Sine of 72 Degrees
Theorem
- $\sin 72 \degrees = \sin \dfrac {2 \pi} 5 = \dfrac {\sqrt{10 + 2 \sqrt 5} } 4$
where $\sin$ denotes the sine function.
Proof
| \(\ds \sin 72 \degrees\) | \(=\) | \(\ds \sqrt {1 - \cos^2 72 \degrees}\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {1 - \paren {\dfrac{\sqrt 5 - 1} 4}^2}\) | Cosine of $72 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {1 - \dfrac {6 - 2 \sqrt 5} {16} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) |
$\blacksquare$
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 $19$