sin30度是1/2（0.5）。sin是三角函数的正炫函数，sin30º即为在直角三角形中该角大小为30º的正炫，值的计算为：该角所对的直角边比上斜边，结果是½。

在高中，我们学过正弦函数，也知道一些特殊角度的正弦数值，如sin90°=1，sin60°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909104666.png","uri":"pgc-image/a75fa853c40240a197402c96080d5863","width":33,"height":49,"darkImgUrl":"https://p26.toutiaoimg.com/origin/pgc-image/0ced9281decd4dd5a3954c7bebc9f1b8","darkImgUri":"pgc-image/0ced9281decd4dd5a3954c7bebc9f1b8","formulaImgStatus":"succeed"}" data-formula="frac{sqrt{3}}{2}">cript>cript>，sin45°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909104711.png","uri":"pgc-image/64f709d911b5442896885c32ecc36ef6","width":33,"height":49,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/7bc184d61d994cd0a7edfcf5670eba8e","darkImgUri":"pgc-image/7bc184d61d994cd0a7edfcf5670eba8e","formulaImgStatus":"succeed"}" data-formula="frac{sqrt{2}}{2}">cript>cript>，sin30°=0.5。但是对于其他非特殊角度的正弦值，我们就不太好计算了。

最近，我在图书馆看到一本国外的科普书《天啊，几何还能这样学》，里面就聊到如何快速手算任意角度的正弦值，方法很新颖，也很有效果，如下图所示。

这种方法的原理是，当角度很小时，该角度所对的圆弧与半径的比值与正弦值非常接近，误差很小。这时，计算正弦值就转换为计算圆弧与半径的比值。

如下图所示，点A为圆心，AB、AD为圆的半径，BD是角A所对应的圆弧，BC垂直AD于C。当角A很小时，圆弧BD与线段BC的长度就非常接近。

假设AB=R，于是sinA°=BC/AB ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909104988.png","uri":"pgc-image/1e4dfc020b8f4a00b5e114bb3c5528ae","width":16,"height":27,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/22dd5e5b831a4d2486f70d0b18e5739a","darkImgUri":"pgc-image/22dd5e5b831a4d2486f70d0b18e5739a","formulaImgStatus":"succeed"}" data-formula="approx">cript>cript> BD/AB=BD/R。

又因为BD=2ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909104946.png","uri":"pgc-image/ea9e19888d984e488af891b78bc1264b","width":13,"height":28,"darkImgUrl":"https://p26.toutiaoimg.com/origin/pgc-image/4c0213efdeae4e778e3d80ddf9d3dfd8","darkImgUri":"pgc-image/4c0213efdeae4e778e3d80ddf9d3dfd8","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript>R（A/360），于是sinA°ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105010.png","uri":"pgc-image/06416d8d612f47cf94d9ddb4d65b560d","width":16,"height":27,"darkImgUrl":"https://p5.toutiaoimg.com/origin/pgc-image/6403b6f373c1456893d2cb52c49e5ce4","darkImgUri":"pgc-image/6403b6f373c1456893d2cb52c49e5ce4","formulaImgStatus":"succeed"}" data-formula="approx">cript>cript>BD/R=ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105059.png","uri":"pgc-image/c3391fceb9e24fe099f5b4785aa64432","width":23,"height":28,"darkImgUrl":"https://p9.toutiaoimg.com/origin/pgc-image/f00669d04b384f36b01f6a305a17c0d4","darkImgUri":"pgc-image/f00669d04b384f36b01f6a305a17c0d4","formulaImgStatus":"succeed"}" data-formula="2pi">cript>cript>（A/360）=A ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105054.png","uri":"pgc-image/9aa90455560a43258c2ac17f034ab303","width":13,"height":28,"darkImgUrl":"https://p26.toutiaoimg.com/origin/pgc-image/148da3f56f324a89b31f58cda4bd6171","darkImgUri":"pgc-image/148da3f56f324a89b31f58cda4bd6171","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript>/180。

