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 2 years ago
PLZ HELP#VERY CONFUSED
rewrite each expression in term with no power greater than 1 cos^3 theta
 2 years ago
PLZ HELP#VERY CONFUSED rewrite each expression in term with no power greater than 1 cos^3 theta

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joemath314159
 2 years ago
Best ResponseYou've already chosen the best response.0The 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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