高中数学高考三角函数重点题型解析及常见试题、答案

高中数学高考三角 函数 excel方差函数excelsd函数已知函数     2 f x m x mx m      2 1 4 2拉格朗日函数pdf函数公式下载 重点题型解析及常见试题、答案 三角函数的主要考点是:三角函数的概念和性质(单调性，周期性，奇偶性，最值)，三角函数的图象，三角恒等变换(主要是求值)，三角函数模型的应用，正余弦 定理 三点共线定理勾股定理的证明证明勾股定理共线定理面面垂直的性质定理 及其应用，平面向量的基本问题及其应用( 题型1 三角函数的最值:最值是三角函数最为重要的内容之一，其主要 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 是利用正余弦函数的有界性，通过三角换元或者是其它的三角恒等变换转化问题( 例1 若是三角形的最小内角，则函数的最大值是( ) xyxxxx,，，sincossincos 11A(,1 B( C(,，2 D(，2 222 ,2sincos12sincosxxxx，,，分析:三角形的最小内角是不大于的，而，换元解，，3 决( ,,,,70,,xtxxx,，,，sincos2sin(),,，,解析:由，令而，得x,344412 12,,t( 2t,12txx,，12sincos，得， 又sincosxx,2 22t,11(2)11,2得ytt,，,，,(1)1，有(选择答案D( 1022，,,，,，y2222 sincosxx,sincosxx点评:涉及到与的问题时，通常用换元解决( ,1,,解法二:， ,，，,，，yxxxxxxsincossincos2sinsin2,,42,, ,1,xy,，2当时，，选D。 max42 ,2ff(0)8,()12,,例2(已知函数(，且( fxaxxbx()2sincos2cos,，6 b (1)求实数a，的值;(2)求函数的最大值及取得最大值时x的值( f(x) ba分析:待定系数求，;然后用倍角 公式 小学单位换算公式大全免费下载公式下载行测公式大全下载excel公式下载逻辑回归公式下载 和降幂公式转化问题( 解析:函数f(x)可化为fxaxbxb()sin2cos2,，，( ,,33f()12, ,，, f(0)8,fb(0)28,, (1)由，可得，，所以fab()126622 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should b,4，( a,43 , (2)， fxxxx()43sin24cos248sin(2)4,，，,，，6 ,,,故当即时，函数取得最大值为( 22xk，,，xkkZ,，,()12fx,,，，662 22ababsincossin,,,,，,，，点评:结论是三角函数中的一个重要公式，它在，， 解决三角函数的图象、单调性、最值、周期以及化简求值恒等式的证明中有着广泛应用，是实现转化的工具，是联系三角函数问题间的一条纽带，是三角函数部分高考命题的重点内容( 题型2 三角函数的图象:三角函数图象从“形”上反应了三角函数的性质，一直是高考所重点考查的问题之一( π,,例3((2009年福建省理科数学高考样卷第8题)为得到函数的图象，yx,，cos2,,3,,只需将函数的图象 yx,sin2 5π5πA(向左平移个长度单位 B(向右平移个长度单位 1212 5π5πC(向左平移个长度单位 D(向右平移个长度单位 66 分析:先统一函数名称，在根据平移的法则解决( π,,,,55,,,,,,,,解析:函数，,，,，，,，,，yxxxxcos2sin2sin2sin2,,,,,,,,332612,,,,,,,, 5π故要将函数的图象向左平移个长度单位，选择答案A( yx,sin212 yyyy 3,,3,,222,,2xxoo2-2-::--,2,2:: ,x3,x,oo3,,,2222 ABDC ,,3(,)yxxxx,，,,tansintansin例4 (2008高考江西文10)函数在区间内的22 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 图象是 分析:分段去绝对值后，结合选择支分析判断( 2tan,tansinxxx当时,,解析:函数(结合选择支yxxxx,，,,,tansintansin,2sin,tansinxxx当时,, 和一些特殊点，选择答案D( 点评:本题综合考察三角函数的图象和性质，当不注意正切函数的定义域或是函数分段不准确时，就会解错这个题目( 题型3 用三角恒等变换求值:其主要方法是通过和与差的，二倍角的三角变换公式解决( π47π,,,,例5 (2008高考山东卷理5)已知，则的值,,,,，,cossin3sin，,,,,656,,,, 是 442323,A( B( C( D( ,5555 7π,,,，将已知条件分拆整合后解决( 分析:所求的，,，sinsin(),,,,66,, 解析: ,,4333434,,,,C (，,，,,，,,，,cossinsincossin,,,,,,,,,6522565,,,, 74,,,,,,所以( ，,,，,,sinsin,,,,,,665,,,, 点评:本题考查两角和与差的正余弦、诱导公式等三角函数的知识，考查分拆与整合的 π4,,数 