Other values of $n$ for which the minimal nesting of radicals is 2 or less include $n=40$ and $n=120$. For instance for $n=40$, we have:

\[ \sin\left(\frac{\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2- \sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \cos\left(\frac{\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \sin\left(\frac{3\pi}{40}\right) = \left(\frac{-\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \cos\left(\frac{3\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \sin\left(\frac{7\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2+\sqrt{2}} - \frac{\sqrt{2}}{8} \sqrt{2- \sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \cos\left(\frac{7\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \sin\left(\frac{9\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2- \sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \cos\left(\frac{9\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \sin\left(\frac{11\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \cos\left(\frac{11\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \cos\left(\frac{11\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \sin\left(\frac{13\pi}{40}\right) = \left(\frac{1+\sqrt{5}}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \cos\left(\frac{13\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2+\sqrt{2}} - \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \sin\left(\frac{17\pi}{40}\right) = \left(\frac{1+\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \cos\left(\frac{17\pi}{40}\right) = \left(\frac{-\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \sin\left(\frac{19\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \cos\left(\frac{19\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \cos\left(\frac{13\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2+\sqrt{2}} - \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \sin\left(\frac{17\pi}{40}\right) = \left(\frac{1+\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \cos\left(\frac{17\pi}{40}\right) = \left(\frac{-\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5-\sqrt{5}} \]

\[ \sin\left(\frac{19\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5+\sqrt{5}} \]

\[ \cos\left(\frac{19\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5+\sqrt{5}} \]

## No comments:

## Post a Comment