Alguns valores precisos de senos e cossenos.

$\sin \frac{\pi}{2}=1$

$\cos \frac{\pi}{2}=0$

$\sin \frac{\pi}{4}={{1}\over{\sqrt{2}}}$

$\cos \frac{\pi}{4}={{1}\over{\sqrt{2}}}$

$\sin \frac{\pi}{8}={{\sqrt{\sqrt{2}-1}}\over{2^{{{3}\over{4}}}}}$

$\cos \frac{\pi}{8}={{\sqrt{\sqrt{2}+1}}\over{2^{{{3}\over{4}}}}}$

$\sin \frac{\pi}{16}={{\sqrt{2^{{{3}\over{4}}}-\sqrt{\sqrt{2}+1}}}\over{2^{{{7}\over{8}}}}}$

$\cos \frac{\pi}{16}={{\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}\over{2^{{{7}\over{8}}}}}$

$\sin \frac{\pi}{32}={{\sqrt{2^{{{7}\over{8}}}-\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}\over{2^{{{15}\over{16}}}}}$

$\cos \frac{\pi}{32}={{\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}\over{2^{{{15}\over{16}}}}}$

$\sin \frac{\pi}{64}={{\sqrt{2^{{{15}\over{16}}}-\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}\over{2^{{{31}\over{32}}}}}$

$\cos \frac{\pi}{64}={{\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}\over{2^{{{31}\over{32}}}}}$

$\sin \frac{\pi}{128}={{\sqrt{2^{{{31}\over{32}}}-\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}\over{2^{{{63}\over{64}}}}}$

$\cos \frac{\pi}{128}={{\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}\over{2^{{{63}\over{64}}}}}$

$\sin \frac{\pi}{256}={{\sqrt{2^{{{63}\over{64}}}-\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}\over{2^{{{127}\over{128}}}}}$

$\cos \frac{\pi}{256}={{\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}\over{2^{{{127}\over{128}}}}}$

$\sin \frac{\pi}{512}={{\sqrt{2^{{{127}\over{128}}}-\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}\over{2^{{{255}\over{256}}}}}$

$\cos \frac{\pi}{512}={{\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}\over{2^{{{255}\over{256}}}}}$

$\sin \frac{\pi}{1024}={{\sqrt{2^{{{255}\over{256}}}-\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}\over{2^{{{511}\over{512}}}}}$

$\cos \frac{\pi}{1024}={{\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}\over{2^{{{511}\over{512}}}}}$

$\sin \frac{\pi}{2048}={{\sqrt{2^{{{511}\over{512}}}-\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}\over{2^{{{1023}\over{1024}}}}}$

$\cos \frac{\pi}{2048}={{\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}\over{2^{{{1023}\over{1024}}}}}$

$\sin \frac{\pi}{4096}={{\sqrt{2^{{{1023}\over{1024}}}-\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}\over{2^{{{2047}\over{2048}}}}}$

$\cos \frac{\pi}{4096}={{\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}\over{2^{{{2047}\over{2048}}}}}$

$\sin \frac{\pi}{8192}={{\sqrt{2^{{{2047}\over{2048}}}-\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}\over{2^{{{4095}\over{4096}}}}}$

$\cos \frac{\pi}{8192}={{\sqrt{2^{{{2047}\over{2048}}}+\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}\over{2^{{{4095}\over{4096}}}}}$

$\sin \frac{\pi}{16384}={{\sqrt{2^{{{4095}\over{4096}}}-\sqrt{2^{{{2047}\over{2048}}}+\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}}\over{2^{{{8191}\over{8192}}}}}$

$\cos \frac{\pi}{16384}={{\sqrt{2^{{{4095}\over{4096}}}+\sqrt{2^{{{2047}\over{2048}}}+\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}}\over{2^{{{8191}\over{8192}}}}}$

$\sin \frac{\pi}{32768}={{\sqrt{2^{{{8191}\over{8192}}}-\sqrt{2^{{{4095}\over{4096}}}+\sqrt{2^{{{2047}\over{2048}}}+\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}}}\over{2^{{{16383}\over{16384}}}}}$

