1 . 关于三角函数有如下的公式:
……①
……②
(
)……③
利用这些公式可以将一些不是特殊角的三角函数转化为特殊角的三角函数来求值,如:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28b234533c85e120f7dd9ff97b93c897.png)
根据上面的知识,你可以选择适当的公式解决下面实际问题:如图所示,直升机在一建筑物
上方
点处测得建筑物顶端
点的俯角
为
,底端
点的俯角
为
,此时直升机与建筑物
的水平距离
为42米,求建筑物
的高.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/befe5d12d0fe4c28a0a4fbafaa9db499.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff4b723500500ef61059ddad360f0200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/570b1b76ebbfd021053f2428b69d6892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0498edee33a5fe35a7df4c08143816a8.png)
利用这些公式可以将一些不是特殊角的三角函数转化为特殊角的三角函数来求值,如:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28b234533c85e120f7dd9ff97b93c897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e680e139402b2a810c3d7bc9872ce25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a4c9b697e3df2b5469b71d3b5e47a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/f0e2468e-ebd1-48e2-8f7d-bb41af47c82d.png?resizew=123)
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2 . 如图,
是
的直径,
切
于点
,
交
于点
,
平分
,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/5642266f-702f-4717-b4df-6495b9b9de1e.png?resizew=149)
(1)求证:
;
(2)若
,
,求
的半径.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21037e170bdbb322558e79c40c00b454.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2947ca8e0cdbeb4aab80ce9e7b63ba98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/5642266f-702f-4717-b4df-6495b9b9de1e.png?resizew=149)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c388d0a3f0a8d9fb0b9576d00af225.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e673ef2d48215ca84a48377f17d6df00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbc39144b305c67d44410d41053a1d28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
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3 . 如图1,在平面直角坐标系中,直线
与抛物线
交于
、
两点,其中
,
.该抛物线与
轴交于点
,与
轴交于另一点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/74bdd37e-64ff-4c9b-83b0-0fed98d62114.png?resizew=464)
(1)求
、
的值及该抛物线的解析式;
(2)如图2.若点
为线段
上的一动点(不与
、
重合).分别以
、
为斜边,在直线
的同侧作等腰直角
和等腰直角
,连接
,试确定
面积最大时
点的坐标.
(3)如图3.连接
、
,在线段
上是否存在点
,使得以
、
、
为顶点的三角形与
相似,若存在,请直接写出点
的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e6e15daf7b14dbff32c390f4984dcfb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58d88bbd34102b55fa928e8ff83f0d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d7a6fd6d651ae341154c2e40928d628.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/906948588ab8a93adbacfd6606495fd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/74bdd37e-64ff-4c9b-83b0-0fed98d62114.png?resizew=464)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)如图2.若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fd17a66a2af938c89e46f22e4d893b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/784b26de6134b6d3f5a81e2882a7d7f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cfce6da93ad3e6053fe4d732f6e456b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3de0022a7f2e122492f1483e4d3cccba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(3)如图3.连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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4 . 如图,一次函数
(
)的图象分别与
轴,
轴相交于点
,
,与反比例函数
(
)的图象相交于点
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/f1733a5a-7c03-43c9-a899-39fcf4ad64b4.png?resizew=166)
(1)求一次函数和反比例函数的表达式;
(2)当
为何值时,
;
(3)当
为何值时,
,请直接写出
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41d49bb2fdf76bbf030f0749462ab208.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1326c3aac61397f6be2d46bfbb8fe917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db8dc0564b187f7adb8184f714ab5da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f494e4db0034036232b9857b95ca8cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a630a706ad5c594a157111613c9fb80d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d549fd5b1e6da722932fe43675d5ff18.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/f1733a5a-7c03-43c9-a899-39fcf4ad64b4.png?resizew=166)
(1)求一次函数和反比例函数的表达式;
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcdb7671e6056d842cdc88fb523467f.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ac67e9a909472ab852d38d2ec66a1e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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5 . 如图:在平行四边形
的边
,
上截取
,
,使得
,连接
,点
,
是线段
上两点,且
,连接
,
.
(1)求证:
;
(2)若
,
,求
的度数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7c581dab991aeadf06d972e47673ea8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.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/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a74fa1d77910529e4eda3b2d70f40a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/25/55e0d838-89da-4cfa-9dda-8e4e571d1bda.png?resizew=225)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c883ea424e7bc9db19d5e35a6ec63134.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/760d7546f3f4f788eabf9c493fcdc867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f898fd9009607f692cddda426d179427.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bcb22ca5b9cdfeebca14bdfad0b8f29.png)
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6 . 先化简,再求值
,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b369c8bb5c7b6a6dcafbe0634f806736.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3053399d2f91ca2f7dda29e7c5d64397.png)
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7 . 在手工课上,小明制作了如图所示的四张卡片(这四张卡片依次分别用字母A,B,C,D表示,四张卡片除正面的文字不同外,其余均相同),现将四张卡片背面朝上,洗匀放好.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/15/a0af7df1-9659-4ad2-9960-bd840e811136.png?resizew=397)
(1)小明从中随机抽取一张卡片,卡片上的字是轴对称图形的概率为__________.
(2)小明从中随机抽取一张卡片不放回,随后小亮再从中随机抽取1张卡片,请用列表法或画树状图法计算两人抽取的卡片恰好组成“光荣”一词的概率.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/15/a0af7df1-9659-4ad2-9960-bd840e811136.png?resizew=397)
(1)小明从中随机抽取一张卡片,卡片上的字是轴对称图形的概率为__________.
(2)小明从中随机抽取一张卡片不放回,随后小亮再从中随机抽取1张卡片,请用列表法或画树状图法计算两人抽取的卡片恰好组成“光荣”一词的概率.
您最近一年使用:0次
8 . 如图,在平行四边形
中,
,点
是
的中点,连接
并延长,交
的延长线于点
,连接
.
(1)求证:四边形
是菱形;
(2)若
,求菱形
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f2665526efab6fdd59a4fe1bfe4e39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/16/0585a7b5-da89-466e-a367-ef2b55eb190c.png?resizew=185)
(1)求证:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24b0b077bfd3700e78db35bea761ae29.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6f341ab26d4158361caf168bc3559c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24b0b077bfd3700e78db35bea761ae29.png)
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9 . 已知二次函数
.
(1)如果二次函数的图象与
轴有两个公共点,求
的取值范围;
(2)如图,二次函数的图象过点
,与
轴交于点
,直线
与这个二次函数图象的对称轴交于点
,求
的面积;
(3)在(2)的条件下,根据图象直接写出使一次函数值小于二次函数值的
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa653894ff179b6f9f797239c74c3c7c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/16/7cef0093-af1f-4602-b49f-47bf9ade55e6.png?resizew=149)
(1)如果二次函数的图象与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)如图,二次函数的图象过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/241ce9bd28046ce9b90f43b391132884.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2b4dcc093218443f71a046b6df94bbe.png)
(3)在(2)的条件下,根据图象直接写出使一次函数值小于二次函数值的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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10 . 实数的计算:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f094bed8b535bdc450e334fff404eae7.png)
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