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== Andre verdier == Verdier for vinkler utenfor området [0°, 45°] kan utledes fra disse verdiene ved bruk av formlene for [[Trigonometriske identiteter#Symmetri, forskyvninger og periodisitet|symmetri i trigonometriske identiteter]]. Merk at 1° = π/180 [[radian]]er. === 0°: fundamental === : <math>\sin 0=0\,</math> : <math>\cos 0=1\,</math> : <math>\tan 0=0\,</math> : <math>\cot 0\mbox{ er undefinert}\,</math> === 3°: 60-sidet polygon === : <math>\sin\frac{\pi}{60}=\sin 3^\circ=\tfrac{1}{16} \left[2(1-\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3+1)\right]\,</math> : <math>\cos\frac{\pi}{60}=\cos 3^\circ=\tfrac{1}{16} \left[2(1+\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3-1)\right]\,</math> : <math>\tan\frac{\pi}{60}=\tan 3^\circ=\tfrac{1}{4} \left[(2-\sqrt3)(3+\sqrt5)-2\right]\left[2-\sqrt{2(5-\sqrt5)}\right]\,</math> : <math>\cot\frac{\pi}{60}=\cot 3^\circ=\tfrac{1}{4} \left[(2+\sqrt3)(3+\sqrt5)-2\right]\left[2+\sqrt{2(5-\sqrt5)}\right]\,</math> === 6°: 30-sidet polygon === : <math>\sin\frac{\pi}{30}=\sin 6^\circ=\tfrac{1}{8} \left[\sqrt{6(5-\sqrt5)}-\sqrt5-1\right]\,</math> : <math>\cos\frac{\pi}{30}=\cos 6^\circ=\tfrac{1}{8} \left[\sqrt{2(5-\sqrt5)}+\sqrt3(\sqrt5+1)\right]\,</math> : <math>\tan\frac{\pi}{30}=\tan 6^\circ=\tfrac{1}{2} \left[\sqrt{2(5-\sqrt5)}-\sqrt3(\sqrt5-1)\right]\,</math> : <math>\cot\frac{\pi}{30}=\cot 6^\circ=\tfrac{1}{2} \left[\sqrt3(3+\sqrt5)+\sqrt{2(25+11\sqrt5)}\right]\,</math> === 9°: 20-sidet polygon === : <math>\sin\frac{\pi}{20}=\sin 9^\circ=\tfrac{1}{8} \left[\sqrt2(\sqrt5+1)-2\sqrt{5-\sqrt5}\right]\,</math> : <math>\cos\frac{\pi}{20}=\cos 9^\circ=\tfrac{1}{8} \left[\sqrt2(\sqrt5+1)+2\sqrt{5-\sqrt5}\right]\,</math> : <math>\tan\frac{\pi}{20}=\tan 9^\circ=\sqrt5+1-\sqrt{5+2\sqrt5}\,</math> : <math>\cot\frac{\pi}{20}=\cot 9^\circ=\sqrt5+1+\sqrt{5+2\sqrt5}\,</math> === 12°: 15-sidet polygon === : <math>\sin\frac{\pi}{15}=\sin 12^\circ=\tfrac{1}{8} \left[\sqrt{2(5+\sqrt5)}-\sqrt3(\sqrt5-1)\right]\,</math> : <math>\cos\frac{\pi}{15}=\cos 12^\circ=\tfrac{1}{8} \left[\sqrt{6(5+\sqrt5)}+\sqrt5-1\right]\,</math> : <math>\tan\frac{\pi}{15}=\tan 12^\circ=\tfrac{1}{2} \left[\sqrt3(3-\sqrt5)-\sqrt{2(25-11\sqrt5)}\right]\,</math> : <math>\cot\frac{\pi}{15}=\cot 12^\circ=\tfrac{1}{2} \left[\sqrt3(\sqrt5+1)+\sqrt{2(5+\sqrt5)}\right]\,</math> === 15°: dodekagon === : <math>\sin\frac{\pi}{12}=\sin 15^\circ=\tfrac{1}{4}\sqrt2(\sqrt3-1)\,</math> : <math>\cos\frac{\pi}{12}=\cos 