解题方法
1 . 如果![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e75cb7615ca33b128496114742a1f2dd.png)
(1)求证:
;
(2)若
为三角形的三个内角,判断
与
的大小关系,并予以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e75cb7615ca33b128496114742a1f2dd.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3e10e76a8cf6e3eb92e57ee971a218a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e263f0291e3df8b6fb866abaf3f4576.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbeb449ec91ac7e5798e3b347fd0d107.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89a25e473dce119ddd92f32fef1dc576.png)
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2 . 固定项链的两端,在重力的作用下项链所形成的曲线是悬链线,1691年,莱布尼茨等得出“悬链线”方程
,其中
为参数.当
时,就是双曲余弦函数
,类似的我们可以定义双曲正弦函数
.它们与正、余弦函数有许多类似的性质.
(1)求证:
;
(2)对
,不等式
恒成立,求实数
的取值范围;
(3)若
,试比较
与
的大小关系,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08144630f70f5bba0c73252569d97841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d848439a448faa1d4cd9fa20ca206215.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe0a24e9c7616bf8afac5a0ffb0aa1fb.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bdd3fb4f930b309f261929ba7a1f055.png)
(2)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe9f3099ed9429dc5b4e38a350e524a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24464f0ea26038d85cc22a1786257605.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a83c8948a168ff2c567aee048cabff9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a2cebaab3423dfb2f2c944dfc43df8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb966b7b2dd6581640bcee2d97dacf77.png)
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3 . 证明:
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8fffe92548da698866a9888e57d472.png)
(2)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c85bf2963df20faabb0e5b2f512e55.png)
(3)已知
,
,求证
.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8fffe92548da698866a9888e57d472.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c85bf2963df20faabb0e5b2f512e55.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c0975b825228b2a91799eee6894987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57802c9ef29020d70c31d10d8c19125f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca26c7f4cf6a970f1f7c400ac2fc53ac.png)
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4 . 固定项链的两端,在重力的作用下项链所形成的曲线是悬链线.1691年,莱布尼茨等得出“悬链线”方程
,其中
为参数.当
时,就是双曲余弦函数
,类似地我们可以定义双曲正弦函数
.它们与正、余弦函数有许多类似的性质.
(1)类比正弦函数的二倍角公式,请写出双曲正弦函数的一个正确的结论:
_____________.(只写出即可,不要求证明);
(2)
,不等式
恒成立,求实数
的取值范围;
(3)若
,试比较
与
的大小关系,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852665ec9c3a65b758898059361f11a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a7c1d3681898e25187a896aeb0c8c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0718c04bdf70989bcc90b902671a692.png)
(1)类比正弦函数的二倍角公式,请写出双曲正弦函数的一个正确的结论:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d8fe1e65b09697538d4dee0746846f4.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe9f3099ed9429dc5b4e38a350e524a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/343e7c30c2a5d166819b28e23fad2203.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/563f464c94feac28033f6f3a271fbe8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a2cebaab3423dfb2f2c944dfc43df8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb966b7b2dd6581640bcee2d97dacf77.png)
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2024-01-27更新
|
938次组卷
|
8卷引用:重庆市缙云教育联盟2024届高三下学期2月月度质量检测数学试题
重庆市缙云教育联盟2024届高三下学期2月月度质量检测数学试题河南省名校联盟2023-2024学年高一下学期3月测试数学试题河南省信阳市信阳高级中学2023-2024学年高一下学期3月月考(一)数学试题福建省宁德市2023-2024学年高一上学期1月期末质量检测数学试题(已下线)压轴题函数与导数新定义题(九省联考第19题模式)讲(已下线)第八章:向量的数量积与三角恒等变换章末重点题型复习(2)-同步精品课堂(人教B版2019必修第三册)(已下线)第8章:向量的数量积与三角恒等变换章末综合检测卷(新题型)-【帮课堂】(人教B版2019必修第三册)(已下线)专题04 三角函数恒等变形综合大题归类 -期末考点大串讲(苏教版(2019))
解题方法
5 . (1)证明:若
,求证:
;
(2)已知
,
均为锐角,且满足
,
,求
值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d677e6f94d57c506bd007617c50a19e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99db5e19a19614908bee34c4ae100286.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/0c2c1ac87d07d6c7a1a62eb333d112d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8628325b9e41358aec8f97a50da7f27a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbfea1e888676a13ad69c72fba0405ea.png)
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6 . 证明:
(1)
.
