两个新的三角命题

文[1]给出了两个用处较大的三角命题: 命题1 △ABC中,设角A,B,C的对边a,b,c成等差数列,则cos(A-C)/(2)＝2*cos(A+C)/(2)且tan(A)/(2)tan(C)/(2)＝(1)/(3).     

两个三角不等式的加强

在△ABC中,则有cot2A+cot2B+cot2C≥1 (1),cot2 A/2+cot2 B/2+cot2 C/2≥9 (2),当且仅当△ABC为正三角形时,(1)式及(2)式取等号.     

一个三角不等式的应用

我们发现:在△ABC中,sinA·sinB≤sin2A+B/2 证明:sinA·sinB=1/2[cos(A-B)-cos(A+B)]≤1/2[1-cos(A+B)]=sin2A+B/2.     

构造单位圆,巧解三角题

有些三角问题 ,若根据已知式的结构 ,挖掘出它的几何背景 ,巧妙地构造单位圆 ,化数为形 ,利用单位圆的直观性 ,便可简捷地求得问题的解 .例 1　已知sinA +sin 3A+sin 5A =a ,cosA +cos 3A +cos 5A =b ,求证 :当b≠ 0时 ,tan 3A =ab.证明　如图 1,因点A′(cosA ,sinA)、B′(cos 3A ,sin 3A)、C′(cos 5A ,sin 5A)均在单位圆上 ,连结OA′、OB′、OC′ ,则有∠A′OB′ =∠B′OC′=2A ,于是｜B′A′｜ =｜B′C′｜ , A′B′C′为等腰三角形 ,其重心必在B′O上 .又 A′B′C′的重心坐标x =13 (cosA +cos 3A +cos 5A) =13 b ,y…     

巧用构造法解三角题

例1 锐角A、B、C满足cos^2A cos^2B cos^2C 2cosAcosBcosC=1,求证：A B C=π。     

一个三角不等式的修正

文[1]证明了以下 命题在△ABC中,对k≥1,有tan(A)/(k)+tan(B)/(2k)+tan(C)/(3k)≥6tan(π)/(6k),(1)当且仅当A＝(π)/(6),B＝(π)/(3)时,等号成立.     

以三角为背景的高考数学题

近几年来 ,以三角为背景的问题已逐渐成为高考数学的热点 .因此 ,掌握它的几种基本类型及解法很有必要 .下面举例说明 .一、求三角形中的一些基本量这里所说的基本量主要指它的边、角、面积等 ,常利用内角和定理 ,正、余弦定理结合三角变换公式来解决 .【例 1】　 ( 1 998年全国高考 )一个直角三角形三内角的正弦值成等比数列 ,其最小内角为(　　 ) .A arccos 5 -12 　　　B arcsin 5 -12C arccos1 -52 D arcsin1 -52解　设最小内角为A ,则另两角为π2 -A ,π2 ,由已知得sin2 ( π2 -A) =sinAsin π2 ,∴cos2 A =sinA ,∴sin2 A +sinA-…     

摆脱思维定势 优化解题途径

已知：cos1A/cos2B＋sin1A/sin2B=1,求证：cos4B/cos2A＋sin4B/sin2A=1. 这是一道纯粹的三角命题,但很多学生往往思维定势,一见到三角就忙着套用三角公式,大大束缚了思维的发展,如果能将题目中式子的形状形式等进行深入地挖掘和联想,就会有柳暗花明又一村的惊喜．     

形式三角矩阵环的双导子

设A,B是有单位元的环,M为(A,B)-双模,研究形式三角矩阵环Tri(A,M,B)的双导子,利用代数方法得到了形式三角矩阵环Tri(A,M,B)的双导子的具体结构形式.     

三角形中的半角正切公式及其应用

1 三角形中的半角正切公式△ABC中∠A,∠B,∠C的对边分别为a,b,c,面积为S,则tan(A/2)=((a~2-((b-c)~2))/(4S));tan(B/2)=((b~2-((a-c)~2))/(4S)); tan(C/2)=((c~2-((a-b)~2))/(4S)).证明由余定理知     

卡思卡特城堡

It is worthy of noting that, whilst Crookston Castle witnessed the earlier and happier portion of Mary's variegated life,     

不同的吃喝玩乐

一、吃和喝吃苹果 eat an apple, 吃药 take medicine,吃糖 have some sweets,吃饭 have one's meals,吃馆子 dine out,吃惊 be surprised/     

漏洞百出的教育

Many college freshmen arrive woefully unprepared to do college work, and as disadvantaged populations continue to grow, the share of the American work force that has made it through college is expected to plummet. Many experts blame that educational failure not just on high schools but also on colleges. School & College, a special report by The Chronicle, looks at efforts to fix the system. What reforms would better prepare students for college? What should schools and colleges be doing differently? How should state and federal officials help?     

只要努力,不要后悔

There are numbers of crossroads on our long and unpredictable life journey where we totally have no idea about which direction to choose. No matter what our decision is, we should not turn back, but face the music and go ahead instead. I am this kind of girl who always does try without regretting, one example is how I dealt with my love.     

浅谈移民问题

Migration occurs behind a variety of reasons and has a great effect on the whole world. People may migrate in order to improve their economic situation, or in order to escape civil strife, persecution, and environmental disasters. The impact of migration is complex, bringing both benefits anddisadvantages. This paper briefly talks about the causes of migration, the allocation of benefits, and the ways in which individual countries and the international community deal with this important subject.     

Sister Carrie, an Adherent of Desires

Sister Carrie is one of the most controversial characters in American literature.Thought as a "fallen woman" firstly,she was defined as a "new woman" by some critics later. However, by digging into the motivaton behind the whole process of Carrie's "success", the relationship between Carrie and her creator (the author), the social conditions of then American, it can be found that Carrie has never been free-standing on her thought and she has never found her real-sdf even after becoming a famous actress. In a society dominated by mass consumerism Carrie is only an adherent of her own desires. She also is a representative of all those country girls flooded into cities, a symbol and a sacrifice of the urbanization of America in a time countryside was overcome by cities.     

Improvement of one-cycle controller by use of proportional integral differential controller

1.IntroductionOne-cyclecontrolmethod,whichwasproposedaboutonedecadeago[1],hasbecomeanattractivemethodinspecialfieldssuchaspowerfactorcorrection[2-6],switchingamplifiers[7,8],etc.Themainideaofthiscontrollerisbasedonintegrationofdiodevoltageinone-cycleandforcesittobeexactlyequaltothereferencevalue.Themainadvantageofthiscontrollerisitsrealtimeabilitytorejectthevariationofinputvoltage[1].Despitethisgreatability,ithasnogoodperformancesinrejectingofloaddisturbanceandfollowingreferencecommands.Espec…     

库克斯顿城堡

城堡, 是欧洲典型的建筑之一, 每个城堡都有一个美丽的传说或是一段历史。从本期开始, 我们将推出“城堡之旅”, 带各位读者去浏览、观赏欧洲各地的城堡, 开始一段文化与历史的旅途!     

胡主席在印度,我却在中国

Today, Sunday morning, I sit thinking and typing at my computer. My thoughts are in my homeland, India and my body is in China. The body is here and mind there for a good reason. Mr. Hu is president and I am resident and both are in each other's countries (what a difference the absence of one small letter makes!). Mr. Hu will return in a few days and I will stay on longer.