问题 解答题
已知坐标平面上三点A(2,0),B(0,2),C(cosα,sinα).
(1)若(
OA
+
OC
)2=7
(O为原点),求向量
OB
OC
夹角的大小;
(2)若
AC
BC
,求sin2α的值.
答案

(1)∵

OA
+
OC
=(2+cosα,sinα),(
OA
+
OC
)2=7

∴(2+cosα)2+sin2α=7,

cosα=

1
2

又B(0,2),C(cosα,sinα),设

OB
OC
的夹角为θ,

则:cosθ=

OB
OC
|
OB
||
OC
|
=
2sinα
2
=sinα=±
3
2

OB
OC
的夹角为
π
6
5
6
π

(2)∵

AC
=(cosα-2,sinα),
BC
=(cosα,sinα-2)

AC
BC
,∴
AC
BC
=0

可得cosα+sinα=

1
2
,①

(cosα+sinα)2=

1
4
,∴2sinαcosα=-
3
4
sin2α=-
3
4

阅读理解

Americans think a great deal about time. From childhood they learn to value(珍惜)time. As children, they are taught to be on time to go to school, to do work and to do everything. When they are having a good time, they say that time flies. When a person is dying, they say he is living on borrowed time. Time is money. Time is knowledge. Time is everything in America. A working American has to work hard for 8 hours a day or 40 hours a week. This is the working time. In his free time, he also works hard for more money. Even Saturday and Sunday are filled with(充满)activities. In the street you can hardly see a man who walks slowly. They walk very fast. In fact, they are running. They love time because time brings them money, comforts and progress. But sometimes they also hate time, for they feel they have become servants(仆人)of the clock.

小题1:A working American has to work___ days a week in his working  

time.   

A. seven      B. six     C. five   

小题2:An American always walks____.

A. sadly       B. slowly    C. fast  

小题3:“He is living on borrowed time” means_____.

A. He has little money.

B. He has no time   

C. He is dying   

小题4:In his free time, an American usually_____.

A. plays cards with their friends

B. stays at home all day and watched TV.

C. gets more money by work.

小题5: Which of the following is NOT TRUE?

A. The Americans usually value time.

B. The Americans are often on time.

C. All Americans love time and don’t hate time.

判断题