CBSE Class 11-science Answered
state and prove De movrie's theorem
Asked by spsberry8 | 29 Dec, 2009, 08:27: AM
Expert Answer
De Moivre's theorem states that for any real number x and any integer n, (cosx + isinx)n = cos(nx) + isin(nx).
It can be proved using the principle of mathematical induction
for n =1
(cosx + isinx) = cos(x) + isin(x) is trivially true
Let the result be true for n =k
(cosx + isinx)k = cos(kx) + isin(kx)
To prove for n =k+1
i.e to prove P(k+1):(cosx + isinx)k+1 = cos((k+1)x) + isin((k+1)x)
(cosx + isinx)k (cosx + isinx) = (cos(kx) + isin(kx))(cosx + isinx) using P(k)
= (coskxcosx- sinkxsinx)+i(coskxsinx-+sinkxcosx)
=cos((k+1)x) + isin((k+1)x)
So P(k+1) is true
Hence result holds for all natural numbers k
Answered by | 29 Dec, 2009, 09:32: AM
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