学思想和运算能力(解题的关键是对的分拆与整合( ,,,，,cossin3,,65,, tan,例6(2008高考浙江理8)若则= cos2sin5,,,，,, 11,22,A( B( C( D( 22 分析:可以结合已知和求解多方位地寻找解题的思路( 112,,tan5sin5,,，,,,,,,方法一:，其中sin,cos，即， ，，255 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firm ly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 3 ,,再由知道，所以， ，,,,2kkZ,,,2ksin1,,，,,,,,,,,，，，，22 ,,,sin,,,,,cos,,,2,,,,,,所以( tantan2tan2k,,,,,,,,,,,,,,,,,22sin,,,,,,,,cos,,,,,2,, 方法二:将已知式两端平方得 2222cos4cossin4sin55sincos,,,,,,，，,,，，， 22 ,,，,sin4sincos4cos0,,,, 2,,，,,,tan4tan40tan2,,, 2sin2cos,,,,tt,0方法三:令，和已知式平方相加得，故， 55,，t sin2cos0,,,,tan2,,即，故( 方法四:我们可以认为点Mcos,sin,,在直线上， xy，,,25，， ,5x,,,,522M而点又在单位圆上，解方程组可得， xy，,1,25,y,,,5, ,ycos2sin5,,，,,,tan2,,,从而(这个解法和用方程组求解实质上是一致的( ,22xsincos1，,,,,, 方法五:只能是第三象限角，排除C(D(，这时直接从选择支入手验证， , 1tan2,,由于计算麻烦，我们假定，不难由同角三角函数关系求出2 255,,,,,,，检验符合已知条件，故选B( sin,cos55 点评:本题考查利用三角恒等变换求值的能力，试题的根源是考生所常见的“已知 1,,,,，,,sincos,0,，求tan,的值(人教A版必修4第三章复习题B组最后，，5 一题第一问)”之类的题目 ，背景是熟悉的，但要解决这个问题还需要考生具有相当的知识迁移能力( 题型4 正余弦定理的实际应用:这类问题通常是有实际背景的应用问题，主要表现在航海和测量上，解决的主要方法是利用正余弦定理建立数学模型( 7E例7((2008高考湖南理19)在一个特定时段内，以点为中心的海里以内海域被设 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 55为警戒水域(点正北海里处有一个雷达观测站(某时刻测得一艘匀速直线行驶EA ,40的船只位于点北偏东且与点相距海里的位置，经过分钟又测得该AAB45402 26,,,船已行驶到点北偏东 (其中，)且与点相距AA45，,090,,,,,sin26 C海里的位置( 1013 (1)求该船的行驶速度(单位:海里/小时); (2)若该船不改变航行方向继续行驶(判断它是否会进入警戒水域，并说明理由( BC,ABC分析:根据方位角画出图形，如图(第一问实际上就是求的长，在中用余弦 BCE定理即可解决;第二问本质上求是求点到直线的距离，即可以用平面解析几何的 方法，也可以通过解三角形解决( 26AC,1013AB,4022解析:(1)如图， ， ， ，,,,,BAC,sin.26 26526,,2090,,,由于，所以 ,,,,cos1().2626 22BCABACABAC,，,,2cos105. ,由余弦定理得 105,155所以船的行驶速度为(海里/小时)( 2 3 A(2)方法一 : 如上面的图所示，以为原点建立平面直角坐标系， BCDBxyCxy,,,设点的坐标分别是，与x轴的交点为( BC,，，，，1122 2由题设有， xyAB,,,40， 112 ,xACCAD,，,,,cos1013cos(45)30,， 2 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firm ly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 5 , yACCAD,，,,,sin1013sin(45)20.,2 20ll所以过点的直线的斜率，直线的方程为( k,,2yx,,240BC,10 |05540|，,l又点到直线的距离，所以船会进入警戒水域( E0,55,d,,,357，， 14， BC,ABCAE解法二: 如图所示，设直线与的延长线相交于点(在中，由余弦定Q 理得， 222222ABBCAC，,4021051013，，，,，310cos，,ABC==( 2ABBC,102402105，， 9102sin1cos1.