Eksakte trigonometriske konstanter

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Eksakte verdier av sinus og cosinus på enhetssirkelen
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Alle verdier av sinus, cosinus og tangens til vinkler med 3-graders inkrementer er det mulig å utlede ved å bruke identitetene for halve vinkler, dobbelte vinkler og sum/differanse med verdiene for 0°, 30°, 36°, og 45°. Merk at 1° = n/180 radianer.

Tabell over konstanter[rediger | rediger kilde]

Verdier utenfor området [0°, 45°] kan utledes fra disse verdiene ved bruk av formlene i Trigonometriske identiteter#Periodisitet, symmetri og forskvyninger.

0°: fundamental[rediger | rediger kilde]

\sin 0=0\,
\cos 0=1\,
\tan 0=0\,
\cot 0\mbox{ er undefinert}\,

3°: 60-sidet polygon[rediger | rediger kilde]

\sin\frac{\pi}{60}=\sin 3^\circ=\tfrac{1}{16} \left[2(1-\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3+1)\right]\, følge A019812 i OEIS
\cos\frac{\pi}{60}=\cos 3^\circ=\tfrac{1}{16} \left[2(1+\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3-1)\right]\, A019896
\tan\frac{\pi}{60}=\tan 3^\circ=\tfrac{1}{4} \left[(2-\sqrt3)(3+\sqrt5)-2\right]\left[2-\sqrt{2(5-\sqrt5)}\right]\, A019901
\cot\frac{\pi}{60}=\cot 3^\circ=\tfrac{1}{4} \left[(2+\sqrt3)(3+\sqrt5)-2\right]\left[2+\sqrt{2(5-\sqrt5)}\right]\, A019985

6°: 30-sidet polygon[rediger | rediger kilde]

\sin\frac{\pi}{30}=\sin 6^\circ=\tfrac{1}{8} \left[\sqrt{6(5-\sqrt5)}-\sqrt5-1\right]\, A019815
\cos\frac{\pi}{30}=\cos 6^\circ=\tfrac{1}{8} \left[\sqrt{2(5-\sqrt5)}+\sqrt3(\sqrt5+1)\right]\, A019893
\tan\frac{\pi}{30}=\tan 6^\circ=\tfrac{1}{2} \left[\sqrt{2(5-\sqrt5)}-\sqrt3(\sqrt5-1)\right]\, A019904
\cot\frac{\pi}{30}=\cot 6^\circ=\tfrac{1}{2} \left[\sqrt3(3+\sqrt5)+\sqrt{2(25+11\sqrt5)}\right]\, A019982

9°: 20-sidet polygon[rediger | rediger kilde]

\sin\frac{\pi}{20}=\sin 9^\circ=\tfrac{1}{8} \left[\sqrt2(\sqrt5+1)-2\sqrt{5-\sqrt5}\right]\, A019818
\cos\frac{\pi}{20}=\cos 9^\circ=\tfrac{1}{8} \left[\sqrt2(\sqrt5+1)+2\sqrt{5-\sqrt5}\right]\, A019890
\tan\frac{\pi}{20}=\tan 9^\circ=\sqrt5+1-\sqrt{5+2\sqrt5}\, A019907
\cot\frac{\pi}{20}=\cot 9^\circ=\sqrt5+1+\sqrt{5+2\sqrt5}\, A019979

12°: 15-sidet polygon[rediger | rediger kilde]

\sin\frac{\pi}{15}=\sin 12^\circ=\tfrac{1}{8} \left[\sqrt{2(5+\sqrt5)}-\sqrt3(\sqrt5-1)\right]\, A019821
\cos\frac{\pi}{15}=\cos 12^\circ=\tfrac{1}{8} \left[\sqrt{6(5+\sqrt5)}+\sqrt5-1\right]\, A019887
\tan\frac{\pi}{15}=\tan 12^\circ=\tfrac{1}{2} \left[\sqrt3(3-\sqrt5)-\sqrt{2(25-11\sqrt5)}\right]\, A019910
\cot\frac{\pi}{15}=\cot 12^\circ=\tfrac{1}{2} \left[\sqrt3(\sqrt5+1)+\sqrt{2(5+\sqrt5)}\right]\, A019976

15°: dodekagon[rediger | rediger kilde]

\sin\frac{\pi}{12}=\sin 15^\circ=\tfrac{1}{4}\sqrt2(\sqrt3-1)\, A019824
\cos\frac{\pi}{12}=\cos 15^\circ=\tfrac{1}{4}\sqrt2(\sqrt3+1)\, A019884
\tan\frac{\pi}{12}=\tan 15^\circ=2-\sqrt3\, A019913
\cot\frac{\pi}{12}=\cot 15^\circ=2+\sqrt3\, A019973

