名校
解题方法
1 . 我国古代数学家僧一行应用“九服晷影算法”在《大衍历》中建立了晷影长l与太阳天顶距θ(
)的对应数表,这是世界数学史上较早的正切函数表.根据三角学知识可知,晷影长l等于表高h与太阳天顶距θ正切值的乘积,即
.对同一“表高”测量两次,第一次和第二次太阳天顶距分别为
,
,第二次的“晷影长”是“表高”的2倍,且
,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b13a855bec1e9c264742bbed4685ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7ba1eba7fba0b56e4e9b4d032e24da.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/5ffb487cd5f8d97cedc263d542caf5bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d3a6ba2ae3161cf7c209a30ea464c1.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2 . 宋朝诗人王镃在《蜻蜓》中写到:“轻绡剪翅约秋霜,点水低飞恋野塘”,描绘了蜻蜓点水的情形,蜻蜓点水会使平静的水面形成水波纹,截取其中一段水波纹,其形状可近似于用函数
的图象来描述,如图所示,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc4f29a473e9757d8f24f627f52e9e15.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3434f3860fb9e47ab2b8ef93eae2a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc4f29a473e9757d8f24f627f52e9e15.png)
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3 . 孤峰塔坐落在与常德城隔江相望的德山孤峰岭.初名“文峰塔”,与北岸笔架城遥相映衬,象征常德人杰地灵,文运昌盛. 常德立德中学高一学生为了测量塔高
,选取与塔底
在同一水平面内的两个测量基点
与
.现测量得
米,在点
处测得塔顶
的仰角分别为
,则孤峰塔高
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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4 . 十七世纪法国数学家、被誉为业余数学家之王的皮埃尔・德・费马提出的一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小”,意大利数学家托里拆利给出了解答,当
的三个内角均小于
时,使得
的点
即为费马点;当
有一个内角大于或等于
时,最大内角的顶点为费马点.已知
,
,
分别是
三个内角
,
,
的对边,且
,若点
为
的费马点,
,则实数
的取值范围为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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5 . 永丰文塔位于湖南省双峰县城永丰镇,修建于清朝同治年间,巍巍七层文塔,塔形呈六角形,塔底用高达五尺八寸的青条石奠基,永丰文塔与双峰书院遥相呼应,象征双峰文运昌隆.如图,某测绘小组为了测量永丰文塔的实际高度
,选取了与塔底
在同一水平面内的两个测量基点
,现测得
,在点
测得塔顶A的仰角为
,则塔高
( )(取
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/259bc0c5c676ec9b44e1cb53f903ead8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c26a46e7879436d532af3f4b6e258a81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a36d0de57c5f7174ced0594ffb320b1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17bfabb8a2fb61ecbc942f399ad4041b.png)
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解题方法
6 . 若
内一点
满足
,则称点
为
的布洛卡点,
为
的布洛卡角.如图,已知
中,
,
,
,点
为的布洛卡点,
为
的布洛卡角.
,且满足
,求
的大小.
(2)若
为锐角三角形.
(ⅰ)证明:
.
(ⅱ)若
平分
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec15e5cb6d4dc2cf6ba0bedd87514448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e781a2489271bfd1597cba1bb6f5887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df81cda12d7601d58b1d9c7c180c4d66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c884a45b56bc34d79273b067c1520b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b05d3b8f5c9df891ef6fbcaf12f43207.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fefcd73e7c22ace3ccd013842cf72a60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(ⅰ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f272ca460306b34bf7e3e99d38dca8b.png)
(ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/988b7e964e313579ab8869d67d5be007.png)
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2024-04-30更新
|
1986次组卷
|
6卷引用:湖南省长沙市长郡中学2024届高考适应考试(三)数学试题
湖南省长沙市长郡中学2024届高考适应考试(三)数学试题河北省部分高中2024届高三下学期二模考试数学试题(已下线)2024年普通高等学校招生全国统一考试数学押题卷(一)(已下线)压轴题07三角函数与正余弦定理压轴题9题型汇总-1(已下线)专题02 第六章 解三角形及其应用-期末考点大串讲(人教A版2019必修第二册)(已下线)专题06 解三角形综合大题归类(2) -期末考点大串讲(苏教版(2019))
名校
解题方法
7 . 青岛胶东国际机场的显著特点之一是弯曲曲线的运用,衡量曲线弯曲程度的重要指标是曲率.考察图所示的光滑曲线
上的曲线段
,其弧长为
,当动点从A沿曲线段
运动到B点时,A点的切线
也随着转动到B点的切线
,记这两条切线之间的夹角为
(它等于
的倾斜角与
的倾斜角之差).显然,当弧长固定时,夹角越大,曲线的弯曲程度就越大;当夹角固定时,弧长越小则弯曲程度越大,因此可以定义
为曲线段
的平均曲率;显然当B越接近A,即
越小,K就越能精确刻画曲线C在点A处的弯曲程度,因此定义曲线
在点
处的曲率计算公式为
,其中
.
