1 . 正弦定理:三角形的各边和它所对角的__________ ,即![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3f7dce380b3e1be1196ed18612f0ce.png)
_____ =____ =____ (R为
外接圆的半径).
点拨:对
的证明如下(R为
外接圆的半径).
证明:设
是
的外接圆,直径
.
如图①,当A为锐角时,连接
,则
.
又因为
,所以
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/23/30639010-fc58-4465-a655-4219e7ffbcc2.png?resizew=354)
如图②,当A为钝角时,连接
,则
.
因为
,可得
,所以
.
当A为直角时,显然有
.
综上所述,不论A是锐角、钝角或直角,总有
.
同理可证
,所以
.
由此可知,三角形各边和它所对角的正弦的比相等,是一个定值,这个定值就是三角形外接圆的直径.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3f7dce380b3e1be1196ed18612f0ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
点拨:对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c30b12f8bf54428846cf378ffd5a23b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
证明:设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aeee1fed5dedd997d80a62b300cd4bd.png)
如图①,当A为锐角时,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37dbe6c3e8e8bb4a350722296a98517f.png)
又因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/150dbfe3d6686d423e70d2d430089de9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e993f6547ee186befb37a24b501e417.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/23/30639010-fc58-4465-a655-4219e7ffbcc2.png?resizew=354)
如图②,当A为钝角时,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37dbe6c3e8e8bb4a350722296a98517f.png)
因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/513e9a98b50b5953543938d868e87257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870cd9f489ae712a24a84d1e196c031b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e993f6547ee186befb37a24b501e417.png)
当A为直角时,显然有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e993f6547ee186befb37a24b501e417.png)
综上所述,不论A是锐角、钝角或直角,总有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e993f6547ee186befb37a24b501e417.png)
同理可证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b9bc8b9d1f2e2d10897247a526a4ef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c30b12f8bf54428846cf378ffd5a23b7.png)
由此可知,三角形各边和它所对角的正弦的比相等,是一个定值,这个定值就是三角形外接圆的直径.
您最近一年使用:0次
21-22高一·全国·课后作业
2 . 正弦定理的表示
正弦定理的常见变形
(1)
(R为
外接圆的半径).
(2)
(R为
外接圆的半径).
(3)三角形的边长之比等于对应角的正弦比,即
.
(4)
.
(5)
.
[微思考]
正弦定理对任意三角形都适用吗?
_____________________
正弦定理的主要功能是什么?
_____________
文字语言 | 在一个三角形中,各边和它所对角的 |
符号语言 | ![]() |
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c3749e009be444e6247c7448fb854a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0305f0ab08b016f9bfb67926829a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(3)三角形的边长之比等于对应角的正弦比,即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb0ac8d3d2603d6fb0a55d3538a40efa.png)
(4)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65eae9d20da1a2ba66ba295b3155af6.png)
(5)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9b76692621bba89bd400968da7004d4.png)
[微思考]
正弦定理对任意三角形都适用吗?
正弦定理的主要功能是什么?
您最近一年使用:0次