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The True Artist

The True Artist
The True Artist

The True Artist

这些译文都是为了感谢我大学本科时最最尊敬的吕俊吕爷爷的,在他身上我们看到了翻译家谦逊的态度以及对译文细致,力求精准的精神,很感谢吕教授一直以来的教诲,从大二到大四,教会了我们很多翻译的精髓,以下是译文赏析课上吕教授的讲评的内容,如有疏漏之处,也请见谅,吕爷爷讲到精彩处总让人感慨万千,竟忘记提笔记下。希望对大家有所裨益。

如果能有幸被吕爷爷看到,希望能再次说一声,吕爷爷,谢谢您,您的课将是我们一生中最宝贵的财富,您的人生态度谦虚正直也将是我对自己一生的要求。吕爷爷,我们好想你。

The True Artist 真正的艺术家

By Norman Bethune 诺尔曼白求恩

The true artist lets himself go. He is natural. He “swims in the stream of his own temperament.”He listens to himself. He respects himself. He comes into the light of every-day like a great leviathan of the deep, breaking the smooth surface of the accepted things, gay, serious, sportive. His appetite for life is enormous. He enters eagerly into the life of man, all men. He becomes all men in himself. The function of the artist is to disturb. His duty is to arouse sleepers, to shake the complacent pillars of the world. He reminds the world of its dark ancestry, shows the world its present and points the way to its new birth. He is at once the product and preceptor of his time. After his passage we are troubled and made unsure of our too-easily accepted realities. He makes uneasy the static, the set and the still. In a world terrified of change, he preaches revolution --the principle of life. He is an agitator, a disturber of the peace--quick, impatient, positive, restless and disquieting. He is the creative spirit working in the soul of man.

真正的艺术家从心所欲,不矫揉造作(秉性自然),他在属于自己气质的溪流中畅游。他倾听自我,尊重自我。他像深海的巨鲸,每日浮出水面,沐浴阳光,打破以往的平静,时而欢快,时而严肃,时而嬉戏玩耍。他对生命充满欲望(希望),急切地走入社会人生,去了解所有人并与他们融为一体。艺术家的作用就是要打破平静,他的责任是唤醒沉睡的人们,震撼那些自鸣得意的社会栋梁。他向世人揭示黑暗的过去,展示社会的现状并指出新生之路。他既是时代的产物,也是时代的先驱。跟随他的脚步,你会变得困惑,对那些认为理所当然的事实开始产生怀疑,他搅动了一潭死水。在一个惧怕变革的世界里,他鼓吹改革这一生活的准则。他是一个鼓动家,是一个平静的破坏者,行动迅捷,性情急切,态度积极,是一个时刻不肯安宁的鼓动家和破坏者。他是活跃在人类灵魂中富有创造力的精灵。

高中常用三角函数公式大全

高中常用三角函数公式 两角和公式 sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-cosAsinB cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB tan(A+B) =tanAtanB -1tanB tanA + tan(A-B) =tanAtanB 1tanB tanA +- cot(A+B) =cotA cotB 1-cotAcotB + cot(A-B) =cotA cotB 1cotAcotB -+ 倍角公式 tan2A =A tan 12tanA 2- Sin2A=2SinA?CosA Cos2A = Cos 2A-Sin 2A=2Cos 2A-1=1-2sin 2A 半角公式 sin(2A )=2 cos 1A - cos(2A )=2 cos 1A + tan(2A )=A A cos 1cos 1+- cot( 2A )=A A cos 1cos 1-+ tan(2 A )=A A sin cos 1-=A A cos 1sin + 诱导公式 sin(-a) = -sina cos(-a) = cosa sin( 2 π-a) = cosa cos(2 π-a) = sina sin(2π+a) = cosa

