anonymous
  • anonymous
PLZ HELP#VERY CONFUSED rewrite each expression in term with no power greater than 1 cos^3 theta
Precalculus
katieb
  • katieb
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anonymous
  • anonymous
@jim_thompson5910
anonymous
  • anonymous
\[\cos ^{3}\theta \]
anonymous
  • anonymous
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:\[=\frac{1}{4}\cos 3\theta+\frac{3}{4}\cos \theta\]

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