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De Moirves theorem notation

Kvothe the Arcane

I had a problem in an Fp2 paper I did last night

Show that

\cos 6 \theta \equiv 32 \cos^6 \theta - 48 \cos^4 \theta + 18 \cos^2 \theta -1

So I noted that

\cos 6 \theta + i\sin 6 \theta = (cos \theta + isin \theta)^6

and used binomial expansion twice to do the thing. It was a trawl of algebra but straight-forward. But my writing for one line actually ended up spanning two (from the MS, I see I could have used

c

for

\cos \theta

and

s

for

\sin \theta

).

My working

In previous years on here, I've seen a method which avoided writing so many terms by considering only the real or imaginary coefficients from the getgo.

It used

\displaystyle \sum

and

\Re

or

\Im

as appropriate but I don't recall how it was notated. Can someone show me below? :smile:

:smile:

(edited 8 years ago)

Original post by Kvothe the arcane

It used

\displaystyle \sum

and

\Re

or

\Im

as appropriate but I don't recall how it was notated. Can someone show me below? :smile:

:smile:

Something like:

Writing

\cos n\theta +i\sin n\theta = (c+is)^n

by De Moivre.

Then,

\displaystyle\cos 6\theta = \Re (c+is)^6\\=\Re\sum_{l=0}^6\binom{n}{l}c^l(is)^{6-l}\\=\sum^6_{\substack{l=0\\ \text{l even}}}\binom{n}{l}c^l(is)^{6-l}[br]

\displaystyle=\binom{6}{0}c^0(is)^6+\binom{6}{2}c^2(is)^4+\binom{6}{4}c^4(is)^2+\binom{6}{6}c^6(is)^0[br]

then as you had.

(edited 8 years ago)

Kvothe the Arcane

OP

@ghostwalker, thanks :smile:

:smile:

.

Nice latex and use of \substack{}
Similar but odd for powers of sin I take it? :smile:

:smile:

Original post by Kvothe the arcane

Similar but odd for powers of sin I take it? :smile:

:smile:

And use

\Im

and not

\Re

.

Kvothe the Arcane

OP

Original post by Zacken

And use

\Im

and not

\Re

.

Naturally

:awesome:

.

Original post by Kvothe the arcane

@ghostwalker, thanks :smile:

:smile:

.

np

Nice latex and use of \substack{}

Had to look that one up :smile:

:smile:

Similar but odd for powers of sin I take it? :smile:

:smile:

Aye.

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