Cosine Cubic Sum
Compute
\[\cos^3 \frac{2 \pi}{7} + \cos^3 \frac{4 \pi}{7} + \cos^3 \frac{8 \pi}{7}.\]
- 1
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- +
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- -
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- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
The triple angle formula states that $ \cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta $. Then
\[\cos^3 \theta = \frac{1}{4} \cos 3 \theta + \frac{3}{4} \cos \theta.\]Hence,
\begin{align*}
\cos^3 \frac{2 \pi}{7} + \cos^3 \frac{4 \pi}{7} + \cos^3 \frac{8 \pi}{7} &= \left( \frac{1}{4} \cos \frac{6 \pi}{7} + \frac{3}{4} \cos \frac{2 \pi}{7} \right) + \left( \frac{1}{4} \cos \frac{12 \pi}{7} + \frac{3}{4} \cos \frac{4 \pi}{7} \right) + \left( \frac{1}{4} \cos \frac{24 \pi}{7} + \frac{3}{4} \cos \frac{8 \pi}{7} \right) \\
&= \frac{1}{4} \left( \cos \frac{6 \pi}{7} + \cos \frac{12 \pi}{7} + \cos \frac{24 \pi}{7} \right) + \frac{3}{4} \left( \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{8 \pi}{7} \right) \\
&= \frac{1}{4} \left( \cos \frac{6 \pi}{7} + \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} \right) + \frac{3}{4} \left( \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7} \right) \\
&= \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7}.\end{align*}Consider the sum
\[S = \operatorname{cis} 0 + \operatorname{cis} \frac{2 \pi}{7} + \operatorname{cis} \frac{4 \pi}{7} + \dots + \operatorname{cis} \frac{12 \pi}{7}.\]Then
\begin{align*}
S \operatorname{cis} \frac{2 \pi}{7} &= \operatorname{cis} \frac{2 \pi}{7} + \operatorname{cis} \frac{4 \pi}{7} + \dots + \operatorname{cis} \frac{12 \pi}{7} + \operatorname{cis} 2 \pi \\
&= \operatorname{cis} \frac{2 \pi}{7} + \operatorname{cis} \frac{4 \pi}{7} + \dots + \operatorname{cis} \frac{12 \pi}{7} + \operatorname{cis} 0 \\
&= S,
\end{align*}so $ S \left( 1 - \operatorname{cis} \frac{2 \pi}{7} \right) = 0 $. Hence, $ S = 0 $.
Taking the real part of $ S $ gives us
\[\cos 0 + \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7} + \cos \frac{8 \pi}{7} + \cos \frac{10 \pi}{7} + \cos \frac{12 \pi}{7} = 0.\]Then
\[1 + \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7} + \cos \frac{6 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{2 \pi}{7} = 0,\]so
\[\cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7} = \boxed{-\frac{1}{2}}.\]