查看“︁三角函數精確值”︁的源代码

'''三角函數精確值'''是利用[[三角恆等式|三角函數的公式]]將特定的[[三角函數]]值加以化簡，並以數學[[方根|根式]]或[[分數]]表示

用[[方根|根式]]或[[分數]]表達的精確[[三角函數]]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。

'''注意'''：以下為相同角度的轉換表：
{{相同角度的轉換表}}

==計算方式==
===基於常識===
例如：0°、30°、45°
[[File:Unit_circle_angles.svg|center|500px|單位圓]]

===經由[[三角恒等式#倍角公式和半角公式|半角公式]]的計算===
例如：15°、22.5°
:<math>\sin\left(\frac{x}{2}\right) =  \pm\, \sqrt{\tfrac{1}{2}(1 - \cos x)}</math>

:<math>\cos\left(\frac{x}{2}\right) =  \pm\, \sqrt{\tfrac{1}{2}(1 + \cos x)}</math>

===利用[[三角恒等式#倍角公式和半角公式|三倍角公式]]求<math>\frac{1}{3}\,</math>角===
例如：10°、20°、7°......等，非三的倍數的角的精確值。
*<math>\sin 3\theta = 3 \sin \theta- 4 \sin^3\theta \,</math>

*<math>\cos 3\theta = 4 \cos^3\theta - 3 \cos \theta \,</math>
把它改為
*<math>\sin \theta = 3 \sin \frac{1}{3}\theta- 4 \sin^3\frac{1}{3}\theta \,</math>
*<math>\cos \theta = 4 \cos^3\frac{1}{3}\theta - 3 \cos \frac{1}{3}\theta \,</math>
把<math>\cos \frac{1}{3}\theta \,</math>當成未知數，<math>\cos \theta \,</math>當成常數項
解[[三次方程|一元三次方程式]]即可求出

例如：<math>\sin\frac{\pi}{9}=\sin 20^\circ=\sqrt[3]{-\frac{\sqrt{3}}{16}+\sqrt{-\frac{1}{256}}}+\sqrt[3]{-\frac{\sqrt{3}}{16}-\sqrt{-\frac{1}{256}}}</math>

===經由合角公式的計算===
例如：21° = 9° + 12°
:<math>\sin(x \pm y) = \sin(x) \cos(y) \pm \cos(x) \sin(y)\,</math>

:<math>\cos(x \pm y) = \cos(x) \cos(y) \mp \sin(x) \sin(y)\,</math>

===經由托勒密定理的計算===
{{see|托勒密定理|弦 (幾何)}}
[[File:Ptolemy Pentagon.svg|right|thumb|Chord(36°) = a/b = 1/f, from [[托勒密定理]] ]]
例如：18°
: <math>\mathrm{crd}\ {36^\circ}=\mathrm{crd}\left(\angle\mathrm{ADB}\right)=\frac{a}{b}=\frac{2}{1+\sqrt{5}}</math>

: <math>\mathrm{crd}\ {\theta}=2\sin{\frac{\theta}{2}}\,</math>

: <math>\sin{18^\circ}=\frac{1}{1+\sqrt{5}}=\tfrac{1}{4}\left(\sqrt5-1\right)</math>

== 三角函數精確值列表 ==
由於三角函數的特性，大於45°角度的三角函數值，可以經由自0°～ 45°的角度的三角函數值的相關的計算取得。

=== 0°: 根本===
: <math>\sin 0=0\,</math>
: <math>\cos 0=1\,</math>
: <math>\tan 0=0\,</math>

=== 3°: 正60邊形 ===
: <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>

=== 6°: 正30邊形===
: <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>

=== 9°: 正20邊形===
: <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>

=== 10° ===
:<math>{}_{\tan10^\circ=-\frac{-1-\sqrt{3}{\rm{i}}}{6}\sqrt[3]{-12\sqrt3 + 36{\rm{i}}}-\frac{-1+\sqrt{3}{\rm{i}}}{6}\sqrt[3]{-12\sqrt3 - 36{\rm{i}}} + \frac{\sqrt3}{3}}\,</math>

=== 12°: [[十五邊形|正十五邊形]]===
: <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>

=== 15°: [[十二邊形|正十二邊形]] ===
: <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>