，,,，,,,ABCABC从而 1010 10402，ABABCsin，10AQ,,,40在中，由正弦定理得，( ,ABQ,sin(45),，ABC2210，210 AE由于AEAQ,,,5540，所以点Q位于点和点之间，且EQAEAQ,,,15( EPBC,BCEPEPE过点作于点，则为点到直线的距离( 在Rt,QPE中， 5, PEQEPQEQEAQCQEABC,，,,，,,,，,，,,sinsinsin(45)15357.5clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 所以船会进入警戒水域( 点评:本题以教材上所常用的航海问题为背景，考查利用正余弦定理解决实际问题的能力，解决问题的关键是根据坐标方位画出正确的解题图( 本题容易出现两个方面的错误，一是对方位角的认识模糊，画图错误;二是由于运算相对繁琐，在运算上出错( 题型5 三角函数与平面向量的结合:三角函数与平面向量的关系最为密切，这二者的结合有的是利用平面向量去解决三角函数问题，有的是利用三角函数去解决平面向量问题，更多的时候是平面向量只起衬托作用，三角函数的基本问题才是考查的重点( 例8(2009年杭州市第一次高考科目教学质量检测理科第18题)已知向量 ,,0，()，令，且的周期为( ,f(x)f(x),a,ba,(2cos,x,cos2,x),b,(sin,x,1) ,,,,,[,,]fx(1) 求的值;(2)写出在上的单调递增区间( f，，,,224,, fx分析:根据平面向量数量积的计算公式将函数的解析式求出来，再根据的周f(x)，， 期为,就可以具体确定这个函数的解析式，下面只要根据三角函数的有关知识解决即可( 解析:(1) ,,sin2,x，cos2,x,2sin(2x，)， ,f(x),a,b,2cos,xsin,x，cos2,x4 ,,,1f(x),2sin(2x，),?f(x)的周期为( ?， ，4 ,,,？f(),sin，cos,1 ( 422 ,f(x),2sin(2x，) (2) 由于， 4 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firm ly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 7 ,,,k,Z当()时，单增， ,，2k,,2x，,，2k,fx，，242 ,,3,,k,Z即()，? ,，k,,x,，k,[,,]x,2288 ,,3,,?在上的单调递增区间为( [,,][,,]fx，，2288 点评:本题以平面向量的数量积的坐标运算为入口，但本质上是考查的三角函数的性质，这是近年来高考命题的一个热点( 例9 (2009江苏泰州期末15题) ,,3,,,已知向量，，，且a,3sin,cos,,b,,2sin,5sin4cos,,,,,2,,，，，，,,2,,,, ( ab, tan,(1)求的值; ,,,,(2)求的值( cos，,,23,, 分析:根据两个平面向量垂直的条件将问题转化为一个三角函数的等式，通过这个等式探究第一问的答案，第一问解决后，借助于这个结果解决第二问( ,,,,,, 解析:(1)?ab,ab,,0，?(而，， a,n,3sicos,,b,,2sin,5sin4cos,,,，，，， ,,22cos0,,ab,,，,,6sin5sincos4cos0,,,,故，由于， 26tan5tan40,,，,,?， 3π41,,,tan0,,,， 2π解得，或(?，， ,,,,,tantan,,2,,23 14故(舍去)(?( ,,,,,tantan23 3π,3π,,,,， 2π(2)?，?( ,(，)π,,2,,24 4,1,由，求得，(舍去)( ,,,,,tan2,tantan2322 ,,525,,,，sincos?， 2525 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should ,,,,ππ251532515，,, ( ,,，,，,cos,,，,coscossinsin,,105252232323,, 点评:本题以向量的垂直为依托，实质上考查的是三角恒等变换(在解题要注意角的范围对解题结果的影响( 题型6 三角形中的三角恒等变换:这是一类重要的恒等变换，其中心点是三角形的内角和是，有的时候还可以和正余弦定理相结合，利用这两个定理实现边与角的互化，然, 后在利用三角变换的公式进行恒等变换，是近年来高考的一个热点题型( 例10((安徽省皖南八校2009届高三第二次联考理科数学17题)三角形的三内角，A ,,,,,, CbB，所对边的长分别为，，，设向量，若， mn//acmcabanabc,,,,，(,),(,) (1)求角B的大小; sinsinAC，(2)求的取值范围( 分析:根据两个平面向量平行的条件将向量的平行关系转化为三角形边的关系，结合余弦定理解决第一问，第一问解决后，第二问中的角就不是独立关系了，可以用其中AC, 的一个表达另一个，就把所要解决的问题归结为一个角的三角函数问题( ,,, 解析:(1)， ？mnccabaab//,()()()？,,,， 222acb，,1,222？,,,？,cacba,1,,cos,BB( 由余弦定理，得( 23ac 2,？，，,？，,ABCAC,(2)， ,3 222,,,？，,，,,，,sinsinsinsin()sinsincoscossinACAAAAA 333 33,,，,， sincos3sin()AAA226 25,,,,？,,？,，,0,AA 3666 13,？,，,？