接下来，我们就能够求出任意角度的正弦值：

一、求sin1°~sin15°。

因为sinA°ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105011.png","uri":"pgc-image/492368fb2ee5473396efd5573b46c2ae","width":16,"height":27,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/3220ec1d02bc41ebab4546834af2e742","darkImgUri":"pgc-image/3220ec1d02bc41ebab4546834af2e742","formulaImgStatus":"succeed"}" data-formula="approx">cript>cript>BD/R=2 ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105093.png","uri":"pgc-image/cd96086c41f647678c986d24edbe0b3f","width":13,"height":28,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/cd0395fb3d40476f8c653c394dd1612d","darkImgUri":"pgc-image/cd0395fb3d40476f8c653c394dd1612d","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript>（A/360）=A ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105067.png","uri":"pgc-image/189035fe16d340b5a693f77cf71ca3d7","width":13,"height":28,"darkImgUrl":"https://p5.toutiaoimg.com/origin/pgc-image/52e110893ab941cb91d202962d74b2fc","darkImgUri":"pgc-image/52e110893ab941cb91d202962d74b2fc","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript>/180，于是：

1、sin1°=1* ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105183.png","uri":"pgc-image/1bbc9955cbb0445491d4393caae0092e","width":13,"height":28,"darkImgUrl":"https://p9.toutiaoimg.com/origin/pgc-image/f26b7ac21e6d4aea8242f20c1e3c13df","darkImgUri":"pgc-image/f26b7ac21e6d4aea8242f20c1e3c13df","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript> /180=0.01745。

（查正弦表可知，sin1°=0.01745……，这时两者小数点5位以内数值都是一致的）

2、sin2°=2* ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105157.png","uri":"pgc-image/e6f825dfe39c4bcebaa4b984246c4d69","width":13,"height":28,"darkImgUrl":"https://p26.toutiaoimg.com/origin/pgc-image/0fca4e8bd794499897f4b0cc794c8302","darkImgUri":"pgc-image/0fca4e8bd794499897f4b0cc794c8302","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript> /180=0.03491。

（查正弦表可知，sin2°=0.03489……，这时两者小数点3位以内数值都是一致的）

3、sin10°=10* ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105267.png","uri":"pgc-image/83e1ca0727a144718a3cc31a9c342d61","width":13,"height":28,"darkImgUrl":"https://p9.toutiaoimg.com/origin/pgc-image/0b63e0587c5a4146af9515f06a3463bc","darkImgUri":"pgc-image/0b63e0587c5a4146af9515f06a3463bc","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript> /180=0.17453。

（查正弦表可知，sin10°=0.17364……，这时两者小数点2位以内数值都是一致的）

3、sin15°=15* ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105274.png","uri":"pgc-image/cd33c329521345b79c21a20df2b4c138","width":13,"height":28,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/f2339acd09284acca10bb9fff2af099a","darkImgUri":"pgc-image/f2339acd09284acca10bb9fff2af099a","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript> /180=0.26180。

（查正弦表可知，sin15°=0.25881……，这时两者小数点1位以内数值都是一致的，这时误差为（0.26180-0.25881）/0.25881=1.15%，这个误差还是相对比较小的，属于可接受范围。）

二、求sin15°~sin30°。

1、sin30°=30* ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105211.png","uri":"pgc-image/24400b19eb3c4b529e97c1f376883b25","width":13,"height":28,"darkImgUrl":"https://p3.toutiaoimg.com/origin/pgc-image/ec197b850d8e4213bc0ca4670054e22c","darkImgUri":"pgc-image/ec197b850d8e4213bc0ca4670054e22c","formulaImgStatus":"succeed"}" data-formula="pi">cript>cript> /180=0.52360。

（我们知道sin30°=0.5，这时依据上述方法算出来的误差为（0.52360-0.5）/0.5=4.72%，这个误差已经很大了，是无法接受的。）

2、这时为了降低误差，我们需要利用勾股定理求出15度角的正弦值（sin15°）。

已知AB=AD，AC垂直于BD，角BAC=角DAC=15°，角BAE=30°，BE垂直于AD，于是BC=CD。

因为角BAE=30°，所以BE= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105320.png","uri":"pgc-image/24ac4c76526a4159b9dffee74db2e6e5","width":15,"height":43,"darkImgUrl":"https://p5.toutiaoimg.com/origin/pgc-image/da03f76f0bf54ae29b532b50d1fda98f","darkImgUri":"pgc-image/da03f76f0bf54ae29b532b50d1fda98f","formulaImgStatus":"succeed"}" data-formula="frac{1}{2}">cript>cript>AB，AE= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105374.png","uri":"pgc-image/cdbad4e785a74da796b36f7cb9e701ac","width":33,"height":49,"darkImgUrl":"https://p5.toutiaoimg.com/origin/pgc-image/6e318a6f78d341579b5ec049f4587203","darkImgUri":"pgc-image/6e318a6f78d341579b5ec049f4587203","formulaImgStatus":"succeed"}" data-formula="frac{sqrt{3}}{2}">cript>cript> AB，ED=AD-AE= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105390.png","uri":"pgc-image/71745234616f4d7b90805a6be828f4e7","width":69,"height":49,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/fd06a89a86b8468da45461d69e407e1c","darkImgUri":"pgc-image/fd06a89a86b8468da45461d69e407e1c","formulaImgStatus":"succeed"}" data-formula="frac{2-sqrt{3}}{2}">cript>cript> AB。

