三角函式精確值是利用三角函式的公式將特定的三角函式值加以化簡，並以數學根式或分數表示。 用根式或分數表達的精確三角函式有時很有用，主要用於簡化的解決某些方程式能進一步化簡。 根據尼雲定理，有理數度數的角的正弦值，其中的有理數僅有 0，pmfrac 12，1。 相同角度的轉換表 角度單位 值 值 值 值 值 值 值 值 轉 0 frac 112 frac 18 frac 16 frac 14 frac 12 frac 34 1 角度 0circ 30circ 45circ 60circ 90circ 180circ 270circ 360circ 弧度 0 frac pi6 frac pi4 frac pi3 frac pi2 pi frac 3pi2 2pi 梯度 0g 33frac 13g 50g 66frac 23g 100g 200g 300g 400g 計算方式 基於常識 例如：0、30、45 經由半形公式的計算 例如：15、225 sinleft frac x2right pm sqrt tfrac 121 cos x cosleft frac x2right pm sqrt tfrac 121 cos x 利用三倍角公式求 frac13 角 例如：10、20、7 等等，非三的倍數的角的精確值。 sin 3theta 3 sin theta 4 sin3theta cos 3theta 4 cos3theta 3 cos theta 把它改為 sin theta 3 sin frac 13theta 4 sin3frac 13theta cos theta 4 cos3frac 13theta 3 cos frac 13theta 把 cos frac 13theta 當成未知數，cos theta 當成常數項 解一元三次方程式即可求出 例如：sinfrac pi9sin 20circsqrt 3frac sqrt 316sqrt frac 1256sqrt 3frac sqrt 316sqrt frac 1256 同樣地，若角度代未知數，則會得到三分之一角公式。 經由尤拉公式的計算 cosfrac thetan Releft sqrt ncosthetaisinthetaright frac 12left sqrt ncosthetaisinthetasqrt ncosthetaisinthetaright sinfrac thetan Imleft sqrt ncosthetaisinthetaright frac 12ileft sqrt ncosthetaisinthetasqrt ncosthetaisinthetaright 例如： sin 1circ frac 12ileft sqrt 3cos 3circisin 3circsqrt 3cos 3circisin 3circright frac 14sqrt 32 iBiggsqrt 3left 2 1sqrt3sqrt 5sqrt5sqrt2 sqrt51sqrt31rightileft 2 1sqrt3sqrt 5sqrt5sqrt2 sqrt51sqrt31right sqrt 3left 2 1sqrt3sqrt 5sqrt5sqrt2 sqrt51sqrt31rightileft 2 1sqrt3sqrt 5sqrt5sqrt2 sqrt51sqrt31rightBigg 由 Wolfram Alpha 驗算： 經由和角公式的計算 例如：21 9 12 sin x pm y sin x cos y pm cos x sin y cos x pm y cos x cos y mp sin x sin y 經由托勒密定理的計算 例如：18 根據托勒密定理，在圓內接四邊形 ABCD 中， a2abb2 left frac abright2frac ab1 mathrm crd 36circmathrm crdleft anglemathrm ADBrightfrac abfrac sqrt 512 mathrm crd theta2sin frac theta2 sin 18circfrac sqrt514 三角函式精確值列表 由於三角函式的特性，大於 45 角度的三角函式值，可以經由自 0～45 的角度的三角函式值的相關的計算取得。 0：根本 sin 00 cos 01 tan 00 1：2 的一半 sin 1circ frac 1sqrt 3 i16sqrt 34sqrt 308sqrt 153sqrt 58sqrt 5sqrt 54sqrt 104sqrt 64sqrt 2left 4sqrt 308sqrt 153sqrt 58sqrt 5sqrt 54sqrt 104sqrt 64sqrt 2right i frac 1sqrt 3 i16sqrt 34sqrt 308sqrt 153sqrt 58sqrt 5sqrt 54sqrt 104sqrt 64sqrt 2left 4sqrt 308sqrt 153sqrt 58sqrt 5sqrt 54sqrt 104sqrt 64sqrt 2right i 使用 Mathematica 驗算，程式碼為 N ArcSin 1 Sqrt 3 I16 Power 4 Sqrt 30 8 Sqrt 15 3 Sqrt 5 8 Sqrt 5 Sqrt 5 4 Sqrt 10 4 Sqrt 6 4 Sqrt 2 4 Sqrt 30 8 Sqrt 15 3 Sqrt 5 8 Sqrt 5 Sqrt 5 4 Sqrt 10 4 Sqrt 6 4 Sqrt 2 I 31 1 Sqrt 3 I16 Power 4 Sqrt 30 8 Sqrt 15 3 Sqrt 5 8 Sqrt 5 Sqrt 5 4 Sqrt 10 4 Sqrt 6 4 Sqrt 2 4 Sqrt 30 8 Sqrt 15 3 Sqrt 5 8 Sqrt 5 Sqrt 5 4 Sqrt 10 4 Sqrt 6 4 Sqrt 2 I 31 100Degree 結果為 1 與原角度無誤差 15：正一百二十邊形 sinleft frac pi120right sinleft 15circright