ and equating the real part to cosnθ and the imaginary part to sin nθ, we get

• If n is rational number, then one of the values of (cosθ + i sinθ )ⁿ is cosnθ + isinnθ . Let n = p/q, where p and q are integers (q > 0) and p, q have no common factor. Then (cosθ  + i sinθ)ⁿ has θ distinct values, one of which is cos nθ  + i sinn θ .
• If z = r (cosθ + isinθ), and n is a positive integer, then

       z^1/n = r^1/n , k = 0, 1, 2, ......, n -1.

       Here if can be noted that any 'n' consecutive values of k will serve the purpose.  

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