cos( 2 π+a) = -sina sin(π-a) = sina cos(π-a) = -cosa sin(π+a) = -sina cos(π+a) = -cosa tgA=tanA =a a cos sin 万能公式 sina=2 )2 (tan 12tan 2a a + cosa=2 2 )2 (tan 1)2(tan 1a a +- tana=2 )2 (tan 12tan 2a a - 其它公式 a?sina+b?cosa=)b (a 22+×sin(a+c) [其中tanc= a b ] a?sin(a)-b?cos(a) = )b (a 22+×cos(a-c) [其中tan(c)=b a ] 1+sin(a) =(sin 2a +cos 2 a )2 1-sin(a) = (sin 2a -cos 2 a )2 公式一: 设α为任意角,终边相同的角的同一三角函数的值相等: sin (2kπ+α)= sinα cos (2kπ+α)= cosα tan (2kπ+α)= tanα cot (2kπ+α)= cotα 公式二: 设α为任意角,π+α的三角函数值与α的三角函数值之间的关系: sin (π+α)= -sinα cos (π+α)= -cosα tan (π+α)= tanα cot (π+α)= cotα 公式三: 任意角α与 -α的三角函数值之间的关系:

必修三 Unit 5 Canada The True North公开课教学设计

高中必修3 Unit5 Canada: The True North The First Period Reading (I) I. 教学目标 1. Language goals a. Learn the use of the following words and phrases: Minister rather than,continent,surround, harbor, measure, aboard, have a gift for, within, manage to do, catch sight of, eagle, acre, urban settle down eastward b. Important sentences: Learn about some sentence patterns and be able to use them freely. 2. Ability goals 1.) Learn how to describe the places that Li Daiyu and Liu Qian visit in Canada. Understand the noun clause used as appositive. 2.) Enable the students to understand the details of the passage about Canada and find the correct answers to the questions in the post-reading. 3. Learning ability goals Improve the students’ re ading ability - searching for information. II. 教学重点 Learn the information about Canada. Master the expressions for describing directions and locations. III. 教学难点 Learn the methods of writing a traveling report about. Learn to read the traveling report according to the traveling route. IV. 教学策略选择与设计 1. Skimming and scanning; 2. Asking-and-answering activities; 3. Listening method.

高中三角函数公式大全

高中三角函数公式大全 2009年07月12日星期日19:27 三角函数公式 两角和公式 sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-cosAsinB cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB tan(A+B) =tanAtanB -1tanB tanA tan(A-B) =tanAtanB 1tanB tanA cot(A+B) =cotA cotB 1-cotAcotB cot(A-B) =cotA cotB 1cotAcotB 倍角公式 tan2A =A tan 12tanA 2Sin2A=2SinA?CosA Cos2A = Cos 2A-Sin 2A=2Cos 2A-1=1-2sin 2A 三倍角公式sin3A = 3sinA-4(sinA)3cos3A = 4(cosA)3-3cosA tan3a = tana ·tan( 3+a)·tan(3-a) 半角公式 sin(2 A )=2cos 1A cos(2 A )=2cos 1A tan(2 A )=A A cos 1cos 1cot(2A )= A A cos 1cos 1tan(2A )=A A sin cos 1=A A cos 1sin 和差化积 sina+sinb=2sin 2b a cos 2 b a

sina-sinb=2cos 2b a sin 2 b a cosa+cosb = 2cos 2b a cos 2 b a cosa-cosb = -2sin 2b a sin 2 b a tana+tanb=b a b a cos cos )sin(积化和差 sinasinb = -2 1[cos(a+b)-cos(a-b)] cosacosb = 2 1[cos(a+b)+cos(a-b)] sinacosb = 2 1[sin(a+b)+sin(a-b)] cosasinb = 2 1[sin(a+b)-sin(a-b)] 诱导公式 sin(-a) = -sina cos(-a) = cosa sin( 2-a) = cosa cos( 2-a) = sina sin( 2+a) = cosa cos(2 +a) = -sina sin(π-a) = sina cos(π-a) = -cosa sin(π+a) = -sina cos(π+a) = -cosa tgA=tanA =a a cos sin 万能公式 sina=2)2 (tan 12tan 2a a cosa=22)2 (tan 1)2(tan 1a a