的圆弧的平均曲率;
(2)已知函数
,求曲线
的曲率的最大值;
(3)已知函数
,若
曲率为0时x的最小值分别为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eefffa1689b5a68786b9a5875f12c0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/505d83f4d34a8cd385577a6ce93a4b11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aea61ddc41f927684c6dfaacdd7f8e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0636a11a086df66133bd50e43481a546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/427eceadd7bb569ff140ea73d650db1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0636a11a086df66133bd50e43481a546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aea61ddc41f927684c6dfaacdd7f8e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bb01270362284437d082c3a2268c6b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/505d83f4d34a8cd385577a6ce93a4b11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07fa72fc4959804b944bfaa93dbe2b21.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d5bca00fa20e6e80480b9d06d2e52ee.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7bce420cf236e5f429afee284239010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(3)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e86f9b172e8232ee105d0436dab312b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c7921ee6a8981f1f4980cdcb0f921bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f3966bd8e4857ccb70afc0fdbab8e87.png)
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2024-04-15更新
|
482次组卷
|
3卷引用:湖南省岳阳县第一中学2024届高三下学期6月适应性考试数学试题
名校
解题方法
8 . 《数书九章》是中国南宋时期杰出数学家秦九韶的著作,全书十八卷共八十一个问题,分为九类,每类九个问题,《数书九章》中记录了秦九韶的许多创造性成就,其中在卷五“三斜求积”中提出了已知三角形三边a,b,c求面积的公式,这与古希腊的海伦公式完全等价,其求法是:“以小斜幂并大斜幂减中斜幂,余半之,自乘于上,以小斜幂乘大斜幂减上,余四约之,为实,一为从隅,开平方得积."若把以上这段文字写成公式,即
.现有
满足
,且
的面积
.给出下列四个结论:①
周长为
;②
三个内角A,C,B满足关系
;③
外接圆半径为
;④
中线CD的长为
,其中,所有正确结论的序号是___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96bd5fefb9a7c618d1ef8d73b3c43cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d44315d3429e4d76842ecfc14f1ca949.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b8364e0de3c661873fe72fb9d5c073e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5eeaa0e92dfcadcfe909e3b8f1251ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e72b4868bdd8d15dff2211cc49cd31b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e916683e0555c8fb27087d015e833ac3.png)
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2024-04-07更新
|
192次组卷
|
2卷引用:湖南省常德市汉寿县第一中学2023-2024学年高一下学期3月月考数学试题
名校
9 . 出土于鲁国故城遗址的“出廓双龙勾玉纹黄玉璜”(图1)的璜身满刻勾云纹,体扁平,呈扇面状,黄身外耧空雕饰“
”型双龙,造型精美.现要计算璜身面积(厚度忽略不计),测得各项数据(图2):
,若
,则璜身(即曲边四边形
)面积近似为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39003d92af881c6ff636e46841c88278.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd88cf7355547a845e856eb40188f15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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10 . 以密位作为角的度量单位,这种度量角的单位制,叫作角的密位制.在角的密位制中,采用四个数码表示角的大小,单位名称密位二字可以省去不写.密位的写法是在百位数与十位数之间画一条短线,如5密位写成“
”,235密位写成“
”,1246密位写成“
”.1周角等于6000密位,写成“
”.在
中,点
在边
上,
是
的内角
的角平分线,
,则
用密位制表示为__________ .
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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