18°: dekagon[rediger | rediger kilde]

\sin\frac{\pi}{10}=\sin 18^\circ=\tfrac{1}{4}\left(\sqrt5-1\right)=\tfrac{1}{2}\varphi^{-1}\, A019827
\cos\frac{\pi}{10}=\cos 18^\circ=\tfrac{1}{4}\sqrt{2(5+\sqrt5)}\, A019881
\tan\frac{\pi}{10}=\tan 18^\circ=\tfrac{1}{5}\sqrt{5(5-2\sqrt5)}\, A019916
\cot\frac{\pi}{10}=\cot 18^\circ=\sqrt{5+2\sqrt 5}\, A019970

21°: summen 9° + 12°[rediger | rediger kilde]

\sin\frac{7\pi}{60}=\sin 21^\circ=\tfrac{1}{16}\left[2(\sqrt3+1)\sqrt{5-\sqrt5}-\sqrt2(\sqrt3-1)(1+\sqrt5)\right]\, A019830
\cos\frac{7\pi}{60}=\cos 21^\circ=\tfrac{1}{16}\left[2(\sqrt3-1)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3+1)(1+\sqrt5)\right]\, A019878
\tan\frac{7\pi}{60}=\tan 21^\circ=\tfrac{1}{4}\left[2-(2+\sqrt3)(3-\sqrt5)\right]\left[2-\sqrt{2(5+\sqrt5)}\right]\, A019919
\cot\frac{7\pi}{60}=\cot 21^\circ=\tfrac{1}{4}\left[2-(2-\sqrt3)(3-\sqrt5)\right]\left[2+\sqrt{2(5+\sqrt5)}\right]\, A019967

22.5°: oktogon[rediger | rediger kilde]

\sin\frac{\pi}{8}=\sin 22.5^\circ=\tfrac{1}{2}(\sqrt{2-\sqrt{2}}),
\cos\frac{\pi}{8}=\cos 22.5^\circ=\tfrac{1}{2}(\sqrt{2+\sqrt{2}})\, A144981
\tan\frac{\pi}{8}=\tan 22.5^\circ=\sqrt{2}-1\,
\cot\frac{\pi}{8}=\cot 22.5^\circ=\sqrt{2}+1\, A014176

24°: summen 12° + 12°[rediger | rediger kilde]

\sin\frac{2\pi}{15}=\sin 24^\circ=\tfrac{1}{8}\left[\sqrt3(\sqrt5+1)-\sqrt2\sqrt{5-\sqrt5}\right]\, A019833
\cos\frac{2\pi}{15}=\cos 24^\circ=\tfrac{1}{8}\left(\sqrt6\sqrt{5-\sqrt5}+\sqrt5+1\right)\, A019875
\tan\frac{2\pi}{15}=\tan 24^\circ=\tfrac{1}{2}\left[\sqrt{2(25+11\sqrt5)}-\sqrt3(3+\sqrt5)\right]\, A019922
\cot\frac{2\pi}{15}=\cot 24^\circ=\tfrac{1}{2}\left[\sqrt2\sqrt{5-\sqrt5}+\sqrt3(\sqrt5-1)\right]\, A019964

27°: summen 12° + 15°[rediger | rediger kilde]

\sin\frac{3\pi}{20}=\sin 27^\circ=\tfrac{1}{8}\left[2\sqrt{5+\sqrt5}-\sqrt2\;(\sqrt5-1)\right]\, A019836
\cos\frac{3\pi}{20}=\cos 27^\circ=\tfrac{1}{8}\left[2\sqrt{5+\sqrt5}+\sqrt2\;(\sqrt5-1)\right]\, A019872
\tan\frac{3\pi}{20}=\tan 27^\circ=\sqrt5-1-\sqrt{5-2\sqrt5}\, A019925
\cot\frac{3\pi}{20}=\cot 27^\circ=\sqrt5-1+\sqrt{5-2\sqrt5}\, A019961

30°: heksagon[rediger | rediger kilde]

\sin\frac{\pi}{6}=\sin 30^\circ=\tfrac{1}{2}\,
\cos\frac{\pi}{6}=\cos 30^\circ=\tfrac{1}{2}\sqrt3\,
\tan\frac{\pi}{6}=\tan 30^\circ=\tfrac{1}{3}\sqrt3\,
\cot\frac{\pi}{6}=\cot 30^\circ=\sqrt3\,