Unit 5 Canada—“The True North”(公开课教学设计)

Unit 5Canada—“The True North”(公开课教学设计) 教学重点 1. Let students read the passage and learn about the geography, population, main cities, natural beauty and natural resources of Canada. 2. Get students to learn different reading skills. 教学难点 1. Develo p students’ reading ability. 2. Enable students to learn about some basic information and talk about Canada. 三维目标 知识目标 1. Get students to learn the useful new words and expressions in this part. 2. Let students learn the knowledge of Canada. 能力目标 1. Develop students’ reading ability and let them learn different reading skills. 2. Let students learn how to read a traveling report and how to use a map. 2. Enable students to learn about some basic information and talk about Canada. 情感目标 1. Stimulate students’ interest in learning about foreign countries. 2. Develop students’ sense of cooperative learning. 教学过程 Step 1 Leading-in and warming up 1. Show a map of Canada to students and talk about Canada. Then ask them the following questions: 1)What kind of country is Canada? 2)How large is it? 3)What else do you know about Canada? Suggested answers: 1)Canada is a multicultural country. 2)It’s the second largest country in the world. It cover s an area of 9 984 670 square kilometers. It is a bit bigger than China. 3)(Students’ answer may vary. Encourage them to tell more information. ) 2. Make a quiz Show the following on the screen. How much do you know about Canada? 1. What language(s) do Canadians speak? A. English. B. English and German. C. English and French. D. English and Spanish.

高中数学三角函数公式大全 (1)

三角函数 1. ①与α(0°≤α<360°)终边相同的角的集合(角α与角β的终边重合): {} Z k k ∈+?=,360 |αββο ②终边在x 轴上的角的集合: {} Z k k ∈?=,180|οββ ③终边在y 轴上的角的集合:{ } Z k k ∈+?=,90180|ο οββ ④终边在坐标轴上的角的集合:{} Z k k ∈?=,90|οββ ⑤终边在y =x 轴上的角的集合:{} Z k k ∈+?=,45180|οοββ ⑥终边在x y -=轴上的角的集合:{} Z k k ∈-?=,45180|οοββ ⑦若角α与角β的终边关于x 轴对称,则角α与角β的关系:βα-=k ο360 ⑧若角α与角β的终边关于y 轴对称,则角α与角β的关系:βα-+=οο180360k ⑨若角α与角β的终边在一条直线上,则角α与角β的关系:βα+=k ο180 ⑩角α与角β的终边互相垂直,则角α与角β的关系:οο90360±+=βαk 2. 角度与弧度的互换关系:360°=2π 180°=π 1°=0.01745 1=57.30°=57°18′ 注意:正角的弧度数为正数,负角的弧度数为负数,零角的弧度数为零. 、弧度与角度互换公式: 1rad =π 180°≈57.30°=57°18ˊ. 1°=180 π≈0.01745(rad ) 3、弧长公式:r l ?=||α. 扇形面积公式:21 1||22 s lr r α==?扇形 4、三角函数:设α是一个任意角,在α的终边上任取(异于原点的)一点P (x,y )P 与原点的距离为r ,则 =αsin r x =αcos ; x y = αtan ; y x =αcot ; x r =αsec ;. αcsc 5、三角函数在各象限的符号:正切、余切 余弦、正割 正弦、余割 6、三角函数线 正弦线:MP; 余弦线:OM; 正切线: AT. SIN \COS 1、2、3、4表示第一、二、三、四象限一半所在区域