15^\circ=\tfrac{1}{4}\sqrt2(\sqrt3+1)\,</math> : <math>\tan\frac{\pi}{12}=\tan 15^\circ=2-\sqrt3\,</math> : <math>\cot\frac{\pi}{12}=\cot 15^\circ=2+\sqrt3\,</math> === 18°: dekagon === : <math>\sin\frac{\pi}{10}=\sin 18^\circ=\tfrac{1}{4}\left(\sqrt5-1\right)=\tfrac{1}{2}\varphi^{-1}\,</math> : <math>\cos\frac{\pi}{10}=\cos 18^\circ=\tfrac{1}{4}\sqrt{2(5+\sqrt5)}\,</math> : <math>\tan\frac{\pi}{10}=\tan 18^\circ=\tfrac{1}{5}\sqrt{5(5-2\sqrt5)}\,</math> : <math>\cot\frac{\pi}{10}=\cot 18^\circ=\sqrt{5+2\sqrt 5}\,</math> === 21°: summen 9° + 12° === : <math>\sin\frac{7\pi}{60}=\sin 21^\circ=\tfrac{1}{16}\left[2(\sqrt3+1)\sqrt{5-\sqrt5}-\sqrt2(\sqrt3-1)(1+\sqrt5)\right]\,</math> : <math>\cos\frac{7\pi}{60}=\cos 21^\circ=\tfrac{1}{16}\left[2(\sqrt3-1)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3+1)(1+\sqrt5)\right]\,</math> : <math>\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]\,</math> : <math>\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]\,</math> === 22.5°: oktogon === : <math>\sin\frac{\pi}{8}=\sin 22.5^\circ=\tfrac{1}{2}(\sqrt{2-\sqrt{2}}),</math> : <math>\cos\frac{\pi}{8}=\cos 22.5^\circ=\tfrac{1}{2}(\sqrt{2+\sqrt{2}})\,</math> : <math>\tan\frac{\pi}{8}=\tan 22.5^\circ=\sqrt{2}-1\,</math> : <math>\cot\frac{\pi}{8}=\cot 22.5^\circ=\sqrt{2}+1\,</math> === 24°: summen 12° + 12° === : <math>\sin\frac{2\pi}{15}=\sin 24^\circ=\tfrac{1}{8}\left[\sqrt3(\sqrt5+1)-\sqrt2\sqrt{5-\sqrt5}\right]\,</math> : <math>\cos\frac{2\pi}{15}=\cos 24^\circ=\tfrac{1}{8}\left(\sqrt6\sqrt{5-\sqrt5}+\sqrt5+1\right)\,</math> : <math>\tan\frac{2\pi}{15}=\tan 24^\circ=\tfrac{1}{2}\left[\sqrt{2(25+11\sqrt5)}-\sqrt3(3+\sqrt5)\right]\,</math> : <math>\cot\frac{2\pi}{15}=\cot 24^\circ=\tfrac{1}{2}\left[\sqrt2\sqrt{5-\sqrt5}+\sqrt3(\sqrt5-1)\right]\,</math> === 27°: summen 12° + 15° === : <math>\sin\frac{3\pi}{20}=\sin 27^\circ=\tfrac{1}{8}\left[2\sqrt{5+\sqrt5}-\sqrt2\;(\sqrt5-1)\right]\,</math> : <math>\cos\frac{3\pi}{20}=\cos 27^\circ=\tfrac{1}{8}\left[2\sqrt{5+\sqrt5}+\sqrt2\;(\sqrt5-1)\right]\,</math> : <math>\tan\frac{3\pi}{20}=\tan 27^\circ=\sqrt5-1-\sqrt{5-2\sqrt5}\,</math> : <math>\cot\frac{3\pi}{20}=\cot 27^\circ=\sqrt5-1+\sqrt{5-2\sqrt5}\,</math> === 30°: heksagon === : <math>\sin\frac{\pi}{6}=\sin 30^\circ=\tfrac{1}{2}\,</math> : <math>\cos\frac{\pi}{6}=\cos 30^\circ=\tfrac{1}{2}\sqrt3\,</math> : <math>\tan\frac{\pi}{6}=\tan 30^\circ=\tfrac{1}{3}\sqrt3\,</math> : <math>\cot\frac{\pi}{6}=\cot 30^\circ=\sqrt3\,</math> === 