1.2应用举例(约4课时)

1.1正弦定理和余弦定理(约3课时)

3．重视加强意识和数学实践能力

学数学的最终目的是应用数学，而如今比较突出的两个问题是，学生应用数学的意识不强，创造能力较弱。学生往往不能把实际问题抽象成数学问题，不能把所学的数学知识应用到实际问题中去，对所学数学知识的实际背景了解不多，虽然学生机械地模仿一些常见数学问题解法的能力较强，但当面临一种新的问题时却办法不多，对于诸如观察、分析、归纳、类比、抽象、概括、猜想等发现问题、解决问题的科学思维方法了解不够。针对这些实际情况，本章重视从实际问题出发，引入数学课题，最后把数学知识应用于实际问题。

1．数学思想方法的重要性

数学思想方法的教学是中学数学教学中的重要组成部分，有利于学生加深数学知识的理解和掌握。

本章重视与内容密切相关的数学思想方法的教学，并且在提出问题、思考解决问题的策略等方面对学生进行具体示范、引导。本章的两个主要数学结论是正弦定理和余弦定理，它们都是关于三角形的边角关系的结论。在初中，学生已经学习了相关边角关系的定性的知识，就是“在任意三角形中有大边对大角，小边对小角”，“如果已知两个三角形的两条对应边及其所夹的角相等,那么这两个三角形全”等。

教科书在引入正弦定理内容时，让学生从已有的几何知识出发，提出探究性问题：“在任意三角形中有大边对大角，小边对小角的边角关系.我们是否能得到这个边、角的关系准确量化的表示呢?”，在引入余弦定理内容时，提出探究性问题“如果已知三角形的两条边及其所夹的角,根据三角形全等的判定方法，这个三角形是大小、形状完全确定的三角形.我们仍然从量化的角度来研究这个问题，也就是研究如何从已知的两边和它们的夹角计算出三角形的另一边和两个角的问题。”设置这些问题，都是为了加强数学思想方法的教学。

本章的中心内容是如何解三角形，正弦定理和余弦定理是解三角形的工具，最后落实在解三角形的应用上。通过本章学习，学生应当达到以下学习目标：

(1)通过对任意三角形边长和角度关系的探索，掌握正弦定理、余弦定理，并能解决一些简单的三角形度量问题。

(2)能够熟练运用正弦定理、余弦定理等知识和方法解决一些与测量和几何计算有关的生活实际问题。

B、C、D，应选A．

[说明]　 此例题用多种方法求解选项，指出3种选择题的技巧．

∴应选D

x轴交点中在原点右边最接近原点的交点，而在原点左边与x轴交点中最

的图象．

∴选D

[说明]　 y=Asin(ωx+j)(A＞0，ω＞0)x∈R的图象可由y=sinx的图象经下列各种顺序变换得到的．

(1)先平移，后伸缩：

①把y=sinx的图象向左(j＞0)或向右(j＜0)沿x轴方向平移|j|个单位；(相位变换)

(周期变换)

③把所有各点纵坐标伸长(A＞1)或缩短(0＜A＜1)到原来的A倍，横坐标不变(振幅变换)

(2)先伸缩，后平移

①把y=sinx图象上各点的横坐标缩短(ω＞1)或伸长(0＜ω＜1)到原

(相位变换)

③把所有各点纵坐标伸长(A＞1)或缩短(0＜A＜1)到原来的A倍横坐标不变(振幅变换)

再把横坐标缩小到原来的一半，纵坐标扩大到原来的4倍，则所得的图象的解析式是　　 [　　 ]

∴选A．

[例17]　 方程sin2x=sinx在区间(0，2π)内解的个数是

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 [　　 ]

A．1　　　　　 B．2　　　　　　 C．3　　　　　　　 D．4

[分析]　 本题有两类解法

(1)求出方程在(0，2π)内的所有解，再数其解的个数．而决定选项，对于选择题，此法一般不用．

(2)在同一坐标系中作出函数y=sin2x和y=sinx的图象，如图2-18所示．

它们在(0，2π)内交点个数，即为所求方程解的个数，从而应选C．

它体现了数、形的结合．

[例18]　 设函数f(x)是定义在R上的周期为3的奇函数，且f(1)=2，则f(5)=____

解：∵f(x)是奇函数，且f(1)=2，∴f(-1)=-2

又∵f(x)是周期为3的函数．　 ∴f(3+x)=f(x)