frac left sqrt 2sqrt2rightleft sqrt 15sqrt3sqrt 102sqrt5right left sqrt 2sqrt2rightleft sqrt 306sqrt5sqrt51right16 cosleft frac pi120right cosleft 15circright frac left sqrt 2sqrt2rightleft sqrt 306sqrt5sqrt51right left sqrt 2sqrt2rightleft sqrt 15sqrt3sqrt 102sqrt5right16 1875：正九十六邊形 sinleft frac pi96right sinleft 1875circright frac12sqrt 2sqrt 2sqrt 2sqrt 2sqrt 3 cosleft frac pi96right cosleft 1875circright frac12sqrt 2sqrt 2sqrt 2sqrt 2sqrt 3 tanleft frac pi96right tanleft 1875circright frac sqrt 2sqrt sqrt sqrt sqrt 3222sqrt sqrt sqrt sqrt sqrt 32222 2：6 的三分之一 sin 2circ frac 12ileft sqrt 3cos 6circisin 6circsqrt 3cos 6circisin 6circright frac 14iBiggsqrt 3left sqrt 2 5sqrt5sqrt3 sqrt51rightileft sqrt 6 5sqrt5sqrt51right sqrt 3left sqrt 2 5sqrt5sqrt3 sqrt51rightileft sqrt 6 5sqrt5sqrt51rightBigg cos 2circ frac 12left sqrt 3cos 6circisin 6circsqrt 3cos 6circisin 6circright frac 14Biggsqrt 3left sqrt 2 5sqrt5sqrt3 sqrt51rightileft sqrt 6 5sqrt5sqrt51right sqrt 3left sqrt 2 5sqrt5sqrt3 sqrt51rightileft sqrt 6 5sqrt5sqrt51rightBigg 225：正八十邊形 sinleft frac pi80right sinleft 225circright frac sqrt 2sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 554484 cosleft frac pi80right cosleft 225circright frac sqrt 2sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 554484 tanleft frac pi80right tanleft 225circright frac sqrt sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 55444sqrt sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 55444 cotleft frac pi80right cotleft 225circright frac sqrt sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 55444sqrt sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 55444 secleft frac pi80right secleft 225circright frac 2sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 55444 cscleft frac pi80right cscleft 225circright frac 2sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 2sqrt sqrt 55444 28125：正六十四邊形 sinleft frac pi64right sinleft 28125circright frac12sqrt 2sqrt 2sqrt 2sqrt 2sqrt 2 cosleft frac pi64right cosleft 28125circright frac12sqrt 2sqrt 2sqrt 2sqrt 2sqrt 2 3：正六十邊形 sinfrac pi60sin 3circtfrac 14 sqrt 8sqrt3sqrt 15sqrt 102sqrt5 cosfrac pi60cos 3circtfrac 14 sqrt 8sqrt3sqrt 15sqrt 102sqrt5 tanfrac pi60tan 3circtfrac 14 left 2sqrt33sqrt52rightleft 2sqrt 2 5sqrt5right 375：正四十八邊形 sinleft frac pi48right sinleft 375circright frac12sqrt 2sqrt 2sqrt 2sqrt 3 cosleft frac pi48right cosleft 375circright frac12sqrt 2sqrt 2sqrt 2sqrt 3 4：12 的三分之一 sin 4circ frac 12ileft sqrt 3cos 12circisin 12circsqrt 3cos 12circisin 12circright frac 14iBiggsqrt 3left sqrt 6 5sqrt5sqrt51rightileft sqrt 2 5sqrt5sqrt3 sqrt51right sqrt 3left sqrt 6 5sqrt5sqrt51rightileft