33°: summen 15° + 18° === : <math>\sin\frac{11\pi}{60}=\sin 33^\circ=\tfrac{1}{16}\left[2(\sqrt3-1)\sqrt{5+\sqrt5}+\sqrt2(1+\sqrt3)(\sqrt5-1)\right]\,</math> : <math>\cos\frac{11\pi}{60}=\cos 33^\circ=\tfrac{1}{16}\left[2(\sqrt3+1)\sqrt{5+\sqrt5}+\sqrt2(1-\sqrt3)(\sqrt5-1)\right]\,</math> : <math>\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]\,</math> : <math>\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]\,</math> === 36°: pentagon ===<!-- This section is linked from [[Pentagram]] --> : <math>\sin\frac{\pi}{5}=\sin 36^\circ=\tfrac14[\sqrt{2(5-\sqrt5)}]\,</math> : <math>\cos\frac{\pi}{5}=\cos 36^\circ=\frac{1+\sqrt5}{4}=\tfrac{1}{2}\varphi\,</math> : <math>\tan\frac{\pi}{5}=\tan 36^\circ=\sqrt{5-2\sqrt5}\,</math> : <math>\cot\frac{\pi}{5}=\cot 36^\circ=\tfrac15[\sqrt{5(5+2\sqrt5)}]\,</math> === 39°: summen 18° + 21° === : <math>\sin\frac{13\pi}{60}=\sin 39^\circ=\tfrac1{16}[2(1-\sqrt3)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3+1)(\sqrt5+1)]\,</math> : <math>\cos\frac{13\pi}{60}=\cos 39^\circ=\tfrac1{16}[2(1+\sqrt3)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3-1)(\sqrt5+1)]\,</math> : <math>\tan\frac{13\pi}{60}=\tan 39^\circ=\tfrac14\left[(2-\sqrt3)(3-\sqrt5)-2\right]\left[2-\sqrt{2(5+\sqrt5)}\right]\,</math> : <math>\cot\frac{13\pi}{60}=\cot 39^\circ=\tfrac14\left[(2+\sqrt3)(3-\sqrt5)-2\right]\left[2+\sqrt{2(5+\sqrt5)}\right]\,</math> === 42°: summen 21° + 21° === : <math>\sin\frac{7\pi}{30}=\sin 42^\circ=\frac{\sqrt6\sqrt{5+\sqrt5}-\sqrt5+1}{8}\,</math> : <math>\cos\frac{7\pi}{30}=\cos 42^\circ=\frac{\sqrt2\sqrt{5+\sqrt5}+\sqrt3(\sqrt5-1)}{8}\,</math> : <math>\tan\frac{7\pi}{30}=\tan 42^\circ=\frac{\sqrt3(\sqrt5+1)-\sqrt2\sqrt{5+\sqrt5}}{2}\,</math> : <math>\cot\frac{7\pi}{30}=\cot 42^\circ=\frac{\sqrt{2(25-11\sqrt5)}+\sqrt3(3-\sqrt5)}{2}\,</math> === 45°: kvadrat === : <math>\sin\frac{\pi}{4}=\sin 45^\circ=\frac{\sqrt2}{2}=\frac{1}{\sqrt2}\,</math> : <math>\cos\frac{\pi}{4}=\cos 45^\circ=\frac{\sqrt2}{2}=\frac{1}{\sqrt2}\,</math> : <math>\tan\frac{\pi}{4}=\tan 45^\circ=1\,</math> : <math>\cot\frac{\pi}{4}=\cot 45^\circ=1\,</math> === 60°: trekant === : <math>\sin\frac{\pi}{3}=\sin 60^\circ=\tfrac{1}{2}\sqrt3\,</math> : <math>\cos\frac{\pi}{3}=\cos 60^\circ=\tfrac{1}{2}\,</math> : <math>\tan\frac{\pi}{3}=\tan 60^\circ=\sqrt3\,</math> : <math>\cot\frac{\pi}{3}=\cot 60^\circ=\tfrac{1}{3}\sqrt3\,</math> der <math> \varphi </math> er [[det gylne snitt]].
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