∴f(-1+3)=f(-1)=-2　 即f(2)=-2

f(2+3)=f(2)=-2　 即f(5)=-2

[例19]　 有一块扇形铁板，半径为R，圆心角为60°，从这个扇形中切割下一个内接矩形，即矩形的各个顶点都在扇形的半径或弧上，求这个内接矩形的最大面积．

[分析]　 本题入手要解决好两个问题．

(1)内接矩形的放置有两种情况，如图2-19所示，应该分别予以处理．

(2)求最大值问题这里应构造函数，怎么选择便于以此表达矩形面积的自变量．

解：如图2-19(1)设∠FOA=θ，则FG＝Rsinθ

又设矩形EFGH的面积为S，那么

又∵0°＜θ＜60°，故当cos(2θ－60°)＝1，即θ=30′时，

如图2-19 (2)，设∠FOA＝θ，则EF＝2Rsin(30°-θ)，在△OFG中，∠OGF＝150°

设矩形的面积为S．

那么S＝EFFG＝4R²sinθsin(30°-θ)

＝2R²［cos(2θ-30°)－cos30°]

又∵0＜θ＜30°，故当cos(2θ-30°)＝1

作出三角函数线，如图2-17

MP=sinθ，OM=cosθ，BS=ctgθ

通过观察和度量得MP＜OM＜BS

从而有sinθ＜cosθ＜ctgθ

∴应选A

∴cosθ＞sinθ

从而可剔除B、D．

再由sinθ＜ctgθ，故可剔除C

故选A

故选A．

每次写作前问自己四个问题：这篇文章的体裁格式是怎样的？主体时态用什么时态？人称用第几人称？可以分几段，之间用什么过渡词、连接词？带着这四个问题去审题，搞清楚文章的主要内容，然后列出提纲。最后丰富自己的提纲就可以了。

第四份资料

单项选择[1] 第一节　单项填空(共15小题；每小题1分，满分15分) 从A、B、C、D四个选项中，选出可以填入空白处的最佳选项。 21.-Do you know Uncle Tom is crazy about gardening? 　 　 -Sure. He ____ his flowers in the garden now. A. may be watering　 　 B. must be watering C. could look after　 　D. ought to watch 22.____ the temperature falling so rapidly, we couldn't go on with the experience. A. Since　 　B. With　 　 C. As　 　 D. For 23. I'd love ___ to your party last Sunday if not because of the unexpected guests. A. to go　 　B. going　 　 C. having gone　 　D.to have gone 24. -Will you two go to the show tomorrow? -No. We're going to a lecture, or at least I'm planning ____. A. it　B. to　 　C. that　 　D. so 25. The rescue team ______to help but failed to get there in time. A. had wanted　B. have planned C. were expecting　D. hoped 26.1 haven't got a single ______from Aunt Mary since I last saw her. A. advice　B. news　C. message　D. information 27.-Is it Pro. King that will give the lecture on radiation? -I can't __ I know it for sure. A. say　 　 B. tell　 　C. speak　 　　D. talk 28. "No, no ", he whispered,____ talking to himself. A. while　 　B. even if　 　 C. as if　 　 D. when 29. - Any phone call for me? -Yes.____ telephoned 10 minutes ago. A. A Doctor Simpson　 　B. The certain doctor C. The Tim Smith　 　 D. A Simpson Doctor 30. ____ is mentioned above, the number of the cattle here has been limited to 200. A. It　B. Which　 　 C. That　 　 D. As 31. -Would you please give him the paper the moment he ____? -No problem. A. arrives　 　 B. appearing　C. came　 　D. reaches 32.I wonder ____ we can solve this hard problem. A. what else　 　B. how else　C. that　 　 D. as if 33.- Excuse me, can you tell me the way to the bank? -Oh, that's ____, it's either the second or the third road on the left. A. the difficult one　 　B. a hard one C. an easy one　 　D. the very one 34. The newly-founded company is faced with lots of trouble, ____ the financial problem ia not the worst. A. which　B. of which　C. to whom　 　D. with which 35. To his surprise, his new play well ____ by the public in China, met a cold welcome abroad. A. received　 　B. accepted　 　C. accepting D. receiving

