《最新年高一数学必修4知识点总结优秀名师资料.doc》由会员分享，可在线阅读，更多相关《最新年高一数学必修4知识点总结优秀名师资料.doc（182页珍藏版）》请在三一文库上搜索。
1、2014年高一数学必修4知识点总结第一章 三角函数 正角:按逆时针方向旋转形成的角,1、任意角负角:按顺时针方向旋转形成的角, ,零角:不作任何旋转形成的角,2、角的顶点与原点重合，角的始边与轴的非负半轴重合，终边落在第几象限，则称为第几象限角( ,x,第一象限角的集合为 ,kkk,，,36036090,第二象限角的集合为 ,kkk,，,，,36090360180,第三象限角的集合为,kkk,，,，,360180360270, ,第四象限角的集合为,kkk,，,，,360270360360, ,kk180,终边在轴上的角的集合为 x,，,kk18090,终边在轴上的角的集合为 y,kk90,终
2、边在坐标轴上的角的集合为 , 2,? ,?、? 2,? ,?、? 2,? ,?、? 2,? ,?、? 2,，,kk360,3、与角,终边相同的角的集合为 ,14、长度等于半径长的弧所对的圆心角叫做弧度( ll,5、半径为的圆的圆心角所对弧的长为，则角的弧度数的绝对值是( rr,180,1,2360,6、弧度制与角度制的换算公式:，( ,157.3,180,lCSCrl,，2,为弧度制lr,7、若扇形的圆心角为，半径为，弧长为，周长为，面积为，则，r，1 112( ,Slrr22228、设是一个任意大小的角，的终边上任意一点的坐标是，它与原点的距离是，xy,rrxy,，,0,，yyxy则，( s
3、in,cos,tan0,x，rrx PT9、三角函数在各象限的符号:第一象限全为正，第二象限正弦为正， v 第三象限正切为正，第四象限余弦为正( OMxsin,cos,tan,10、三角函数线:，( A22222211、角三角函数的基本关系:;1sincos1,，,sin1cos,cos1sin,，sin,sin,( 2tan,sintancos,cos，,cos,tan,12、函数的诱导公式: 1sin2sink,，,，cos2cosk,，,，tan2tankk,，,( ，2sinsin,，,，coscos,，,，tantan,，,( ，3sinsin,coscos,tantan,，( ，4
4、sinsin,coscos,tantan,，( ，口诀:函数名称不变，符号看象限( ,，(，( 5sincoscossin6sincoscossin,，,，,，,2222,口诀:正弦与余弦互换，符号看象限( ,yx,，sin,yx,，sin,13、?的图象上所有点向左(右)平移个单位长度，得到函数的图象;再将函数，1yx,，sin,的图象上所有点的横坐标伸长(缩短)到原来的倍(纵坐标不变)，得到函数的图象;再将，,yx,，sin,函数的图象上所有点的纵坐标伸长(缩短)到原来的倍(横坐标不变)，得到函数，yx,，sin,的图象( ，1?数yx,sin的图象上所有点的横坐标伸长(缩短)到原来的倍(
5、纵坐标不变)，得到函数 ,yx,sin,的图象;再将函数yx,sin,的图象上所有点向左(右)平移个单位长度，得到函数,yx,，sin,yx,，sin,的图象;再将函数的图象上所有点的纵坐标伸长(缩短)到原来的倍(横，2 坐标不变)，得到函数的图象( yx,，sin,，14、函数的性质: yx,，,sin0,0,，2,1,?振幅:;?周期:;?频率:;?相位:;?初相:( ,f,x，,2,函数，当时，取得最小值为 ;当时，取得最大值为，则yx,，,sin,xx,yxx,y，1min2max,2,y,ASinx， , A,0 , , 0 , T,11,xxxx,yy,，yy，maxminmaxm
6、in2112222，( ,2,y,ACosx， , A,0 , , 0 , T,15 周期问题 ,，,y,ASinx， , A,0 , , 0 , T,，, ,y,ACosx， , A,0 , , 0 , T,，,2,，y,ASinx， ，b , A,0 , , 0 , b ,0 , T,，2,y,ACosx， ，b , A,0 , , 0 , b,0 , T,y,Atanx， , A,0 , , 0 , T,，,y,Acotx， , A,0 , , 0 , T,，, ,y,Atanx， , A,0 , , 0 , T,，,，y,Acotx， , A,0 , , 0 , T,15、正弦函数、余
7、弦函数和正切函数的图象与性质: 函 yx,cos yx,tan yx,sin数 性 质 图象 ,xxkk, ,，,定义域 RR ,2,1,1,1,1 ,值域 R 3 ,时，时， 当当xk,，2k,xkk,2,，2,xk,，2,;当 ;当 xk,2y,1y,1,最值 既无最大值也无最小值 maxmax2时，( 时，( k,k,y,1y,1，minmin,周期性 2,2, 奇偶性 奇函数 偶函数 奇函数 ,，在 2,2kk,，,22，上是在2,2kkk,，, 在kk,上是增函数;在 ,，k,，,22,增函数;在 2,2kk,，单调性 ,3， 2,2kk，上是增函数( k,，,上是减函数( k,，2
8、2，k,上是减函数( ，对称中心kk,0, ,，, 对称中心k,kk,0，,，,对称中心 ,0k,，2,对称性 ,2,xkk,，,对称轴 ,，2xkk,对称轴 无对称轴 ，第二章 平面向量 16、向量:既有大小，又有方向的量( 数量:只有大小，没有方向的量( 0有向线段的三要素:起点、方向、长度( 零向量:长度为的向量( 1单位向量:长度等于个单位的向量( 平行向量(共线向量):方向相同或相反的非零向量(零向量与任一向量平行( 相等向量:长度相等且方向相同的向量( 17、向量加法运算: ?三角形法则的特点:首尾相连( ?平行四边形法则的特点:共起点( C ,ababab,，,，?三角形不等式:
9、( ,a ,abba，,，?运算性质:?交换律:; b , ,abcabc，,，aaa，,，,00?结合律:;?( ，,abCC, 4 ,?坐标运算:设，则( axy,bxy,abxxyy，,，,，1122121218、向量减法运算: ?三角形法则的特点:共起点，连终点，方向指向被减向量( ,?坐标运算:设，则( axy,bxy,abxxyy,，11221212,设、两点的坐标分别为，则( ,xy,xy,xxyy,，1122121219、向量数乘运算: ,a,a?实数与向量的积是一个向量的运算叫做向量的数乘，记作( ,?; ,aa,0,aa,0,aa,0?当时，的方向与的方向相同;当时，的方向
