# 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

$\blacksquare$