Cosine of 72 Degrees/Proof 2
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Theorem
- $\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = \dfrac {\sqrt 5 - 1} 4$
Proof
\(\ds \cos 72 \degrees\) | \(=\) | \(\ds 2 \cos 36 \degrees - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \paren {\dfrac \phi 2}^2 - 1\) | Cosine of $36 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^2} 2 - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi + 1} 2 - 1\) | Square of Golden Mean equals One plus Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi - 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac {1 - \phi} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{-1} } 2\) | Reciprocal Form of One Minus Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\sqrt 5 - 1} 4\) | Definition 2 of Golden Mean, and algebra |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $19$