$\cos \frac{\pi}{32768}={{\sqrt{2^{{{8191}\over{8192}}}+\sqrt{2^{{{4095}\over{4096}}}+\sqrt{2^{{{2047}\over{2048}}}+\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}}}\over{2^{{{16383}\over{16384}}}}}$

${\small \sin \frac{\pi}{65536}={{\sqrt{2^{{{16383}\over{16384}}}-\sqrt{2^{{{8191}\over{8192}}}+\sqrt{2^{{{4095}\over{4096}}}+\sqrt{2^{{{2047}\over{2048}}}+\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}}}}\over{2^{{{32767}\over{32768}}}}}}$

${\small \cos \frac{\pi}{65536}={{\sqrt{2^{{{16383}\over{16384}}}+\sqrt{2^{{{8191}\over{8192}}}+\sqrt{2^{{{4095}\over{4096}}}+\sqrt{2^{{{2047}\over{2048}}}+\sqrt{2^{{{1023}\over{1024}}}+\sqrt{2^{{{511}\over{512}}}+\sqrt{2^{{{255}\over{256}}}+\sqrt{2^{{{127}\over{128}}}+\sqrt{2^{{{63}\over{64}}}+\sqrt{2^{{{31}\over{32}}}+\sqrt{2^{{{15}\over{16}}}+\sqrt{2^{{{7}\over{8}}}+\sqrt{2^{{{3}\over{4}}}+\sqrt{\sqrt{2}+1}}}}}}}}}}}}}}}\over{2^{{{32767}\over{32768}}}}}}$

$\sin \frac{\pi}{3}={{\sqrt{3}}\over{2}}$

$\cos \frac{\pi}{3}={{1}\over{2}}$

$\sin \frac{\pi}{6}={{1}\over{2}}$

$\cos \frac{\pi}{6}={{\sqrt{3}}\over{2}}$

$\sin \frac{\pi}{12}={{\sqrt{2-\sqrt{3}}}\over{2}}$

$\cos \frac{\pi}{12}={{\sqrt{\sqrt{3}+2}}\over{2}}$

$\sin \frac{\pi}{24}={{\sqrt{2-\sqrt{\sqrt{3}+2}}}\over{2}}$

$\cos \frac{\pi}{24}={{\sqrt{\sqrt{\sqrt{3}+2}+2}}\over{2}}$

$\sin \frac{\pi}{48}={{\sqrt{2-\sqrt{\sqrt{\sqrt{3}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{48}={{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{96}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{96}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{192}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{192}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{384}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{384}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{768}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{768}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{1536}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{1536}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{3072}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{3072}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{6144}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{6144}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{12288}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{12288}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{24576}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{24576}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{49152}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{49152}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\sin \frac{\pi}{98304}={{\sqrt{2-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}}\over{2}}$

$\cos \frac{\pi}{98304}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{3}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}+2}}\over{2}}$

$\cos \frac{\pi}{5}={{\sqrt{5}+1}\over{4}}$

$\sin \frac{\pi}{10}={{\sqrt{3-\sqrt{5}}}\over{2^{{{3}\over{2}}}}}$

$\cos \frac{\pi}{10}={{\sqrt{\sqrt{5}+5}}\over{2^{{{3}\over{2}}}}}$

$\sin \frac{\pi}{20}={{\sqrt{2^{{{3}\over{2}}}-\sqrt{\sqrt{5}+5}}}\over{2^{{{5}\over{4}}}}}$

$\cos \frac{\pi}{20}={{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}}\over{2^{{{5}\over{4}}}}}$

$\sin \frac{\pi}{40}={{\sqrt{2^{{{5}\over{4}}}-\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}}}\over{2^{{{9}\over{8}}}}}$

$\cos \frac{\pi}{40}={{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}}\over{2^{{{9}\over{8}}}}}$

$\sin \frac{\pi}{80}={{\sqrt{2^{{{9}\over{8}}}-\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}}}\over{2^{{{17}\over{16}}}}}$

$\cos \frac{\pi}{80}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}}\over{2^{{{17}\over{16}}}}}$

