You can say,
Also, you can expand this,
and
Then,
By binomal formula.
Just so you remember,
....
I need help with this question:
By applying de Moivre's theorem and by also expanding (cos a + i sin a)^5, express cos 5a as a polynomial in cos a
Nothing jumps out, and for there must be an easier way that expanding the whole thing. Plus I don't fully understand the question
Thanks
Hello, chancey!
DeMoivre's Theorem: .
By applying DeMoivre's theorem and by also expanding ,
express as a polynomial in .
From the theorem, we have: .
Expand the right side:
.
And we have: .
. .
Equate real components: . **
Replace with
. .
Therefore: .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
** .If we equate the 'imaginary' components, we have:
. . . .
. . which simplifies to: .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Another bonus . . .
Divide top and bottom by
.
Therefore: .