(2)已知
,
,求证:
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1fc2d308d990e5771657c9f56a0936b.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fbdc8633a22f3b9fb3a789d3818657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45a173784888adf2946382fa093ba53a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/129f958b08c51df454111d41c6db204f.png)
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解题方法
7 . 记
的内角
所对的边分别为
,已知
.
(1)求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e717c243991f038d7bc21a0fdad985b.png)
(2)若
的面积
,求
的最大值,并证明:当
取最大值时,
为直角三角形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce7af7c5df749c6fa9bbe87faa72c66d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88f2599ca8b6b683e57a82699c8b1ebb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55abde5108e7846f496584016ce82286.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e717c243991f038d7bc21a0fdad985b.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3a88d9c428cc72bdf012746e2781a64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2022-12-06更新
|
753次组卷
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3卷引用:安徽省皖优联盟2022-2023学年高三上学期12月第二次阶段性联考数学试题
名校
解题方法
8 . 在
中, 角
的对边分别为
, 若
.
(1)求证:
;
(2)对
, 请你给出一个
的值, 使不等式
成立或不成立,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03290185439a3e7332f41ea038f43eaf.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46abdcf989e4e121de989f73340a55f9.png)
(2)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0abad0f9bfeee920a0444badf1701a2f.png)
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9 . 已知△ABC中,角A,B,C所对的边分别为a,b,c,
.
(1)求证:B为钝角;
(2)若△ABC同时满足下列4个条件中的3个:①
;②
;③
;④
.请证明使得△ABC存在的这3个条件仅有一组,写出这组条件并求b的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e11b926db0bfd6261de42f88d9fcfcb.png)
(1)求证:B为钝角;
(2)若△ABC同时满足下列4个条件中的3个:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ea81a4761aa43976c2b9be0b0dd16b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdf7e7beb7ca1ffd445c7501bd5e3dc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/413f9851aad373d782ae62b308f1de85.png)
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解题方法
10 . 悬索桥(如图)的外观大漂亮,悬索的形状是平面几何中的悬链线.
年莱布尼兹和伯努利推导出某链线的方程为
,其中
为参数.当
时,该方程就是双曲余弦函数
,类似的我们有双曲正弦函数
.
![](https://img.xkw.com/dksih/QBM/2022/1/23/2900721970536448/2907279254913024/STEM/8a914e2499134cf68207c8add767fe65.png?resizew=325)
(1)从下列三个结论中选择一个进行证明,并求函数
的最小值;
①
;
②
;
③
.
(2)求证:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/046db679c09a10434e81f7a01c55e243.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ad2f5a11d7437f506adab0996961269.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0099b9b80ed478824fa95677ebe9d5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3321510a9eb73909a36c084a8630e89.png)
![](https://img.xkw.com/dksih/QBM/2022/1/23/2900721970536448/2907279254913024/STEM/8a914e2499134cf68207c8add767fe65.png?resizew=325)
(1)从下列三个结论中选择一个进行证明,并求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3634cf0ca04b381dec8fcfee8805bdac.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ff61bdd9ed784248cfdcc965ce06db0.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e40ff30f6f7fca28159dedeff7168c74.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95c3de984177769fa426e10eb14cd82c.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0645c3c42e19271f86a10b1fe9dbb0f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b39ee39c38f49390a03be161109a2b4.png)
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2022-02-01更新
|
1286次组卷
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7卷引用:湖南省株洲市第二中学2021-2022学年高一下学期“同济大学”杯数理化联赛数学试题
湖南省株洲市第二中学2021-2022学年高一下学期“同济大学”杯数理化联赛数学试题重庆市2023届高三下学期3月月度质量检测数学试题江苏省苏州市2021-2022学年高一上学期期末数学试题湖南省株洲市南方中学2022-2023学年高一下学期期末数学试题(已下线)重难点突破02 函数的综合应用(九大题型)(已下线)压轴题函数与导数新定义题(九省联考第19题模式)讲(已下线)压轴题三角函数新定义题(九省联考第19题模式)讲