在三角形BED中， ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105387.png","uri":"pgc-image/cb4127f9b4b24b85bff89ac2ade813bf","width":408,"height":49,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/e37e85ebdcf24f15b691f173c444d822","darkImgUri":"pgc-image/e37e85ebdcf24f15b691f173c444d822","formulaImgStatus":"succeed"}" data-formula="BD^{2}=BE^{2}+ED^{2}=frac{1}{4}AB^{2}+frac{7-4sqrt{3}}{4}AB^{2}">cript>cript> 。

可计算得出： ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105478.png","uri":"pgc-image/8711f612a780444c94073bec12b089b5","width":259,"height":52,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/5d9ae383892d42f1adbb1c6ca2b43fab","darkImgUri":"pgc-image/5d9ae383892d42f1adbb1c6ca2b43fab","formulaImgStatus":"succeed"}" data-formula="BC=frac{1}{2}BD=frac{sqrt{2-sqrt{3}}}{2}AB">cript>cript>。

于是sin15°=BC/AB= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105420.png","uri":"pgc-image/a61d90a8ba164cff8bc29334b1e8edf5","width":89,"height":52,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/9ed928eb687c4ea1a8e80f173cd51ba9","darkImgUri":"pgc-image/9ed928eb687c4ea1a8e80f173cd51ba9","formulaImgStatus":"succeed"}" data-formula="frac{sqrt{2-sqrt{3}}}{2}">cript>cript>=0.258819……=0.259（取3位小数）

又sin30°=0.5，所以（sin30°-sin15°）/15=0.01608=0.016（取3位小数）

15°~30°，我们认为角度在增加，正弦值也随之成比例增加，即角度每增加1°，正弦值随之增加0.016。

3、于是：

sin15°=0.259

sin16°=0.259+0.016=0.275

……

sin20°=0.259+0.016*5=0.339

（查正弦表可知，sin20°=0.342……，误差很小）

……

sin25°=0.259+0.016*10=0.419

（查正弦表可知，sin25°=0.422……，误差很小）

……

三、求sin30°~sin45°。

1、sin30°=0.5，sin45°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105590.png","uri":"pgc-image/a3e42b64cfdf441ba7cba3d472c85373","width":107,"height":49,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/730bd1230e3e45ec899e5d0faa4e7d16","darkImgUri":"pgc-image/730bd1230e3e45ec899e5d0faa4e7d16","formulaImgStatus":"succeed"}" data-formula="frac{sqrt{2}}{2}=0.707">cript>cript>（取3位小数）

2、（sin45°-sin30°）/15=0.01608=0.0138=0.014。（取3位小数）

3、30°~45°，我们认为角度在增加，正弦值也随之成比例增加，即角度每增加1°，正弦值随之增加0.014。

4、于是：

sin30°=0.5

sin31°=0.5+0.014=0.514

……

sin35°=0.5+0.014*5=0.570

（查正弦表可知，sin35°=0.5735……，误差很小）

……

sin40°=0.5+0.014*10=0.640

（查正弦表可知，sin40°=0.6427……，误差很小）

……

sin44°=0.5+0.014*14=0.696

（查正弦表可知，sin44°=0.69465……，误差很小）

四、求sin45°~sin90°。

1、如下图，在直角三角形ABC中，角C是直角，角A+角B=90°。

2、于是 ：ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105518.png","uri":"pgc-image/22221aa2096945028fb0a42b39b148c1","width":175,"height":28,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/b5d796f4b6a34183872790067719b6b4","darkImgUri":"pgc-image/b5d796f4b6a34183872790067719b6b4","formulaImgStatus":"succeed"}" data-formula="AC^{2}+BC^{2}=AB^{2}">cript>cript> ，等式两边同时除以 ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105519.png","uri":"pgc-image/b7058426f275493893591d6d30cbeff9","width":41,"height":28,"darkImgUrl":"https://p5.toutiaoimg.com/origin/pgc-image/84b1f7995fb9445aba80c763dea2f558","darkImgUri":"pgc-image/84b1f7995fb9445aba80c763dea2f558","formulaImgStatus":"succeed"}" data-formula="AB^{2}">cript>cript> 。