高三三角函数公式大全

第一部分三角函数公式 2两角和与差的三角函数 cos(α+β)=cosα2cosβ-sinα2sinβ cos(α-β)=cosα2cosβ+sinα2sinβ sin(α±β)=sinα2cosβ±cosα2sinβ tan(α+β)=(tanα+tanβ)/(1-tanα2tanβ) tan(α-β)=(tanα-tanβ)/(1+tanα2tanβ) 2和差化积公式: sinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2] sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2] cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2] cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2] 2积化和差公式: sinα2cosβ=(1/2)[sin(α+β)+sin(α-β)] cosα2sinβ=(1/2)[sin(α+β)-sin(α-β)] cosα2cosβ=(1/2)[cos(α+β)+cos(α-β)] sinα2sinβ=-(1/2)[cos(α+β)-cos(α-β)] 2倍角公式: sin(2α)=2sinα2cosα=2/(tanα+cotα) cos(2α)=(cosα)^2-(sinα)^2=2(cosα)^2-1=1-2(sinα)^2 tan(2α)=2tanα/(1-tan^2α) cot(2α)=(cot^2α-1)/(2cotα) sec(2α)=sec^2α/(1-tan^2α) csc(2α)=1/2*secα2cscα 2三倍角公式: sin(3α) = 3sinα-4sin^3α = 4sinα2sin(60°+α)sin(60°-α) cos(3α) = 4cos^3α-3cosα = 4cosα2cos(60°+α)cos(60°-α) tan(3α) = (3tanα-tan^3α)/(1-3tan^2α) = tanαtan(π/3+ α)tan(π/3-α)

高中三角函数公式大全

高中三角函数公式大全 三角函数公式 两角和公式 sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-cosAsinB cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB tan(A+B) =tanAtanB -1tanB tanA + tan(A-B) =tanAtanB 1tanB tanA +- cot(A+B) =cotA cotB 1-cotAcotB + cot(A-B) =cotA cotB 1cotAcotB -+ 倍角公式 tan2A =A tan 12tanA 2- Sin2A=2SinA?CosA Cos2A = Cos 2A-Sin 2A=2Cos 2A-1=1-2sin 2A 三倍角公式 sin3A = 3sinA-4(sinA)3 cos3A = 4(cosA)3-3cosA tan3a = tana ·tan(3π+a)·tan(3 π-a) 半角公式 sin(2A )=2cos 1A - cos(2A )=2cos 1A + tan(2 A )=A A cos 1cos 1+- cot( 2A )=A A cos 1cos 1-+ tan(2A )=A A sin cos 1-=A A cos 1sin + 和差化积 sina+sinb=2sin 2b a +cos 2b a - sina-sinb=2cos 2b a +sin 2 b a - cosa+cosb = 2cos 2b a +cos 2b a - cosa-cosb = -2sin 2b a +sin 2 b a - tana+tanb=b a b a cos cos )sin(+ 积化和差 sinasinb = -21[cos(a+b)-cos(a-b)] cosacosb = 2 1[cos(a+b)+cos(a-b)] sinacosb = 21[sin(a+b)+sin(a-b)] cosasinb = 2 1[sin(a+b)-sin(a-b)] 万能公式 sina=2)2(tan 12tan 2a a + cosa=22)2(tan 1)2(tan 1a a +- tana=2 )2 (tan 12tan 2a a - 其它公式 a?sina+b?cosa=)b (a 22+×sin(a+c) [其中tanc=a b ] a?sin(a)-b?cos(a) = )b (a 22+×cos(a-c) [其中tan(c)=b a ]