单项选择[2] 第一节　单项填空(共15小题；每小题1分，满分15分) 从A、B、C、D四个选项中，选出可以填入空白处的最佳选项。 21.- ____ at the new school? -Fine, I've got used to the life there and I've made some friends. A. How are you　B. How are you doing C. Are you getting on well　D. How do you do 22. -Mary has a very low opinion of George. -It can' t be any worse than ____ of her. A. he　B. his　 　 C. him　 　 D. he does 23. I'll be home late, but please don't stay ____ for me. A. up　 　 B. in　 　 C. over　 　 　D. on 24.- What are you going to do next? -To write a science story about outer space ___ by Science and Technology Press. A. published　 　　B. being published C. to be published　 　 D. publishing 25. -How do you like the film? -There was nothing special--- it was only ____. A. average　 　　B. usual　 　 C. normal　 　D. common 26. In salad bars the waiter does not bring you salad. You have to____ yourself, usually to as much as you want. A. bring　 　B. serve　C. help D. supply 27. Our TV sets sell well, but ten years ago no one could have imagined such a big share in the market that they ____. A. were having　 　B. had had　 　C. were to have　D. had 28. Let us suppose that you are in____ position of____ parent. Would you allow your child to do such a thing? A. a; a　 　 B. a; the　 　C. the; a　 　D. the; the 29.- He has to finish the experiment today. He must be in the lab,____? -That s quite possible. A. mustn't he　B. won't he　C. doesn't he　 　D. isn't he 30. However, at times this balance in nature is disturbed, ____ a number of possibly unforeseen effects. A. lying in　 　 B. calling in C. taking in　D. resulting in 31. I wrote to my parents once a week,____ I was too busy with my teaching work. A. it　B. unless　 　C. when　 　 D. as 32. -Where will you start your work after graduation? -Mmm, it's not been decided yet. I ____ continue my study for a higher degree. A. need　 　B. must　 　C. would　 　 D. might 33. The truth, Sir, is that the old man____ across the street when my car hit him. A. was to walk B. had been walkingC. walked D. was walking 34. So many uses have been discovered for this wood ____ the supply of white pine is becoming smaller and smaller. A. then B. what　 　 C. that　 　D. which 35.-You seemed to have been impressed by his songs. -Well, not exactly so. It was his way of singing____ his voice that really impressed me. A. rather than　B. as well as　 　C. but also D. together with

单项选择[3] 第一节　单项填空(共15小题；每小题1分，满分15分) 从A、B、C、D四个选项中，选出可以填入空白处的最佳选项。 21. I smoke out of ____ habit, not for ____ pleasure; for I have been in the habit for twenty years. A. a; the　 　B. the; a　 　C. x; the　 　D. x; x 22. His job is to sell the ____ carvings in the ____ department of the company. A. wooden; sales　 　 B. wood; sales　 　 　 C. wood; sale　 　 　　D. wooden; sale 23. My father ____ my brother for driving but he wanted to pick ____ the knowledge of the computer during his spare time. A. advised; up　 　　B.persuaded; out　 　 　　C.intended; up　 　 　D.managed; out 24. -Whose advice do you think I should take? -_______. A.You speak　B.That's itC.It's up to you　D.You got it 25. ____ the youth ____ the rising sun at 8 or 9 o'clock am, Mao Zedong expressed his great hope for the young men. A. Comparing; withB. Compared; to C. Compared; with D. Comparing; to 26. Not far from the club, there was a girden, ____ owner seated in it playing bridge with his children every afternoon. A. whose　B. its　C. which　D. that 27. I haven' t seen Mary these past few days. I m afraid she ____ herself for some time. A. isn't feeling　 　　B. hasn' t been feeling　 　 C. hadn't been feeling　 　 D. wasn't feeling 28. Doing your homework is a sure way to improve yor test scores, and this is especially true ____ it comes to classroom tests. A. before　B. since　 　 C. when　 　D. after 29. A love marriage, however, does not necessarily ____ much sharing of interests and responsibilities. A. take over　 　B. result in　 　C. hold on　 　 D. keep to 30. The teacher came to the classroom and demanded ____ what had happened. A. his students to tell him　 　 　B. being told C. telling him　 　 　D. to be told 31. There were many students and teachers at the party yesterday. I looked for two seats to sit down with my friend, Tim, but there were ____. A. neither　 　B. nothing　 　 C. none　 　 　D. no 32. It is in Steven Spielberg' s first film, Jaws, ____ a big white shark attacks swimmers ____ are spending their holidays in a small village by the sea. A. where; who　 　B. which; that　 　 C. that; that　D. where; that 33. -What do you think of the dinner in Mr Brown's? -Oh, great! We have never had a better one. It's a dinner of twenty ____. A. drinks　 　 B. people　 　C. tables　 　D. courses 34. The captain ordered the crew to throw ____ the side of the ship all heavy guns and even stores in order to raise the boat in the water. A. on　 　 B. through　 　 C. across　 　 　D. over 35. Knowing how long the test would last, the students who finished ____ back and waited until the end of the exam. A. settle　 　　B. settled　 　 C. settling　 　 D. to settle