3、可得 ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105592.png","uri":"pgc-image/a9a77e41e3b74a8bb4093b3fe78d822f","width":171,"height":28,"darkImgUrl":"https://p9.toutiaoimg.com/origin/pgc-image/02d3b7b8f0c64acfa26046f86c1f095e","darkImgUri":"pgc-image/02d3b7b8f0c64acfa26046f86c1f095e","formulaImgStatus":"succeed"}" data-formula="sinB^{2}+sinA^{2}=1">cript>cript> 。

4、于是 ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105697.png","uri":"pgc-image/f4eac39b4c3a47799226720ad33ca78e","width":256,"height":28,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/a989bcf6b9b549088c3a0282167c7adf","darkImgUri":"pgc-image/a989bcf6b9b549088c3a0282167c7adf","formulaImgStatus":"succeed"}" data-formula="sinA^{2}=1-sin（90-A）^{2}">cript>cript>

5、于是：sin46°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105681.png","uri":"pgc-image/a40ca9d00aac48ca80775d2e29d374e2","width":254,"height":30,"darkImgUrl":"https://p9.toutiaoimg.com/origin/pgc-image/05db2146e2244fd4919a428d47fd10bc","darkImgUri":"pgc-image/05db2146e2244fd4919a428d47fd10bc","formulaImgStatus":"succeed"}" data-formula="sqrt{1-sin44^{2}}=sqrt{1-0.696^{2}}">cript>cript> =0.718

（查正弦表可知，sin46°=0.71933……，误差很小）

……

sin50°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105759.png","uri":"pgc-image/c44914f4d78d41a291864c9e212e2d74","width":243,"height":30,"darkImgUrl":"https://p3.toutiaoimg.com/origin/pgc-image/e77687aad69a4e738e2c9d63ccc8d476","darkImgUri":"pgc-image/e77687aad69a4e738e2c9d63ccc8d476","formulaImgStatus":"succeed"}" data-formula="sqrt{1-sin40^{2}}=sqrt{1-0.64^{2}}">cript>cript> =0.768

（查正弦表可知，sin50°=0.76604……，误差很小）

……

sin65°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105728.png","uri":"pgc-image/37c0431c07434a4bb5ea1e4f66f844de","width":254,"height":30,"darkImgUrl":"https://p6.toutiaoimg.com/origin/pgc-image/ce89cceda62648d4a85e5f627e07a982","darkImgUri":"pgc-image/ce89cceda62648d4a85e5f627e07a982","formulaImgStatus":"succeed"}" data-formula="sqrt{1-sin25^{2}}=sqrt{1-0.419^{2}}">cript>cript>=0.908

（查正弦表可知，sin65°=0.90630……，误差很小）

……

sin75°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105731.png","uri":"pgc-image/65bc1ddcb17c41ffbf571075ee894caf","width":254,"height":30,"darkImgUrl":"https://p3.toutiaoimg.com/origin/pgc-image/1145021c724f4bc8aae274c876b99c2a","darkImgUri":"pgc-image/1145021c724f4bc8aae274c876b99c2a","formulaImgStatus":"succeed"}" data-formula="sqrt{1-sin15^{2}}=sqrt{1-0.259^{2}}">cript>cript> =0.966

（查正弦表可知，sin75°=0.96592……，误差很小）

……

sin80°= ntent="{"url":"https://www.muyuhao.com/wp-content/uploads/2022/05/2022050909105875.png","uri":"pgc-image/ed06042232ce409680d166b19536aebe","width":275,"height":30,"darkImgUrl":"https://p3.toutiaoimg.com/origin/pgc-image/4a164beec5e340179aac85d5f7ed536f","darkImgUri":"pgc-image/4a164beec5e340179aac85d5f7ed536f","formulaImgStatus":"succeed"}" data-formula="sqrt{1-sin10^{2}}=sqrt{1-0.17453^{2}}">cript>cript>=0.985

（查正弦表可知，sin80°=0.98480……，误差很小）

……