10、与的方向相反;当时，( ,a,0,?运算律:?;?;?,abab，,，( ,aa,，,，aaa，,?坐标运算:设axy,，则,axyxy,( ，,aa,020、向量共线定理:向量与b共线，当且仅当有唯一一个实数，使( ba,，,abb,0axy,b,0设，其中，则当且仅当时，向量、共线( bxy,xyxy,0，11122122,a21、平面向量基本定理:如果、是同一平面内的两个不共线向量，那么对于这一平面内的任意向量，有ee12,且只有一对实数、，使(不共线的向量、作为这一平面内所有向量的一组基底) ,aee,，,ee12111222,xy,xy,22、分点坐标公式:设点是线段上的一点，、的坐
11、标分别是，当时，,，1122121212xxyy，,1212,1时，就为中点公式。)点的坐标是(当 ,11，,23、平面向量的数量积: ,0ababab,cos0,0,0180,?(零向量与任一向量的数量积为( ，,aaaabab,babab,0bb?性质:设和都是非零向量，则?(?当与同向时，;当与反向,22aaaa,abab,abab,aaa,时，;或(?( ,ababab,abcacbc，,，,abba,?运算律:?;?;?( ，,axy,bxy,?坐标运算:设两个非零向量，则abxxyy,，( ，11221212,22222axy,，axy,，axy,axy,bxy,若abxxyy,，
12、,0，则，或( 设，则( ，112212125 ,，,xxyyab,1212a,a,设、都是非零向量，是与的夹角，则cos,( axy,bb,bxy,，,11222222abxyxy，1122测试题 一、选择题 1(若三点共线，则有( ) ABaCb(2,3),(3,),(4,)ab,，,1023ab,ab,20A( B( C( D( ab,3,50,2,2(设，已知两个向量， ，OP,cos,sin,1，则向量长度的最大值是( ) ，OP,2，sin,2,cos,PP212323A. B. C. D. 2323(下列命题正确的是( ) A(单位向量都相等 B(若与是共线向量，与是共线向量，则
13、与是共线向量( ) abbcac,C(，则ab,0 |a，b|,|a,b|,bD(若与是单位向量，则 ab,1a0000,0ab，,34(已知均为单位向量，它们的夹角为60，那么( ) ab,710134A( B( C( D( ,ab,1,4,ab,25(已知向量abab，满足且则与的夹角为 , A( B( C( D( 64326(若平面向量与向量平行，且，则b,( ) ba,(2,1)|b|,25A( B( C( D(或 (4,2)(,4,2)(6,3)(4,2)(,4,2)二、填空题 ,|1,|2,abcab,，ca,ab1(若，且，则向量与的夹角为 ( ,a,(1,2)b,(2,3)c,
14、(4,1)abcc2(已知向量，若用和表示，则=_。 ,a,1b,20(35)ab，,()mab,60abm3(若,，与的夹角为，若，则的值为 ( ,ABCBCD,，,ABCD24(若菱形的边长为，则_。 ,(2,3)(,4,7)abab5(若=，=，则在上的投影为_。 ,2ab,a,(cos,sin),6(已知向量，向量，则的最大值是 ( b,(3,1)6 7(若，试判断则?ABC的形状_( ABC(1,2),(2,3),(2,5),8(若，则与垂直的单位向量的坐标为_。 aa,(2,2),9(若向量则 。 |1,|2,|2,abab,|ab，,10(平面向量中，已知，且，则向量_。 b,1
15、ab ,5b,a,ba,(4,3)第三章 三角恒等变换 24、两角和与差的正弦、余弦和正切公式: ?;?; coscoscossinsin,，coscoscossinsin,，,，?;?; sinsincoscossin,sinsincoscossin,，,，tantan,tan,? (); tantantan1tantan,，,，1tantan，,tantan,，tan，,? (tantantan1tantan,，,，,)( ,，1tantan,25、二倍角的正弦、余弦和正切公式: 222sin22sincos,?( ,1,sin2,sin,，cos,2sin,cos,(sin,cos,)2
16、222cos2cossin2cos112sin,? ,221cos2cos,1cos2sin，,升幂公式 ,22cos21,，1cos2,22cos,sin,降幂公式，( ,22:万能公式2tan,tan2, ?( 221tan,2tan1tan,半角公式:,22sin ;cos ,26、 1，cos1,cos22cos,;sin,1tan1tan ，222222 1,cossin1,cos tan,21，cos1，cossin (后两个不用判断符号，更加好用) ,27、合一变形把两个三角函数的和或差化为“一个三角函数，一个角，一次方”的 y,Asin(,x，,)，B,22,，,，,，sinc
17、ossin,tan,形式。，其中( ，,28、三角变换是运算化简的过程中运用较多的变换，提高三角变换能力，要学会创设条件，灵活运用三角公式，掌握运算，化简的方法和技能(常用的数学思想方法技巧如下: (1)角的变换:在三角化简，求值，证明中，表达式中往往出现较多的相异角，可根据角与角之间的和差，倍7 半，互补，互余的关系，运用角的变换，沟通条件与结论中角的差异，使问题获解，对角的变形如: ,2,4,2,?是的二倍;是的二倍;是的二倍;是的二倍; ,224o30,ooooo?;问: ; ; 1545306045sin,cos,12122,?;?;?;等等 ，,(,),2,(，,)，,(,),(，,