单项选择[4] 第一节　单项填空(共15小题；每小题1分，满分15分) 从A、B、C、D四个选项中，选出可以填入空白处的最佳选项。 21. What ______ public is concerned about is whether medical workers and scientists will be able to find for this new disease in a short time. A.不填;不填　B.不填;the　 　 C. the;不填　 　D.the;a 22._____ good, these cookies sell well. A.Smelled　 　 B. Smelling　C. Being smelled D. To smell 23.-Haven't I told you that I like the beef well done? -Yes, but I _____ it longer than I usually do. A. was cooking B.will cook　C. cooked D. had cooked 24. We' ve published large quantities of books. This year___ we've published three million. A. only　 　　B. just　 　 　C. alone　 　　D. merely 25. He ______. It's a shame my mobile phone was power-off. A. ought to call me　B. could have called me　 C. may be calling me　 　D.might call me 26.-Since you can't find a better job, why don't you stick to the present one? 　 　 -Well,_____. A.I believe not　 　　B.I don't care C. Never mind　 　 D. I might as well 27.The wedding party is to be held on May 18,_____ is, ten days_____ today. A. which; fromB.that;except C.that;from　D.as;since 28.You don't have to share the umbrellas. I'll give you one_____. A.every B.each　 　C.all　D.both 29.-Tom, keep your_____ on your studies. 　 　 - Sure, mum. I'll keep that in_____. A. mind;mind　B.heart;headC. brain; mind D.heart;brain 30.According to the Ministry of Health, by May 1, 2003, the total SARS deaths on the Chinese mainland_____to 170. A.has come　 　B. had reached　C.increased　 　D. had climbed 31. The door opened and in _____. A. he came　B. came he　C. did he come D. did Mr Smith come 32. After months of voyage, Columbus arrived in_____ later proved a new continent. A.where　B. what　C.which　 　 D.that 33._____ I toured Zhang Jiajie, I was deeply impressed with its beautiful scenery. A.For the fust time　 　B. At first　 　 C.It was the first time　 　 　D. The first time 34._____ private cars are bringing us convenience, they may also cause more traffic accidents and pollution. A.While　 　 B.As　 　 　C.If　 　 　 D.Since 35. Having ended the major combat (主战), President Bush_____ rebuilding Iraq. A. started out B. went all out　C. turned to　D. went on

单项选择[5] 第一节　单项填空(共15小题；每小题1分，满分15分) 从A、B、C、D四个选项中，选出可以填入空白处的最佳选项。 21. ---Good morning. Grand Hotel. 　 　---Hello, I'd like to book a room for the nights of the 18th and 19th. A. What can I do for you?　B. Just a minute, please. C. What's the matter?　D. At your service. 22. You’d better choose someone______ you think ____ kind and friendly to help you. A. whom; to beB. who; is C. that; is　 　D. which; to be 23. Readers can __quite well without knowing the exact meaning of each word. A. get along B. get over C. get in　D. get through 24. The first textbooks written for teaching English as a foreign language ____ in the 16th century. A. came about　B. came out C. came along　D. came up 25. I shall always ____ your friendship greatly for friendship is more important than money. A. value B. pay high respect C. respect　 　D. store 26. Later they discovered, ____ was news to him, that Pamela was the headmaster’s daughter. A. that　 　 B. which　 　C. what　 　D. who 27. Was it in this very room ____ he made up his mind ____ he should break away from his family? A. that; which　B. which; that C. which; which D. that; that 28. Tim Bemers-Lee is generally considered ____ the World Wide Web, on which all the information is shared by all. A.to have founded B.having foundedC. founding D. to found 29. ____has been said before, it is not easy to learn a foreign language. A. As　 　B. It　 　　C. What　 　　D. As it 30. The employee must have been dismissed by the employer last month, ____ he? A. haven't　 　　B. hasn't　 　C. didn't　 　 D. wasn't 31.---I didn't go to work yesterday because my car broke down. 　---You ____ mine. I wasn't using it then. A. might borrow B. could have borrowed C. must have borrowed　 　 D. ought to borrow 32. I admired my classmate Lisa very much. ____ her prettiness, she was smart and helpful. A. Except for　B. But for　C. Apart from　D. In spite of 33. I haven’t seen her ____ the time ____ she had the accident. A. for; that　B. since; when　 　C. since; as　D. for; when 34. ---I'm going to Guilin next week. Do you have anything ____ there? 　---No, but thank you all the same. A. to be taken　B. to take　 　C. taking　 　　D. taken 35. The ____ is that the new machine will arrive tomorrow. A. chance　 　　B. luck　 　 　 C. happening　 　 D. accident