sqrt 2 5sqrt5sqrt3 sqrt51rightBigg cos 4circ frac 12left sqrt 3cos 12circisin 12circsqrt 3cos 12circisin 12circright frac 14Biggsqrt 3left sqrt 6 5sqrt5sqrt51rightileft sqrt 2 5sqrt5sqrt3 sqrt51right sqrt 3left sqrt 6 5sqrt5sqrt51rightileft sqrt 2 5sqrt5sqrt3 sqrt51rightBigg 45：正四十邊形 sinleft frac pi40right sinleft 45circright frac 12 sqrt 2sqrt 2sqrt frac 5sqrt 52 cosleft frac pi40right cosleft 45circright frac 12sqrt 2sqrt 2sqrt frac 5sqrt 52 5：15 的三分之一、正三十六邊形 sinfrac pi36sin 5circ frac 2 2sqrt 3mathrm i2 sqrt 32 sqrt 2 sqrt 6 2sqrt 3 frac 1 sqrt 3mathrm i sqrt 32 sqrt 2 sqrt 6 2sqrt 38 5625：正三十二邊形 sinleft frac pi32right sinleft 5625circright frac12sqrt 2sqrt 2sqrt 2sqrt 2 cosleft frac pi32right cosleft 5625circright frac12sqrt 2sqrt 2sqrt 2sqrt 2 6：正三十邊形 sinfrac pi30sin 6circtfrac 18 left sqrt 6 5sqrt5sqrt51right cosfrac pi30cos 6circtfrac 18 left sqrt 2 5sqrt5sqrt3 sqrt51right tanfrac pi30tan 6circtfrac 12 left sqrt 2 5sqrt5sqrt3 sqrt51right cotfrac pi30cot 6circtfrac 12 left sqrt 5022sqrt53sqrt3sqrt 15right secfrac pi30sec 6circsqrt3sqrt 52sqrt5 cscfrac pi30csc 6circ2sqrt5sqrt 156sqrt5 75：正二十四邊形 sinfrac pi24sin 75circtfrac 14 sqrt 82sqrt62sqrt2 cosfrac pi24cos 75circtfrac 14 sqrt 82sqrt62sqrt2 tanfrac pi24tan 75circsqrt6sqrt22sqrt3 cotfrac pi24cot 75circsqrt6sqrt22sqrt3 secfrac pi24sec 75circsqrt 166sqrt610sqrt28sqrt3 cscfrac pi24csc 75circsqrt 166sqrt610sqrt28sqrt3 9：正二十邊形 sinfrac pi20sin 9circtfrac 14 sqrt 82sqrt 102sqrt5 cosfrac pi20cos 9circtfrac 14 sqrt 82sqrt 102sqrt5 tanfrac pi20tan 9circsqrt51sqrt 52sqrt5 cotfrac pi20cot9circsqrt51sqrt 52sqrt5 10：正十八邊形 tan10circfrac 1sqrt 3rm i6sqrt 312sqrt3 36 rm ifrac 1sqrt 3rm i6sqrt 312sqrt3 36 rm i frac 1sqrt3 1125：正十六邊形 sinfrac pi16sin 1125circfrac 12sqrt 2sqrt 2sqrt 2 cosfrac pi16cos 1125circfrac 12sqrt 2sqrt 2sqrt 2 tanfrac pi16tan 1125circsqrt 42sqrt 2sqrt 21 cotfrac pi16cot 1125circsqrt 42sqrt 2sqrt 21 12：正十五邊形 sinfrac pi15sin 12circtfrac 18 left sqrt 2 5sqrt5sqrt3 sqrt51right cosfrac pi15cos 12circtfrac 18 left sqrt 6 5sqrt5sqrt51right tanfrac pi15tan 12circtfrac 12 left sqrt3 3sqrt5sqrt 2 2511sqrt5right 15：正十二邊形 sinfrac pi12sin 15circfrac 14sqrt2left sqrt31right cosfrac pi12cos 15circfrac 14sqrt2left sqrt31right tanfrac pi12tan 15circ2sqrt3 cotfrac pi12cot 15circ2sqrt3 18：正十邊形 sinfrac pi10sin 18circtfrac 14left sqrt51righttfrac 12varphi1 cosfrac pi10cos 18circtfrac 14sqrt 2left 5sqrt5right tanfrac pi10tan 18circtfrac 15sqrt 5left 52sqrt5right 20：正九邊形、60 的三分之一 sinfrac pi9sin 20circsqrt 3frac sqrt 316sqrt frac 1256sqrt 3frac sqrt 316sqrt frac 1256 2frac 43left sqrt 3isqrt 3sqrt 3isqrt 3right cosfrac pi9cos 20circ 2frac 43left