18、),(,),(,，,),42444(2)函数名称变换:三角变形中，常常需要变函数名称为同名函数。如在三角函数中正余弦是基础，通常化切为弦，变异名为同名。 (3)常数代换:在三角函数运算，求值，证明中，有时需要将常数转化为三角函数值，例如常数“1”的代换变形有: 22oo 1,sin,，cos,tan,cot,sin90,tan45(4)幂的变换:降幂是三角变换时常用方法，对次数较高的三角函数式，一般采用降幂处理的方法。常用降幂1，cos,公式有: ; 。降幂并非绝对，有时需要升幂，如对无理式常用升幂化为有理式，常用升幂公式有: ; ; (5)公式变形:三角公式是变换的依据，应熟练掌握三角公式的
19、顺用，逆用及变形应用。 ,1，tan1,tan,_,_ 如:; ; 1,tan,1，tan,; tan,，tan,_1,tan,tan,_; tan,tan,_1，tan,tan,_22tan,1,tan, ; ; ooootan20，tan40，3tan20tan40, ; sin,，cos, = ; asin,，bcos,tan, = ;(其中 ;) 1，cos,1,cos, ; ; (6)三角函数式的化简运算通常从:“角、名、形、幂”四方面入手; 基本规则是:见切化弦，异角化同角，复角化单角，异名化同名，高次化低次，无理化有理，特殊值与特殊角的三角函数互化。 oo如: ; sin50(1
20、，3tan10),tan,cot, 。 易错点提示: 8 1. 在解三角问题时，你注意到正切函数、余切函数的定义域了吗,你注意到正弦函数、余弦函数的有界性了吗, 2. 在三角中，你知道1等于什么吗,( 这些统称为1的代换) 常数 “1”的种种代换有着广泛的应用( 3. 你还记得三角化简的通性通法吗,(切割化弦、降幂公式、用三角公式转化出现特殊角. 异角化同角，异名化同名，高次化低次) 4. 你还记得在弧度制下弧长公式和扇形面积公式吗,() ,sintanxxx,5(常见三角不等式:(1)若x,(0,)，则. 2,(2) 若x,(0,)，则. (3) . 1sincos2,，,xx|sin|co
21、s|1xx，,2测试题 一、选择题 1(下列转化结果错误的是 ( ) 103,A( 6730化成弧度是rad B. 化成度是-600度 837,150C(化成弧度是rad D. 化成度是15度 612,2(已知是第二象限角，那么是 ( ) ,2A(第一象限角 B. 第二象限角 C. 第二或第四象限角 D(第一或第三象限角 21,sin,3(已知，则化简的结果为 ( ) sin,0,tan,0cos,cos,cos,A( B. C( D. 以上都不对 ,y,cos(2x，)4(函数的图象的一条对称轴方程是 ( ) 2,x,x,x,A( B. C. D. x, 248,3x,(,0)sinx,5(
22、已知，,则tan2x= ( ) 52772424,A( B. C. D. 24247711,tan(，),tan(,),tan(，),6(已知，则的值为 ( ) 24349 2A( B. 1 C. D. 2 22cosx，sinxf(x),7(函数的最小正周期为 ( ) cosx,sinx,2,A(1 B. C. D. ,2x,8(函数的单调递增区间是 ( ) y,cos(,)234242，A( B. 2k,2k,，,(k,Z)4k,4k,，,(k,Z),3333，2828，C( D. 2k,，,2k,，,(k,Z)4k,，,4k,，,(k,Z),3333，,x,9(函数，的最大值为 ( )
23、y,3sinx，cosx2233A(1 B. 2 C. D. 210(若均为锐角，且，则的大小关系为 ( ) ,与,、,2sin,sin(,，,)A( B. C. D. 不确定 ,二、填空题 11、函数的最大值是3，则它的最小值_ yax,，sin1,abab，,ab12、若，则、的关系是_ 13、若函数f()是偶函数，且当,0时，有f()=cos3+sin2，则当,0时，f()的表达式为 . ,y,sin(2x，)14(把函数先向右平移个单位，然后向下平移2个单位后所得的函数解析式为32_ ,2tan(，),21，3sin,cos,2cos,15(已知，则=_ ,410 (英文版 ) eas
24、ily blame, to prevent the broken window effect. Supervise the leading cadres to play an exemplary role, take the lead in the strict implementation of the and , lead to safeguard the solemnity and authority of the party discipline, ensure that the party discipline and the laws and regulations for imp
25、lementation in place. Throughout the discipline in the daily supervision and management , strengthen supervision and inspection, from the thorough investigation of violations of discipline behavior. Strengthen to key areas, key departments and key projects as well as the masses reflect the concentra
26、tion of the units and departments for supervision. - strengthening supervision, discipline inspection and supervision of cadres to set an example for compliance with the and is a man must be hexyl, blacksmith needs its own hardware. Discipline inspection organs as the executor of the party disciplin
