正五邊形 - 修訂歷史

丁志仁：/* sin18° */

2022-04-04T05:58:30Z

sin18°

2022-04-04T05:55:17Z

方法二：

2022-04-04T05:52:30Z

五頂點座標

2022-04-04T05:48:37Z

五頂點座標

2022-04-04T05:47:16Z

五頂點座標

2022-04-04T05:41:52Z

五頂點座標

2022-04-04T05:32:14Z

五頂點座標

2022-04-04T05:31:49Z

五頂點座標

2022-04-04T05:30:11Z

五頂點座標

2022-04-04T05:28:53Z

五頂點座標

@@ 行 45： / 行 45： @@
 =sin90°cos54°-cos90°sin54°<br>
 =cos54°<br>
-=4(cos18°)^3-3cos18°<br>
+=4(cos18°)<sup>3</sup>-3cos18°<br>
-sin18°cos18°=4(cos18°)^3-3cos18°<br>
+sin18°cos18°=4(cos18°)<sup>3</sup>-3cos18°<br>
-cos18°(4(cos18°)^2-3-2sin18°)=0<br>
+cos18°(4(cos18°)<sup>2</sup>-3-2sin18°)=0<br>
-(1-sin18°^2)-2sin18°-3=0<br>
+(1-sin18°<sup>2</sup>)-2sin18°-3=0<br>
-(sin18°)^2+2sin18°-1=0<br>
+(sin18°)<sup>2</sup>+2sin18°-1=0<br>
 解之得sin18°=(√5-1)/4

@@ 行 18： / 行 18： @@
 :<math>t = \sqrt{\frac {5-\sqrt 5}{2}}\cdot R = 2\sqrt{5-\sqrt{20}}\cdot r</math>
 :<math>R = \sqrt{\frac {2}{5-\sqrt 5}} \approx 0.8507 \cdot t</math>
-:高<math> = \frac{\sqrt{5+2\sqrt{5}}}{2} \cdot</math>t<math>\approx 1.539 \cdot</math>t
+:高<math> = \frac{\sqrt{5+2\sqrt{5}}}{2} \cdot</math>t<math>\approx 1.539 \cdot t</math>
-:寬<math> = \frac{1+\sqrt5}{2} \cdot </math>t<math>\approx 1.618 \cdot</math>t
+:寬<math> = \frac{1+\sqrt5}{2} \cdot </math>t<math>\approx 1.618 \cdot t</math>
-:對角線長<math> = R\ {\sqrt { \frac {5+\sqrt{5}}{2}} } = 2R\cos 18^\circ = 2R\cos\frac{\pi}{10} \approx 1.902 R,</math>
+:對角線長<math> = R\ {\sqrt { \frac {5+\sqrt{5}}{2}} } = 2R\cos 18^\circ = 2R\cos\frac{\pi}{10} \approx 1.902 R</math>
 ===sin18°===

@@ 行 17： / 行 17： @@
 :<math>r = \frac{(\sqrt{5}+1)}{4}</math>R
 :<math>t = \sqrt{\frac {5-\sqrt 5}{2}}\cdot R = 2\sqrt{5-\sqrt{20}}\cdot r</math>
-:<math>R = \sqrt{\frac {2}{5-\sqrt 5}} = \approx 0.8507 \cdot t</math>
+:<math>R = \sqrt{\frac {2}{5-\sqrt 5}} \approx 0.8507 \cdot t</math>
 :高<math> = \frac{\sqrt{5+2\sqrt{5}}}{2} \cdot</math>t<math>\approx 1.539 \cdot</math>t
 :寬<math> = \frac{1+\sqrt5}{2} \cdot </math>t<math>\approx 1.618 \cdot</math>t

@@ 行 16： / 行 16： @@
 :<math>r = \frac{t}{2\tan ( \pi /5)}=\frac{t}{2\sqrt{5-\sqrt{20}}}\approx 0.6882 \cdot t</math>
 :<math>r = \frac{(\sqrt{5}+1)}{4}</math>R
-:<math>t = \sqrt{ \frac {5-\sqrt 5}{2}}\cdot R = 2\sqrt{5-\sqrt{20}}\cdot r</math>
+:<math>t = \sqrt{\frac {5-\sqrt 5}{2}}\cdot R = 2\sqrt{5-\sqrt{20}}\cdot r</math>
-:<math>R = \approx 1.1756 \cdot t</math>
+:<math>R = \sqrt{\frac {2}{5-\sqrt 5}} = \approx 0.8507 \cdot t</math>
 :高<math> = \frac{\sqrt{5+2\sqrt{5}}}{2} \cdot</math>t<math>\approx 1.539 \cdot</math>t
 :寬<math> = \frac{1+\sqrt5}{2} \cdot </math>t<math>\approx 1.618 \cdot</math>t