sqrt 31isqrt 3sqrt 31isqrt 3right 21：9 與 12 的和 sinfrac 7pi60sin 21circtfrac 14sqrt 8sqrt3sqrt 15sqrt 102sqrt5 cosfrac 7pi60cos 21circtfrac 14sqrt 8sqrt3sqrt 15sqrt 102sqrt5 tanfrac 7pi60tan 21circtfrac 14left 2left 2sqrt3rightleft 3sqrt5rightrightleft 2sqrt 2left 5sqrt5rightright 36017，mathbf left 21frac 317rightcirc，mathbf left frac 36017rightcirc：正十七邊形 operatorname cos2piover17frac 1sqrt 17sqrt 342sqrt 172sqrt 173sqrt 17sqrt 342sqrt 172sqrt 342sqrt 1716 225：正八邊形 sinfrac pi8sin 225circtfrac 12 left sqrt 2sqrt 2 right cosfrac pi8cos 225circtfrac 12 left sqrt 2sqrt 2 right tanfrac pi8tan 225circsqrt 21 24：12 的二倍 sinfrac 2pi15sin 24circtfrac 18left sqrt3 sqrt51sqrt2sqrt 5sqrt5right cosfrac 2pi15cos 24circtfrac 18left sqrt6sqrt 5sqrt5sqrt51right tanfrac 2pi15tan 24circtfrac 12left sqrt 2 2511sqrt5sqrt3 3sqrt5right 1807，mathbf left 25frac 57rightcirc，mathbf left frac 1807rightcirc：正七邊形 cosfrac pi7cosfrac 1807circcos 25frac 57circfrac 16frac 1sqrt 3 i24sqrt 32884sqrt 3 ifrac 1sqrt 3 i24sqrt 32884sqrt 3 i 27：12 與 15 的和 sinfrac 3pi20sin 27circtfrac 18left 2sqrt 5sqrt5sqrt2sqrt51right cosfrac 3pi20cos 27circtfrac 18left 2sqrt 5sqrt5sqrt2left sqrt51rightright tanfrac 3pi20tan 27circsqrt51sqrt 52sqrt5 30：正六邊形 sinfrac pi6sin 30circtfrac 12 cosfrac pi6cos 30circtfrac 12sqrt3 tanfrac pi6tan 30circtfrac 13sqrt3 33：15 與 18 的和 sinfrac 11pi60sin 33circtfrac 14sqrt 8sqrt3sqrt 15sqrt 102sqrt5 cosfrac 11pi60cos 33circtfrac 14sqrt 8sqrt3sqrt 15sqrt 102sqrt5 tanfrac 11pi60tan 33circtfrac 14left 2sqrt3sqrt51rightleft 2sqrt 52sqrt53sqrt5right cotfrac 11pi60cot33circtfrac 14left 2sqrt3sqrt51rightleft 2sqrt 52sqrt53sqrt5right 36：正五邊形 sinfrac pi5sin 36circtfrac14left sqrt 2left 5sqrt5rightright cosfrac pi5cos 36circfrac 1sqrt54tfrac 12varphi tanfrac pi5tan 36circsqrt 52sqrt5 39：18 與 21 的和 sinfrac 13pi60sin 39circtfrac1 4sqrt 8sqrt3sqrt 15sqrt 102sqrt5 cosfrac 13pi60cos 39circtfrac1 4sqrt 8sqrt3sqrt 15sqrt 102sqrt5 tanfrac 13pi60tan 39circtfrac14left left 2sqrt3rightleft 3sqrt5right2rightleft 2sqrt 2left 5sqrt5rightright 42：21 的 2 倍 sinfrac 7pi30sin 42circfrac sqrt6sqrt 5sqrt5sqrt518 cosfrac 7pi30cos 42circfrac sqrt2sqrt 5sqrt5sqrt3left sqrt51right8 tanfrac 7pi30tan 42circfrac1 2left sqrt3sqrt 15sqrt 102sqrt5right cotfrac 7pi30cot 42circfrac1 2left 3sqrt3sqrt 15sqrt 5022sqrt5right secfrac 7pi30sec 42circsqrt 52sqrt5sqrt3 secfrac 7pi30sec 42circsqrt 156sqrt5sqrt52 45：正方形 sinfrac pi4sin 45circfrac sqrt22frac 1sqrt2 cosfrac pi4cos 45circfrac sqrt22frac 1sqrt2 tanfrac pi4tan 45circ1 48 sin 48circfrac 14sqrt 7sqrt 5sqrt 6 5sqrt 5 54：27 與 27 的和 sinfrac 3pi10sin 54circfrac sqrt514 cosfrac 3pi10cos 54circfrac sqrt 102sqrt 54 tanfrac 