27、e, and supervisor of the defenders, for its supervision must be more strictly, discipline inspection and supervision of cadres to firmly establish the awareness of Party Constitution, sense of discipline and rules consciousness, politics loyalty, sense obey. Action speak Ji Ordinance to set an examp
28、le of the regulations of the rule of law, strengthen supervision and accept the supervision of the firmness and consciousness, do comply with and . To firmly establish the discipline must first be disciplined, the supervisor will be subject to the 11 supervision of concept, and consciously safeguard
29、 and implement party compasses party, take the lead in practicing three strict real strict, so loyal, clean, play. To be good at learning, the Constitution and the as morality, politics and brought to fruition; to implement , do not want to, dare not, not with disciplinary ruler to supervision; to d
30、iscipline a ruler, often the control inspection, and consciously in the ideological red line to draw the row Ming Good accumulation is indeed the bottom line, so that the heart has fear, said to have quit, the line has ended. Attached: indifferent to heart, calmly to the table in our life, there are
31、 many unpredictable things will happen, some good, some bad things, we cannot control is powerless to stop, but with time, you will find in life sometimes turns out to be not good, some bad things finally turned out to be a good thing, but then we muddy however did not know, this is the life teach u
32、s things. 1, life can be complex, can also be simple. Want simple life of precipitation, to have enough time to reflect, to make Become more perfect. Life is the most important thing is not to win, but the struggle; not to have conquered, but to have fought well. 2, the plain is the background of li
33、fe. Live a plain life, give up on themselves is not a coward, but the wise answers; not disillusioned after the heart, such as ashes, but experience the storm after the enlightenment; not unrewarding perfunctorily, but calm attitude of life of unrestrained self-confidence. Plain living, there is no
34、noise noisy, no earthly troubles, more did not fill in the discontent of desire, 12 some just a calm, a calm. 3, memory of heart will not good things to erase the, life is a When no movie, pain is a beginning, the struggle is a kind of process, death is a kind of ending. Give up this giving up is th
35、e helpless, do not give up the abandoned, do not give up this giving up is ignorance, do not give up should not give up is persistent. 4, a thing figured is heaven, think impassability is hell. Since the living, to live better. Sometimes we because of too narrow-minded, too care around the chores an
