Quadruple Angle Formulas/Cosine/Factor Form
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Theorem
- $\cos 4 \theta = \paren {\cos \theta - \cos \dfrac \pi 8} \paren {\cos \theta - \cos \dfrac {3 \pi} 8} \paren {\cos \theta - \cos \dfrac {5 \pi} 8} \paren {\cos \theta - \cos \dfrac {7 \pi} 8}$
Proof
\(\ds z^8 + 1\) | \(=\) | \(\ds \prod_{k \mathop = 0}^3 \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} 8 + 1}\) | Complex Algebra Examples: $z^8 + 1$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^4 + z^{-4}\) | \(=\) | \(\ds \prod_{k \mathop = 0}^3 \paren {z - 2 \cos \dfrac {\paren {2 k + 1} \pi} 8 + z^{-1} }\) | dividing both sides by $z^4$ |
Setting $z = e^{i \theta}$:
\(\ds e^{4 i \theta} + e^{-4 i \theta}\) | \(=\) | \(\ds \prod_{k \mathop = 0}^3 \paren {e^{i \theta} - 2 \cos \dfrac {\paren {2 k + 1} i \pi} 8 + e^{-i \theta} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \dfrac {e^{4 i \theta} + e^{-4 i \theta} } 2\) | \(=\) | \(\ds \prod_{k \mathop = 0}^3 \paren {2 \frac {e^{i \theta} + e^{-i \theta} } 2 - 2 \cos \dfrac {\paren {2 k + 1} i \pi} 8}\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \cos 4 \theta\) | \(=\) | \(\ds \prod_{k \mathop = 0}^3 \paren {2 \cos \theta - 2 \cos \dfrac {\paren {2 k + 1} i \pi} 8}\) | Euler's Cosine Identity | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos 4 \theta\) | \(=\) | \(\ds \prod_{k \mathop = 0}^3 \paren {\cos \theta - \cos \dfrac {\paren {2 k + 1} i \pi} 8}\) | simplifying |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Exercise $8$