必修三UnitCanadaTheTrueNorth公开课教育教学设计

必修三-Unit--Canada-The-True-Nort h公开课教学设计

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高中必修3 Unit5 Canada:The True North The First Period Reading (I) I. 教学目标 1. Language goals a. Learn the use of the following words and phrases: Minister rather than,continent,surround, harbor, measure, aboard, have a gift for, within, manage to do, catch sight of, eagle, acre, urban settle down eastward b. Important sentences: Learn about some sentence patterns and be able to use them freely. 2. Ability goals 1.) Learn how to describe the places that Li Daiyu and Liu Qian visit in Canada. Understand the noun clause used as appositive. 2.) Enable the students to understand the details of the passage about Canada and find the correct answers to the questions in the post-reading. 3. Learning ability goals Improve the students’ reading ability - searching for information. II. 教学重点 Learn the information about Canada. Master the expressions for describing directions and locations. III. 教学难点 Learn the methods of writing a traveling report about. Learn to read the traveling report according to the traveling route. IV. 教学策略选择与设计 1.Skimming and scanning; 2.Asking-and-answering activities; 3.Listening method. V. 学情分析

高中三角函数公式大全及经典习题解答

高中三角函数公式大全及经典习题解答 -CAL-FENGHAI-(2020YEAR-YICAI)_JINGBIAN

用心辅导中心 高二数学 三角函数 知识点梳理: ⒈L 弧长=αR=nπR 180 S 扇=21L R=2 1R 2 α=3602R n ?π ⒉正弦定理: A a sin =B b sin =C c sin = 2R (R 为三角形外接圆半径) ⒊余弦定理:a 2=b 2+c 2-2bc A cos b 2=a 2+c 2-2ac B cos c 2 =a 2 +b 2 -2ab C cos bc a c b A 2cos 2 22-+= ⒋S ⊿=2 1a a h ?=2 1ab C sin =2 1bc A sin =2 1ac B sin =R abc 4=2R 2A sin B sin C sin =A C B a sin 2sin sin 2=B C A b sin 2sin sin 2=C B A c sin 2sin sin 2=pr =))()((c p b p a p p --- (其中)(2 1c b a p ++=, r 为三角形内切圆半径) ⒌同角关系: ⑴商的关系:①θtg =x y =θ θ cos sin =θθsec sin ? ② θθθ θθcsc cos sin cos ?=== y x ctg ③θθθtg r y ?== cos sin ④θθθθcsc cos 1sec ?== =tg x r ⑤θθθctg r x ?== sin cos ⑥θθθθsec sin 1csc ?== =ctg y r ⑵倒数关系:1sec cos csc sin =?=?=?θθθθθθctg tg ⑶平方关系:1csc sec cos sin 222222=-=-=+θθθθθθctg tg ⑷)sin(cos sin 22?θθθ++= +b a b a (其中辅助角?与点(a,b ) 在同一象限,且a b tg =?) ⒍函数y=++?)sin(?ωx A k 的图象及性质:(0,0>>A ω)

A TRIP ON “THE TRUE NORTH课文文本

A TRIP ON “THE TRUE NORTH” Paragraph 1 Li Daiyu and her cousin Liu Qian were on a trip to Canada to visit their cousins in Montreal on the Atlantic coast. Rather than take the aeroplane all the way, they decided to fly to Vancouver and then take the train from west to east across Canada. The thought that they could cross the whole continent was exciting. Paragraph 2 Their friend, Danny Lin, was waiting at the airport. He was going to take them and their baggage to catch “The True North”, the cross-Canada train. On the way to the station, he chatted about their trip. “You’re going to see some great scenery. Going eastward, you’ll pass mountains and thousands of lakes and forests, as well as wide rivers and large cities. Some people have the idea that you can cross Canada in less than five days, but they forget the fact that Canada is 5,500 kilometres from coast to coast. Here in Vancouver, you’re in Canada’s warmest part. People say it is Canada’s most beautiful city, surrounded by mountains and the Pacific Ocean. Skiing in the Rocky Mountains and sailing in the harbour make Vancouver one of Canada’s most popular cities to live in. Its population is increasing rapidly. The coast north of Vancouver has some of the oldest and most beautiful forests in the world. It is so wet there that the trees are extremely tall, some measuring over 90 metres. ” Paragraph 3 That afternoon aboard the train, the cousins settled down in their seats. Earlier that day, when they crossed the Rocky Mountains, they managed to catch sight of some mountain goats and even a grizzly bear and an eagle. Their next stop was Calgary, which is famous for the Calgary Stampede. Cowboys from all over the world come to compete in the Stampede. Many of them have a gift for riding wild horses and can win thousands of dollars in prizes. Paragraph 4 After two days’ travel, the girls began to realize that Canada is quite empty. At school, they had learned that most Canadians live within a few hundred kilometres of the USA border, and Canada’s population is only slightly over thirty million, but now they were amazed to see such an empty country. They went through a wheat-growing province and saw farms that covered thousands of acres. After dinner, they were back in an urban area, the busy port city of Thunder Bay at the top of the Great Lakes. The girls were surprised at the fact that ocean ships can sail up the Great Lakes. Because of the Great Lakes, they learned, Canada has more fresh water than any other country in the world. In fact, it has one-third of the world’s total fresh water, and much of it is in the Great Lakes. Paragraph 5 That night as they slept, the train rushed across the top of Lake Superior, through the great forests and southward towards Toronto.

