drasy22 Group Title PLZ HELP#VERY CONFUSED rewrite each expression in term with no power greater than 1 cos^3 theta one year ago one year ago

• This Question is Open
1. drasy22

@jim_thompson5910

2. drasy22

$\cos ^{3}\theta$

3. joemath314159

The only way I can think of doing this is using complex numbers. Using Euler's formula we know that:$\cos \theta =\frac{e^{i\theta}+e^{-i\theta}}{2}$Therefore:$\cos^3\theta=\left(\frac{e^{i\theta}+e^{-i\theta}}{2}\right)^3$$=\frac{1}{2^3}\left(e^{3i\theta}+3e^{2i\theta}e^{-i\theta}+3e^{i\theta}e^{-2i\theta}+e^{-3i\theta}\right)$$=\frac{1}{2^3}\left(e^{3i\theta}+3e^{i\theta}+3e^{-i\theta}+e^{-3i\theta}\right)$$=\frac{1}{2^3}\left(e^{3i\theta}+e^{-3i\theta}\right)+\frac{3}{2^3}\left(e^{i\theta}+e^{-i\theta}\right)$$\frac{1}{4}\left(\frac{e^{3i\theta}+e^{-3i\theta}}{2}\right)+\frac{3}{4}\left(\frac{e^{i\theta}+e^{-i\theta}}{2}\right)$Using Euler's formula again backwards yields:$=\frac{1}{4}\cos 3\theta+\frac{3}{4}\cos \theta$