33°: summen 15° + 18°[rediger | rediger kilde]

\sin\frac{11\pi}{60}=\sin 33^\circ=\tfrac{1}{16}\left[2(\sqrt3-1)\sqrt{5+\sqrt5}+\sqrt2(1+\sqrt3)(\sqrt5-1)\right]\, A019842
\cos\frac{11\pi}{60}=\cos 33^\circ=\tfrac{1}{16}\left[2(\sqrt3+1)\sqrt{5+\sqrt5}+\sqrt2(1-\sqrt3)(\sqrt5-1)\right]\, A019866
\tan\frac{11\pi}{60}=\tan 33^\circ=\tfrac{1}{4}\left[2-(2-\sqrt3)(3+\sqrt5)\right]\left[2+\sqrt{2(5-\sqrt5)}\right]\, A019931
\cot\frac{11\pi}{60}=\cot 33^\circ=\tfrac{1}{4}\left[2-(2+\sqrt3)(3+\sqrt5)\right]\left[2-\sqrt{2(5-\sqrt5)}\right]\, A019955

36°: pentagon[rediger | rediger kilde]

\sin\frac{\pi}{5}=\sin 36^\circ=\tfrac14[\sqrt{2(5-\sqrt5)}]\, A019845
\cos\frac{\pi}{5}=\cos 36^\circ=\frac{1+\sqrt5}{4}=\tfrac{1}{2}\varphi\, A019863
\tan\frac{\pi}{5}=\tan 36^\circ=\sqrt{5-2\sqrt5}\, A019934
\cot\frac{\pi}{5}=\cot 36^\circ=\tfrac15[\sqrt{5(5+2\sqrt5)}]\, A019952

39°: summen 18° + 21°[rediger | rediger kilde]

\sin\frac{13\pi}{60}=\sin 39^\circ=\tfrac1{16}[2(1-\sqrt3)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3+1)(\sqrt5+1)]\, A019848
\cos\frac{13\pi}{60}=\cos 39^\circ=\tfrac1{16}[2(1+\sqrt3)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3-1)(\sqrt5+1)]\, A019860
\tan\frac{13\pi}{60}=\tan 39^\circ=\tfrac14\left[(2-\sqrt3)(3-\sqrt5)-2\right]\left[2-\sqrt{2(5+\sqrt5)}\right]\, A019937
\cot\frac{13\pi}{60}=\cot 39^\circ=\tfrac14\left[(2+\sqrt3)(3-\sqrt5)-2\right]\left[2+\sqrt{2(5+\sqrt5)}\right]\, A019949

42°: summen 21° + 21°[rediger | rediger kilde]

\sin\frac{7\pi}{30}=\sin 42^\circ=\frac{\sqrt6\sqrt{5+\sqrt5}-\sqrt5+1}{8}\, A019851
\cos\frac{7\pi}{30}=\cos 42^\circ=\frac{\sqrt2\sqrt{5+\sqrt5}+\sqrt3(\sqrt5-1)}{8}\, A019857
\tan\frac{7\pi}{30}=\tan 42^\circ=\frac{\sqrt3(\sqrt5+1)-\sqrt2\sqrt{5+\sqrt5}}{2}\, A019940
\cot\frac{7\pi}{30}=\cot 42^\circ=\frac{\sqrt{2(25-11\sqrt5)}+\sqrt3(3-\sqrt5)}{2}\, A019946

45°: kvadrat[rediger | rediger kilde]

\sin\frac{\pi}{4}=\sin 45^\circ=\frac{\sqrt2}{2}=\frac{1}{\sqrt2}\,
\cos\frac{\pi}{4}=\cos 45^\circ=\frac{\sqrt2}{2}=\frac{1}{\sqrt2}\,
\tan\frac{\pi}{4}=\tan 45^\circ=1\,
\cot\frac{\pi}{4}=\cot 45^\circ=1\,

60°: trekant[rediger | rediger kilde]

\sin\frac{\pi}{3}=\sin 60^\circ=\tfrac{1}{2}\sqrt3\,
\cos\frac{\pi}{3}=\cos 60^\circ=\tfrac{1}{2}\,
\tan\frac{\pi}{3}=\tan 60^\circ=\sqrt3\,
\cot\frac{\pi}{3}=\cot 60^\circ=\tfrac{1}{3}\sqrt3\,

der  \varphi er det gylne snitt.

Se også[rediger | rediger kilde]

Referanser[rediger | rediger kilde]

Eksterne lenker[rediger | rediger kilde]