Canada The True North教案

Canada The True North教案 必修3第5单元Reading2教案设计 一、 学生分析 教学对象为高一学生,智力发展趋于成熟。他们的认知能力有了进一步的发展,渐渐形成用英语获取信息、处理信息、分析问题和解决问题的能力,因此在教学中我特别注重提高学生用英语进行思维和表达的能力。逐渐由大量输入信息转向试着自己输出信息,形成自己学习技能和策略,学会把语言和现实生活联系起来。 通过任务型课堂活动和学习,让学生自主参与课堂活动,形成主体。勇于表现自己和阐述自己的观点。把课堂知识拓展到课外,通过多渠道获取信息。 二、 教材分析 这是本单元的第四课时。在这之前学生对加拿大的一些基本信息有一定的了解,例如:领土面积,风土人情和多元文化等等。他们急切的想知道加拿大的一些美丽城市。想通过书本领略其风光。本课时主要注重学生的听说读写能力的培养和训练。 三、

教学目标 ) 语言知识: 单词:理解和运用以下生词:broad,distance,downtown,nearby,dawn等 语法;复习和运用名词从句介绍本国风景和异国风情 2) 语言技能: 听:听懂介绍一个地方的片段,抓住时间、地点、名胜的描述 说:能用得体的语言描述风景区并且符合逻辑 读:Scanning,skimming,carefulreading,generalization,inf erence等阅读技能训练 写:能运用atfirst,then,next,…atlast简要且富有逻辑地描述风景区 3)学习策略: 自主学习、英语思维、信息处理和有效交际 4)情感态度: 分享自己的旅游经历,体验用英语交流的成功与喜悦,培养合作互动精神 四、教学重难点

所有三角函数公式

诱导公式 常用的诱导公式有以下几组: 公式一: 设α为任意角,终边相同的角的同一三角函数的值相等: sin(2kπ+α)=sinα (k∈Z) cos(2kπ+α)=cosα (k∈Z) tan(2kπ+α)=tanα (k∈Z) cot(2kπ+α)=cotα (k∈Z) 公式二: 设α为任意角,π+α的三角函数值与α的三角函数值之间的关系:sin(π+α)=-sinα cos(π+α)=-cosα tan(π+α)=tanα cot(π+α)=cotα 公式三: 任意角α与-α的三角函数值之间的关系: sin(-α)=-sinα cos(-α)=cosα tan(-α)=-tanα cot(-α)=-cotα 公式四: 利用公式二和公式三可以得到π-α与α的三角函数值之间的关系:sin(π-α)=sinα cos(π-α)=-cosα tan(π-α)=-tanα cot(π-α)=-cotα 公式五: 利用公式一和公式三可以得到2π-α与α的三角函数值之间的关系:sin(2π-α)=-sinα cos(2π-α)=cosα tan(2π-α)=-tanα cot(2π-α)=-cotα 公式六: π/2±α及3π/2±α与α的三角函数值之间的关系: sin(π/2+α)=cosα cos(π/2+α)=-sinα tan(π/2+α)=-cotα cot(π/2+α)=-tanα