$\sin \frac{\pi}{160}={{\sqrt{2^{{{17}\over{16}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}}}\over{2^{{{33}\over{32}}}}}$

$\cos \frac{\pi}{160}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}}\over{2^{{{33}\over{32}}}}}$

$\sin \frac{\pi}{320}={{\sqrt{2^{{{33}\over{32}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}}}\over{2^{{{65}\over{64}}}}}$

$\cos \frac{\pi}{320}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}}\over{2^{{{65}\over{64}}}}}$

$\sin \frac{\pi}{640}={{\sqrt{2^{{{65}\over{64}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}}}\over{2^{{{129}\over{128}}}}}$

$\cos \frac{\pi}{640}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}}\over{2^{{{129}\over{128}}}}}$

$\sin \frac{\pi}{1280}={{\sqrt{2^{{{129}\over{128}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}}}\over{2^{{{257}\over{256}}}}}$

$\cos \frac{\pi}{1280}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}}\over{2^{{{257}\over{256}}}}}$

$\sin \frac{\pi}{2560}={{\sqrt{2^{{{257}\over{256}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}}}\over{2^{{{513}\over{512}}}}}$

$\cos \frac{\pi}{2560}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}}\over{2^{{{513}\over{512}}}}}$

$\sin \frac{\pi}{5120}={{\sqrt{2^{{{513}\over{512}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}}}\over{2^{{{1025}\over{1024}}}}}$

$\cos \frac{\pi}{5120}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}}\over{2^{{{1025}\over{1024}}}}}$

$\sin \frac{\pi}{10240}={{\sqrt{2^{{{1025}\over{1024}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}}}\over{2^{{{2049}\over{2048}}}}}$

$\cos \frac{\pi}{10240}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}}\over{2^{{{2049}\over{2048}}}}}$

$\sin \frac{\pi}{20480}={{\sqrt{2^{{{2049}\over{2048}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}}}\over{2^{{{4097}\over{4096}}}}}$

$\cos \frac{\pi}{20480}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}+2^{{{2049}\over{2048}}}}}\over{2^{{{4097}\over{4096}}}}}$

$\sin \frac{\pi}{40960}={{\sqrt{2^{{{4097}\over{4096}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}+2^{{{2049}\over{2048}}}}}}\over{2^{{{8193}\over{8192}}}}}$

$\cos \frac{\pi}{40960}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}+2^{{{2049}\over{2048}}}}+2^{{{4097}\over{4096}}}}}\over{2^{{{8193}\over{8192}}}}}$

${\small \sin \frac{\pi}{81920}={{\sqrt{2^{{{8193}\over{8192}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}+2^{{{2049}\over{2048}}}}+2^{{{4097}\over{4096}}}}}}\over{2^{{{16385}\over{16384}}}}}}$

${\small \cos \frac{\pi}{81920}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}+2^{{{2049}\over{2048}}}}+2^{{{4097}\over{4096}}}}+2^{{{8193}\over{8192}}}}}\over{2^{{{16385}\over{16384}}}}}}$

${\small \sin \frac{\pi}{163840}={{\sqrt{2^{{{16385}\over{16384}}}-\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}+2^{{{2049}\over{2048}}}}+2^{{{4097}\over{4096}}}}+2^{{{8193}\over{8192}}}}}}\over{2^{{{32769}\over{32768}}}}}}$

${\small \cos \frac{\pi}{163840}={{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}+5}+2^{{{3}\over{2}}}}+2^{{{5}\over{4}}}}+2^{{{9}\over{8}}}}+2^{{{17}\over{16}}}}+2^{{{33}\over{32}}}}+2^{{{65}\over{64}}}}+2^{{{129}\over{128}}}}+2^{{{257}\over{256}}}}+2^{{{513}\over{512}}}}+2^{{{1025}\over{1024}}}}+2^{{{2049}\over{2048}}}}+2^{{{4097}\over{4096}}}}+2^{{{8193}\over{8192}}}}+2^{{{16385}\over{16384}}}}}\over{2^{{{32769}\over{32768}}}}}}$