# DeMoivre's Theorem

• Dec 15th 2008, 09:55 AM
wick
DeMoivre's Theorem
I was asked to find both sin4() and cos4() in terms of sin() and cos()
using DeMoivre's Theorem.

I couldnt figure out how to type the actual symbol for theta on here so i just used ()....

I have the Theorem in front of me and understand it (pretty much), but am struggling getting past the question above.
I need some help please! Its been driving me crazy....
• Dec 15th 2008, 10:33 AM
running-gag
Hi

$\displaystyle (\cos \theta + i \,\sin \theta)^4 = \cos(4 \theta) + i \sin (4 \theta)$

Developing the left member
$\displaystyle \cos^4 \theta + 4i \cos^3 \theta \,\sin \theta - 6 \cos^2 \theta \sin^2 \theta - 4i \cos \theta \sin^3 \theta + \sin^4 \theta = \cos(4 \theta) + i \sin (4 \theta)$

Separating real and imaginary parts
$\displaystyle \cos(4 \theta) = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta$

$\displaystyle \sin (4 \theta) = 4 \cos^3 \theta \,\sin \theta - 4 \cos \theta \sin^3 \theta$

Using $\displaystyle \sin^2 \theta = 1 - \cos^2 \theta$

$\displaystyle \cos(4 \theta) = 8\cos^4 \theta - 8 \cos^2 \theta + 1$