sin(π/2-α)=cosα cos(π/2-α)=sinα tan(π/2-α)=cotα cot(π/2-α)=tanα sin(3π/2+α)=-cosα cos(3π/2+α)=sinα tan(3π/2+α)=-cotα cot(3π/2+α)=-tanα sin(3π/2-α)=-cosα cos(3π/2-α)=-sinα tan(3π/2-α)=cotα cot(3π/2-α)=tanα (以上k∈Z) 注意:在做题时,将a看成锐角来做会比较好做。 诱导公式记忆口诀 上面这些诱导公式可以概括为: 对于π/2*k ±α(k∈Z)的三角函数值, ①当k是偶数时,得到α的同名函数值,即函数名不改变; ②当k是奇数时,得到α相应的余函数值,即sin→cos;cos→sin;tan→cot,cot→tan. (奇变偶不变) 然后在前面加上把α看成锐角时原函数值的符号。 (符号看象限) 例如: sin(2π-α)=sin(4·π/2-α),k=4为偶数,所以取sinα。 当α是锐角时,2π-α∈(270°,360°),sin(2π-α)<0,符号为“-”。 所以sin(2π-α)=-sinα 上述的记忆口诀是: 奇变偶不变,符号看象限。 公式右边的符号为把α视为锐角时,角k·360°+α(k∈Z),-α、180°±α,360°-α 所在象限的原三角函数值的符号可记忆 水平诱导名不变;符号看象限。 # 各种三角函数在四个象限的符号如何判断,也可以记住口诀“一全正;二正弦(余割);三两切;四余弦(正割)”. 这十二字口诀的意思就是说: 第一象限内任何一个角的四种三角函数值都是“+”; 第二象限内只有正弦是“+”,其余全部是“-”; 第三象限内切函数是“+”,弦函数是“-”;

三角函数公式大全(很详细)

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图2 在直角坐标系中定义三角函数示意图在直角坐标系中,如下定义六个三角函数: ?正弦函数 r ?余弦函数 ?正切函数 ?余切函数 ?正割函数

?余割函数 2 转化关系2.1 倒数关系 2.2 平方关系 2 和角公式 3 倍角公式、半角公式3.1 倍角公式 3.2 半角公式

3.3 万能公式 4 积化和差、和差化积4.1 积化和差公式

证明过程 首先,sin(α+β)=sinαcosβ+sinβcosα(已证。证明过程见《和角公式与差角公式的证明》)因为sin(α+β)=sinαcosβ+sinβcosα(正弦和角公式) 则 sin(α-β) =sin[α+(-β)] =sinαcos(-β)+sin(-β)cosα =sinαcosβ-sinβcosα 于是 sin(α-β)=sinαcosβ-sinβcosα(正弦差角公式) 将正弦的和角、差角公式相加,得到 sin(α+β)+sin(α-β)=2sinαcosβ 则 sinαcosβ=sin(α+β)/2+sin(α-β)/2(“积化和差公式”之一) 同样地,运用诱导公式cosα=sin(π/2-α),有 cos(α+β)= sin[π/2-(α+β)] =sin(π/2-α-β) =sin[(π/2-α)+(-β)] =sin(π/2-α)cos(-β)+sin(-β)cos(π/2-α) =cosαcosβ-sinαsinβ 于是 cos(α+β)=cosαcosβ-sinαsinβ(余弦和角公式) 那么 cos(α-β) =cos[α+(-β)] =cosαcos(-β)-sinαsin(-β) =cosαcosβ+sinαsinβ cos(α-β)=cosαcosβ+sinαsinβ(余弦差角公式) 将余弦的和角、差角公式相减,得到 cos(α+β)-cos(α-β)=-2sinαsinβ 则