3pi10tan 54circfrac sqrt 2510sqrt 55 cotfrac 3pi10cot 54circsqrt 52sqrt 5 60：等邊三角形 sinfrac pi3sin 60circfrac sqrt32 cosfrac pi3cos 60circfrac 12 tanfrac pi3tan 60circsqrt3 cotfrac pi3cot 60circfrac sqrt33frac 1sqrt3 675：75 與 60 的和 sinfrac 3pi8sin 675circtfrac 12sqrt 2sqrt 2 cosfrac 3pi8cos 675circtfrac 12sqrt 2sqrt 2 tanfrac 3pi8tan 675circsqrt 21 cotfrac 3pi8cot 675circsqrt 21 72：36 的二倍 sinfrac 2pi5sin 72circtfrac 14sqrt 2left 5sqrt5right cosfrac 2pi5cos 72circtfrac 14left sqrt51right tanfrac 2pi5tan 72circsqrt 52sqrt 5 cotfrac 2pi5cot 72circtfrac 15sqrt 5left 52sqrt5right 75 30 與 45 的和 sinfrac 5pi12sin 75circtfrac 14left sqrt6sqrt2right cosfrac 5pi12cos 75circtfrac 14left sqrt6sqrt2right tanfrac 5pi12tan 75circ2sqrt3 cotfrac 5pi12cot 75circ2sqrt3 81 sin 81circfrac 12sqrt frac 12Big 4sqrt 2 5sqrt 5Big 90：根本 sin frac pi2sin 90circ1 cos frac pi2cos 90circ0 cot frac pi2cot 90circ0 列表 在下表中，i 為虛數單位，omegaexp frac pi i3frac 12frac 12 isqrt 3。 n sinleft frac 2pinright cosleft frac 2pinright tanleft frac 2pinright 1 0 1 0 2 0 0 3 frac 12sqrt 3 4 1 0 pminfty 5 frac 14left sqrt 102sqrt 5right frac 14left sqrt 51right sqrt 52sqrt 5 6 frac 12sqrt 3 frac 12 sqrt 3 7 frac 12sqrt frac 13left 7omega2sqrt 3frac 721sqrt 32omegasqrt 3frac 721sqrt 32right frac 16left 1sqrt 3frac 721sqrt 32sqrt 3frac 721sqrt 32right 8 frac 12sqrt 2 frac 12sqrt 2 1 9 frac 14left sqrt 34 1sqrt 3sqrt 34 1sqrt 3right 10 frac 14left sqrt 102sqrt 5right frac 14left sqrt 51right sqrt 52sqrt 5 11 12 frac 12 frac 12sqrt 3 frac 13sqrt 3 13 14 frac 124sqrt 3left 112sqrt 314336sqrt 5549064193sqrt 314336sqrt 5549064193right frac 124sqrt 3left 80sqrt 314336sqrt 5549064193sqrt 314336sqrt 5549064193right sqrt frac 112sqrt 314336sqrt 5549064193sqrt 314336sqrt 554906419380sqrt 314336sqrt 5549064193sqrt 314336sqrt 5549064193 15 frac 18left sqrt 15sqrt 3sqrt 102sqrt 5right frac 18left 1sqrt 5sqrt 306sqrt 5right frac 12left 3sqrt 3sqrt 15sqrt 5022sqrt 5right 16 frac 12left sqrt 2sqrt 2right frac 12left sqrt 2sqrt 2right sqrt 21 17 frac 14sqrt 8sqrt 2left 15sqrt 17sqrt 342sqrt 172sqrt 173sqrt 17sqrt 17038sqrt 17right frac 116left 1sqrt 17sqrt 342sqrt 172sqrt 173sqrt 17sqrt 342sqrt 172sqrt 342sqrt 17right 18 frac 14left sqrt 34sqrt 14sqrt 3sqrt 34sqrt 14sqrt 3right frac 14left sqrt 344sqrt 3sqrt 344sqrt 3right 19 20 frac 14left sqrt 51right frac 14left sqrt 102sqrt 5right frac 15left sqrt 2510sqrt 5right 21 22 23 24 frac 14left sqrt 6sqrt 2right frac 14left sqrt 6sqrt 2right 2sqrt 3 相關 參見 可作圖多邊形 十七邊形 參考文獻 π3 60π6 30π12 15π24 75 π4 45π8 225π16 1125π32 5625 π5 36π10 18π20 9 π7π14 π9 20π18 10 π11 π13 π15 12π30 6 π17 π19 π23 註釋 Category 三角學 Category 三角函式