=== 18°: [[十邊形|正十邊形]] ===
: <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>

=== 20°: [[九邊形|正九邊形]]   和 60°的三分之一(<math>\frac{1}{3}\,</math>  60°) ===
: <math>\sin\frac{\pi}{9}=\sin 20^\circ=\sqrt[3]{-\frac{\sqrt{3}}{16}+\sqrt{-\frac{1}{256}}}+\sqrt[3]{-\frac{\sqrt{3}}{16}-\sqrt{-\frac{1}{256}}}=</math>
:: <math>2^{-\frac{4}{3}}(\sqrt[3]{i-\sqrt{3}}-\sqrt[3]{i+\sqrt{3}})</math>
: <math>\cos\frac{\pi}{9}=\cos 20^\circ=</math>
:: <math>2^{-\frac{4}{3}}(\sqrt[3]{1+i\sqrt{3}}+\sqrt[3]{1-i\sqrt{3}})</math>

=== 21°: 9° 與 12°的[[加法|和]] ===
: <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>(21\frac{3}{17}^{\circ}) , \,</math>(360/17)°：[[十七邊形|正17邊形]]===
:<math>\operatorname{cos}{2\pi\over17}=\frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}}{16}.</math>

=== 22.5°: [[八邊形|正八邊形]] ===
: <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>

=== 24°: 兩倍的 12° 角 ===
: <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>
=== 25(6/7)°，(180/7)°：[[七邊形|正七邊形]]===
: <math>\cos\frac{\pi}{7}=\cos\frac{180}{7}^\circ=\cos 25\frac{6}{7}^\circ=\frac{1}{6}+\frac{1-\sqrt{3} i}{24}\sqrt[3]{28-84\sqrt{3} i}+\frac{1+\sqrt{3} i}{24}\sqrt[3]{28-84\sqrt{3} i}</math>

=== 27°: 12° 與 15° 的和===
: <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>

=== 30°: [[六邊形|正六邊形]] ===
: <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>

=== 33°: 15° 與 18° [[加法|之和]]===
: <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>

=== 36°: [[五邊形|正五邊形]] ===
: <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>

=== 39°: 18°角加21°角 ===
: <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>

=== 42°: 21°的[[2|兩]][[倍數|倍]] ===
: <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>

=== 45°: [[正方形]] ===
: <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>

==相關==
{{see|三角函數|三角恆等式}}
{{see|:en:Generating trigonometric tables}}
==參見==
*[[可作图多边形]]
*[[:en:Trigonometric_number|三角數]]
*[[十七邊形]]

==參考文獻==
* {{MathWorld|title=Constructible polygon|urlname=ConstructiblePolygon}}
* {{MathWorld|title=Trigonometry angles|urlname=TrigonometryAngles}}
** [http://mathworld.wolfram.com/TrigonometryAnglesPi3.html π/3 (60°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi6.html π/6 (30°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi12.html π/12 (15°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi24.html π/24 (7.5°)]
** [http://mathworld.wolfram.com/TrigonometryAnglesPi4.html π/4 (45°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi8.html π/8 (22.5°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi16.html π/16 (11.25°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi32.html π/32 (5.625°)]
** [http://mathworld.wolfram.com/TrigonometryAnglesPi5.html π/5 (36°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi10.html π/10 (18°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi20.html π/20 (9°)]
** [http://mathworld.wolfram.com/TrigonometryAnglesPi7.html π/7] — ''π/14''
** [http://mathworld.wolfram.com/TrigonometryAnglesPi9.html π/9 (20°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi18.html π/18 (10°)]
** [http://mathworld.wolfram.com/TrigonometryAnglesPi11.html π/11]
** [http://mathworld.wolfram.com/TrigonometryAnglesPi13.html π/13]
** [http://mathworld.wolfram.com/TrigonometryAnglesPi15.html π/15 (12°)] — [http://mathworld.wolfram.com/TrigonometryAnglesPi30.html π/30 (6°)]
** [http://mathworld.wolfram.com/TrigonometryAnglesPi17.html π/17]
** ''π/19''
** [http://mathworld.wolfram.com/TrigonometryAnglesPi23.html π/23]
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[[Category:三角学]]