解题方法
1 . 如图,在直角坐标系中,设单位圆O与x轴的非负半轴相交于点
,以x轴的非负半轴为始边分别作任意角
,
,它们的终边分别与单位圆相交于点
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/13/a11c83f7-4a15-43e3-9fda-c5c7e25e104a.png?resizew=185)
(1)请在图中作出以x轴的非负半轴为始边时角
的终边
(与单位圆交于点P),并说明AP与
的长度关系;
(2)根据第(1)问的发现,证明两角差的余弦公式;
(3)由两角差的余弦公式推导两角差的正弦公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8950c7bc835103d52ceffab14b6b31a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/13/a11c83f7-4a15-43e3-9fda-c5c7e25e104a.png?resizew=185)
(1)请在图中作出以x轴的非负半轴为始边时角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd927b4b5a7875528c1b54aa4bb8b2dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae97d7f57b159b72a23eb909b74d7c3.png)
(2)根据第(1)问的发现,证明两角差的余弦公式;
(3)由两角差的余弦公式推导两角差的正弦公式.
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解题方法
2 . 如图,
是坐标原点,
,
是单位圆上的两点,且分别在第一和第三象限;
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/30/8d287f9e-1872-468a-b29c-dd57ae58883e.png?resizew=150)
(1)证明:
;
(提示:设
为
的终边,
为
的终边,则
,
两点的坐标可表示为
和
)
(2)求
的范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/30/8d287f9e-1872-468a-b29c-dd57ae58883e.png?resizew=150)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff8eb79da2ae1202feebf45ba5e795c.png)
(提示:设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e9f7d1272b7344346b58b660aa260a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9faa86fd7ec41cacc3ff1859a9b1fc94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a05658511cc8728f4a77fbed890a637a.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc5f2d9131427515d94aa48abc44d51.png)
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2023-04-29更新
|
168次组卷
|
2卷引用:四川省成都东部新区养马高级中学2022-2023学年高一下学期期中考试数学试题
3 . 在直角坐标系
中,以
为始边分别作角
,
,其终边分别与单位圆交于点
,
.
(1)证明:
;
(2)已知
,
为锐角,
,
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e5af20b2f8c1fba4470f9650989e51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4e4c210b2342523b23a43e0a5fd4f63.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e82e61189eee22b8a316b16ead9fed5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ccd1dfdc167165bcad456709247c723.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800db22e042a298041eda8b0c72abb7c.png)
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2023-04-04更新
|
172次组卷
|
2卷引用:贵州省贵阳市三新改革联盟校2022-2023学年高一下学期4月联考数学试题
解题方法
4 . 写出两角差的余弦公式,并利用单位圆以及向量的数量积证明该公式.
您最近一年使用:0次
名校
解题方法
5 . 十字测天仪广泛应用于欧洲中世纪晚期的航海领域,主要用于测量太阳等星体的方位,便于船员确定位置.如图1所示,十字测天仪由杆AB和横档CD构成,并且E是CD的中点,横档与杆垂直并且可在杆上滑动.十字测天仪的使用方法如下:如图2,手持十字测天仪,使得眼睛可以从A点观察.滑动横档CD使得A,C在同一水平面上,并且眼睛恰好能观察到太阳,此时视线恰好经过点D,DE的影子恰好是AE.然后,通过测量AE的长度,可计算出视线和水平面的夹角
(称为太阳高度角),最后通过查阅地图来确定船员所在的位置.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/2/f340736c-a23f-4b3e-a3ef-710244b4e922.png?resizew=359)
(1)若在某次测量中,横档
的长度为20,测得太阳高度角
,求影子AE的长;
(2)若在另一次测量中,
,横档
的长度为20,求太阳高度角的正弦值;
(3)在杆AB上有两点
,
满足
.当横档CD的中点E位于
时,记太阳高度角为
,其中
,
都是锐角.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5157b42da58d55daad27d98b2fec15ff.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/2/f340736c-a23f-4b3e-a3ef-710244b4e922.png?resizew=359)
(1)若在某次测量中,横档
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b528818e98c5c2ddf301048b4228d2.png)
(2)若在另一次测量中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af0e847821c95966efc534f26fbe4f6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(3)在杆AB上有两点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2059d1b10017e04aa35812c0354049b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e819b87f90651d89fcd258c276294e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cafc046509e3ca71090d8a1de862efa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/943b765718479c160ba61ec5c6f8c5f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e29bf5652f0d4f764c3606efcdb445f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dadb2357b7a648d3a69c7a84dbdffcc0.png)
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2023-04-26更新
|
1479次组卷
|
6卷引用:四川省成都市第七中学2022-2023学年高一下学期期中考试数学试题
20-21高一·江苏·课后作业
解题方法
6 . 求证:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/370cd955f4d1f8d1a829425eaa64e6c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f34a4cf3e88ba292fec8aaf926979801.png)
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2021-10-30更新
|
188次组卷
|
3卷引用:7.2 三角函数概念
7 . 以单位圆为工具,根据三角函数定义证明:
,
.(提示,同圆中,等角所对弦长相等.)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac64bd6e38ccf03ef6593ac5b5bbd6b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6739328fd4604784c78e0f5d88726c6e.png)
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8 . (1)写出下列两组诱导公式:
①关于
与
的诱导公式;
②关于
与
的诱导公式.
(2)从上述①②两组诱导公式中任选一组,用任意角的三角函数定义给出证明.
①关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47796eaacdc3edda0feeaf87daf274ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
②关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2256b0609359cf7ebd52153066926177.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(2)从上述①②两组诱导公式中任选一组,用任意角的三角函数定义给出证明.
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2019-02-09更新
|
334次组卷
|
2卷引用:【市级联考】山东省滨州市2018-2019学年高一上学期期末考试数学试题
名校
9 . (1)已知点
在角
的终边上,且
,求
和
的值;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab8eeec247a225410b66f4be1d3a264f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/349252bd781a0b4c464812caf3f8ea26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fb49947eb370e18720a191b18796c6f.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e91e91d2f96849ff6dcec784849d666.png)
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10 . 如图,在四边形
中,
,
,
为
的中点,连接
,过点
作
交
于点
,连接
,已知
.
(1)求证:
;
(2)若
,求
的长度;
(3)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fc1ba2766874730daaad7d3bb7a8ee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e0a957a55460c72673c0f2ee90dbb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53a08d81353a3ea431787f550b9d9b7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40fe2cc764de5fa58e9d90d750c5c528.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33f0279b8c6e1097c2686173474227f8.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa80cfd38f59e5cbbb30628d03113689.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b867df58301f823a4c2e11587f01fc5d.png)
![](https://img.xkw.com/dksih/QBM/2017/9/15/1774464467001344/1775818797252608/STEM/0868c767ca3f415b9dfaa4c5ddbfcfef.png?resizew=105)
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