Cosine of 72 Degrees
Theorem
- $\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = \dfrac {\sqrt 5 - 1} 4$
where $\cos$ denotes the cosine function.
Proof 1
| \(\ds \cos 72 \degrees\) | \(=\) | \(\ds \map \cos {90 \degrees - 18 \degrees}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sin 18 \degrees\) | Cosine of Complement equals Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sqrt 5 - 1} 4\) | Sine of $18 \degrees$ |
$\blacksquare$
Proof 2
| \(\ds \cos 72 \degrees\) | \(=\) | \(\ds 2 \cos 36 \degrees - 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \paren {\dfrac \phi 2}^2 - 1\) | 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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $92 \ \text {(b)}$