Triple Angle Formulas/Cosine/2 cos 3 theta + 1
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Theorem
- $2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9}$
Proof
\(\ds z^6 + z^3 + 1\) | \(=\) | \(\ds \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1}\) | Complex Algebra Examples: $z^6 + z^3 + 1$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^3 + z^0 + z^{-3}\) | \(=\) | \(\ds \paren {z - 2 \cos \dfrac {2 \pi} 9 + z^{-1} } \paren {z - 2 \cos \dfrac {4 \pi} 9 + z^{-1} } \paren {z - 2 \cos \dfrac {8 \pi} 9 + z^{-1} }\) |
Setting $z = e^{i \theta}$:
\(\ds e^{3 i \theta} + 1 + e^{-3 i \theta}\) | \(=\) | \(\ds \paren {e^{i \theta} - 2 \cos \dfrac {2 \pi} 9 + e^{-i \theta} } \paren {e^{i \theta} - 2 \cos \dfrac {4 \pi} 9 + e^{-i \theta} } \paren {e^{i \theta} - 2 \cos \dfrac {8 \pi} 9 + e^{-i \theta} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \dfrac {e^{3 i \theta} + e^{-3 i \theta} } 2 + 1\) | \(=\) | \(\ds \paren {2 \dfrac {e^{i \theta} + e^{-i \theta} } 2 - 2 \cos \dfrac {2 \pi} 9} \paren {2 \dfrac {e^{i \theta} + e^{-i \theta} } 2 - 2 \cos \dfrac {4 \pi} 9} \paren {2 \dfrac {e^{i \theta} + e^{-i \theta} } 2 - 2 \cos \dfrac {8 \pi} 9}\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \cos 3 \theta + 1\) | \(=\) | \(\ds \paren {2 \cos \theta - 2 \cos \dfrac {2 \pi} 9} \paren {2 \cos \theta - 2 \cos \dfrac {4 \pi} 9} \paren {2 \cos \theta - 2 \cos \dfrac {8 \pi} 9}\) | Euler's Cosine Identity |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Example $6$: $(3.15)$