36、d penny wise and pound foolish, not worth the candle. Some things to attract trouble and worry, completely depends on how we look at and deal with it. Dont always take everything back to things, and dont get into a blind alley, dont want to face, dont be narrow-minded. Poke to care, is a kind of ope
37、n-minded, a free and easy. 5, I am not afraid of others behind me a knife, I afraid to look back and see stab me, is my intention to treat people; I am not afraid of the truth to tell the best friend, Im afraid he turned to it as a joke to tell dont 6, when we are in a positive frame of mind, you wi
38、ll find many good things; and when we are in a negative state of mind, you will find many depressed things; life happy and worry, all is you of life attitude, optimistic, good luck; loss of sink, Eritrea company. When you are in adversity, may wish to change a point of view to think everything over
39、to the good Think, because good mentality decided the fate of the! 7, people are tired, rest; heart tired, calm. Grow up, mature, this society read. Tired and sad, squat down, to their a hug. Because the world no one can 13 sympathize with you, have mercy on you. You cry, tears is your own; you pain
40、, no one can understand. Then you only tears to smile. 8, each people have youth, Each youth are a story, the life of the world never gets easier, I want what, wish the world all know, as has been the same; now want anything, for fear that others know, or like to lose the same. 9, the heart move, ev
41、erything in the world is followed by birth, Rangrang, important thing is often the most difficult to open ones mouth, because words will reduce its importance; to let strangers people care about your life in the good things, the original is not easy 10, do not blame, do not laugh at who, also dont e
42、nvy who. Like a person is a kind of feeling, not like a person is true. The truth is easy to explain, I feel Is unspeakable. The best travel life is that you in a strange place found a long lost touched. 11, happy life not in the bustling in, and in the peace of mind; no matter how many grievances,
43、how uncomfortable, and ultimately to heal themselves or their own, others may got you to comfort, but never know your heart is how wanjianchuanxin. 12, maam, like a movie, learn to appreciate, learn to be grateful, learn tolerance, and goodness, helping others. Instead of accusing the society, as into one; and an exception is better to give than to what 13, dont envy him A sum of, dont lose your life and the life, respectively is: the former is a we experienced cannot escape in a day finally will last minute, w