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高中三角函数公式大全 诱导公式29个 sin(α+2kπ)=sinα cos(α+2kπ)=cosα tan(α+2kπ)=tanα sin(α+π)=-sinα cos(α+π)=-cosα tan(α+π)=tanα cot(α+π)=cotα sin(-α)=-sinα cos(-α)=cosα tan(-α)=-tanα cot(-α)=-cotα sin(π-α)=sinα cos(π-α)=-cosα cot(π-α)=-cotα tan(π-α)=-tanα sin(2π-α)=sinα cos(2π-α)=cosα sin(π/2-α)=cosα cos(π/2-α)=sinα tan(π/2-α)=cotα sin(π/2+α)=cosα cos(π/2+α)=-sinα tan(π/2+α)=-cotα sin(3π/2-α)=-cosα cos(3π/2-α)=-sinα tan(3π/2-α)=cotα sin(3π/2+α)=-cosα cos(3π/2+α)=sinα tan(3π/2+α)=-cotα 两角和差公式6个 cos(α+β)=cosαcosβ-s inαsinβcos(α-β)=cosαcosβ+sinαsinβsin(α+β)=sinαcosβ+cosαcosβ

sin(α-β)=sinαcosβ-sinαcosβ tan(α+β)=(tanα+tanβ)/(1-tanαtanβ) tan(α-β)=(tanα-tanβ)/(1+tanαtanβ) 倍角公式5个 sin(2α)=2sinαcosα cos(2α)=cos^2α-sin^2α =2cos^2α-1 =1-2sin^2α tan(2α)=2tanα/(1-tan^2α) 半角公式5个 sin^2(α/2)=(1-cosα)/2 cos^2(α/2)=(1+cosα)/2 tan^2(α/2)=(1-cosα)/(1+cosα) tan(α/2)=(1-cosα)/sinα =sinα/(1+cosα) 万能公式2个 sin(2α)=2tanα/(1+tan^2α) cos(2α)=(1-tan^2α)/(1+tan^2α) 积化和差4个 sinαcosβ=[sin(α+β)+sin(α-β)]/2 cosαsinβ=[sin(α+β)-sin(α-β)]/2 cosαcosβ=[cos(α+β)+cos(α-β)]/2 sinαsinβ=-[cos(α+β)-cos(α-β)]/2 和差化积4个 sinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2] sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2] cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2] cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2] 同角关系8个 sin^2α+cos^2α=1 sec^2α=1/cos^2α=tan^2α+1 csc^2α=1/sin^2α=cot^2α+1 tanα=sinα/cosα cotα=cosα/sinα tanαcotα=1 sinαcscα=1 cosαsecα=1

必修三-Unit-5-Canada-The-True-North公开课教学设计

必修三-Unit-5-Canada-The-True-Nor th公开课教学设计

高中必修3 Unit5 Canada:The True North The First Period Reading (I) I. 教学目标 1. Language goals a. Learn the use of the following words and phrases: Minister rather than,continent,surround, harbor, measure, aboard, have a gift for, within, manage to do, catch sight of, eagle, acre, urban settle down eastward b. Important sentences: Learn about some sentence patterns and be able to use them freely. 2. Ability goals 1.) Learn how to describe the places that Li Daiyu and Liu Qian visit in Canada. Understand the noun clause used as appositive. 2.) Enable the students to understand the details of the passage about Canada and find the correct answers to the questions in the post-reading. 3. Learning ability goals Improve the students’ reading ability - searching for information. II. 教学重点 Learn the information about Canada. Master the expressions for describing directions and locations. III. 教学难点 Learn the methods of writing a traveling report about. Learn to read the traveling report according to the traveling route. IV. 教学策略选择与设计 1.Skimming and scanning; 2.Asking-and-answering activities; 3.Listening method. V. 学情分析

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