①.在…中间, 在…当中, 是…中间的一个

　　　　 a. The great leader prefers to live among the working people.

　　　　 b. Divide the money among the six of you.

　　　　 c. The town lies among the mountains.

　　　　 d. I was among the first to arrive.

　　　　 e. Shelley is among the world’s greatest poets.

　　　　 f. Paris is among the largest cities in the world.

　　 ②.(和反身代词等连用)相互间, 共同, 一起

　　　　 a. They quarreled among themselves for half an hour.

　　　　 b. The three boys discussed the matter among themselves.

　　　　 c. Among them, they collected over a hundred pounds.

　　　　 a. They traveled along the west coast.

　　　　 b. There are trees all along the river banks.

　　 ①.表示反对关系:

　　　　 a. I will speak against anything that I know to be wrong.

　　　　 b. Many of the students are against the plan.

　　 ②.(表位置)靠着, 顶着, 迎着, 衬着

　　　　 a. The piano stood against the wall.

　　　　 b. He could hear the rain pattering against the windows.

　　　　 c. The trees were black against the morning sky.

　　　　 d. He struck her head against the branch of a tree.

③.预防, 防备, 准备…时用

　　　　 a. We are all taking medicine against the disease.

　　　　 b. They bought some warm clothes against the cold winter.

　　　　 c. We have to save some money against old age.

　　 ④.表示对比关系

　　　　 a. He was elected by a majority of 20 votes against 10.

　　 ⑤.词组或固定搭配: against one’s will违反自己的意愿; go against违反, 不利于

　　 ①.(指时间)在…之后

　　　　 a. What shall we do after graduation ?

　　　　 b. He was still weak after his long illness.

　　　　 c. After sending that letter I realized how stupid I was.

　　　　 d. After having had twelve months’ practice, I decided to make one more effort.

　　 ②.(指空间)在…后面

　　　　 a. He entered the room after his father.

　　　　 b. He called after me three or four times.

　　 ③.照…的样子, 跟着

　　　　 a. Will you please make the box after this model ?

　　　　 b. The boy was named after his uncle.

　　 ④.词组或固定搭配: after all毕竟, 究竟; after dark天黑之后; be after想得到, 寻求, 追求; go after追求, 设法得到; look after照顾, 照看; one after another一个接一个地

①.穿过, 跨过(从一边到另一边)

　　　　 a. There is no bridge across the river.

　　　　 b. He hurried across the bridge to Waterloo.

　　 ②.在(河, 街等的)另一边

　　　　 a. The bus stop is just across the road.

　　　　 b. Across the street was a café full of people.

　　 ③.交叉

　　　　 a. He sat with his arms across the chest.

　　　　 b. At one point the railway line goes across the road.

　　 ④.在…全境, 在…境内

　　　　 a. There are hundreds of such shops across the country.

　　　　 b. The measures are already taken in a few places across the United States.

　 　⑤.词组或固定搭配: come acorss偶然碰到

①.在…上方

　　　　 a. The moon was now above the trees in the east.

　　　　 b. The shelf should be six feet above the level of the floor.

　　 ②.在.. .之上, 比…高, 高于, 超过

　　　　 a. The temperature was only a degree above zero.

　　　　 b. He is a head above me.

　　　　 c. Above 200 people were there when the fire broke out.

　　　　 d. All children above six years of age must go to school.

　　 ③.词组或固定搭配: above all首先, 最重要的是

　　 ①.关于

　　　　 a. What are you talking about ?

　　　　 b. I read about it in today’s newspaper.

　　　　 c. When I first saw you together, I was so happy about it.

　　　　 d. You are entirely wrong about the whole matter.

　　　　 e. He said he had to see me about something important.

　　　　 f. I have no doubt about that.

　　 ②.在…周围, 围绕

　　　　 a. There was a white fence about the house.

　　　　 b. I went closer, and he put his arms about me.

　　 ③.在…范围内, 到处

　　　　 a. After tea I wandered alone about the town.

　　　　 b. He walked for three hours about the London street.

　　 ④.在…附近, 在身边

　　　　 a. The streets about the park are very clean.

　　　　 b. She likes to have her children about her.

　　 ⑤.在身上, 在身边

　　　　 a. Do you have two hundred yuan about you ?

　　　　 b. I happen to have a pencil about me.

　　 ⑥.词组或固定搭配: see about负责处理; set about开始着手做某事

10.(2006全国Ⅰ)的三个内角为，求当A为何值时，取得最大值，并求出这个最大值　

解: 由A+B+C=π, 得 = － , 所以有cos =sin　

cosA+2cos =cosA+2sin =1－2sin² + 2sin

=－2(sin － )²+

当sin = , 即A=时, cosA+2cos取得最大值为

[探索题]是否存在锐角α、β,使①α+2β=, ②同时成立?若存在,求出α、β,若不存在,请说明理由.

解:假设存在,由①得

由②代入上式得, 又②

是方程的两个根,解得

.

∵α、β是锐角,　 ∴,tanβ=1.

,代入①得.即存在,使①②式同时成立.

9. 求证:

证法1:左边=

证法2:右边=

由合比定理得

8.求。

解:原式=

注：在化简三角函数式过程中，除利用三角变换公式，还需用到代数变形公式，如本题平方差公式。