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
![382_De moivre theorem.png](https://www.expertsmind.com/CMSImages/382_De%20moivre%20theorem.png)
- 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
z1/n = r1/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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