,，, sin()1,sinsin3AAC262 点评:本题从平面向量的平行关系入手，实质考查的是余弦定理和三角形中的三角恒等变换，解决三角形中的三角恒等变换要注意三角形内角和定理和角的范围对结果的影响( clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firm ly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 9 题型7 用平面向量解决平面图形中的问题:由于平面向量既有数的特征(能进行类似数的运算)又具有形的特征，因此利用平面向量去解决平面图形中的问题就是必然的了，这在近年的高考中经常出现(考试大纲明确指出用会用平面向量解决平面几何问题( G,ABOOAOB例11. 如图，已知点 是的重心，点在上，点在上，且过PQPQ 11,ABOGOPmOA, 的重心，，，试证明为常数，并求出这个常，OQnOB,mn 数( 分析:根据两向量共线的充要条件和平面向量基本定理，把题目中需要的向量用基向量表达出来，本题的本质是点共线，利用这个关系寻找所满足的方程( mn,PGQ,, ,,,,,,,,,,,,,,,,,,,, ABMOAa,OBb,OPma,解析:令，，则，，设的中点为， 显然OQnb, ,,,,,,,,,,,,,,,,,,121PG,ABCOMab,，().OGOMab,,,，()，因为是的重心，所以(由、233 ,,,,,,,,,,,,,,,, G,PG、三点共线，有、共线，所以，有且只有一个实数，使 ，QPGGQ,,GQ ,,,,,,,,,,,,,,,,,111PGOGOPabmamab,,,，,,,，()()而， 333 ,,,,,,,,,,,,,,,,,111GQOQOGnbabanb,,,,，,,，,()()， 333 ,,,,1111,，,,，,,()[()]mabanb所以( 3333 11,,,m,,,,,33,ab又因为、不共线，由平面向量基本定理得，消去， ,11,,,(n,),33, 113mnmn,，3，,3整理得，故(结论得证(这个常数是( mn 【点评】平面向量是高中数学的重要工具，它有着广泛的应用，用它解决平面几何问题clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 是一个重要方面，其基本思路是根据采用基向量或坐标把所要解决的有关的问题表达出来，再根据平面向量的有关知识加以处理(课标区已把几何证明选讲列入选考范围，应引起同学们的注意( 题型8 用导数研究三角函数问题:导数是我们在中学里引进的一个研究函数的重要工具，利用导数探讨三角函数问题有它极大的优越性，特别是单调性和最值( ,,22例12. 已知函数，若函数在区间上(,]fx()fxxtxxx()cos2sincossin,，,126是增函数，求实数的取值范围( t ,,分析:函数的导数在大于等于零恒成立( (,]fx，，126 ,,,,,解析:函数在区间(,]上是增函数，则等价于不等式在区间(,]fx()fx()0,126126 ,,,上恒成立，即在区间(,]上恒成立， 从而fxxtx()2sin22cos20,,，,126 ,,,,tx,tan2(,](,]在区间上恒成立， 而函数在区间上为增函数，yx,tan2126126 ,,,(,]y,，,tan(2)3t,3所以函数在区间上的最大值为，所以yx,tan2max1266 为所求( 点评:用导数研究函数问题是导数的重要应用之一，是解决高中数学问题的一种重要的 2fxtxxtx,，,，，sin2cos21sin(2),思想意识(本题如将化为的形式，fx()，， 则,与有关，讨论起来极不方便，而借助于导数问题就很容易解决( t 题型9 三角函数性质的综合应用:将三角函数和其它的 知识点 高中化学知识点免费下载体育概论知识点下载名人传知识点免费下载线性代数知识点汇总下载高中化学知识点免费下载 相结合而产生一些综合性的试题，解决这类问题往往要综合运用我们的数学知识和数学思想，全方位的多方向进行思考( 2例13. 设二次函数，已知不论，为何实数，恒有,,fxxbxcbcR()(,),，，, 和( f(sin)0,,f(2cos)0，,, bc，,,1(1)求证: ; c,3(2)求证:; 8bc(3)若函数f(sin),的最大值为，求，的值( f10,分析:由三角函数的有界性可以得出，再结合有界性探求( ，， ,,,1sin1,f(sin)0,,f(1)0,解析:(1)因为且恒成立，所以，又因为 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firm ly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 11 且恒成立，所以， 从而知，f(2cos)0，,,f(1)0,12cos3,，,,f(1)0, 10，，,bcbc，,,1，即( (2)由且恒成立得， 即 ，930，，,bcf(2cos)0，,,f(3)0,12cos3,，,, bc,,,19330,,，,ccc,3将代如得，即( 11，，cc222(3)， ,,,,fccc(sin)sin(1)sin(sin)(),，,,，,,，,22 18,，,bc,1，csin1,,,因为,2，所以当时， 由 ， 解得 [(sin)]8f,,,max210，，,bc, b,,4c,3，( f(sin)0,,,bc，,,1点评:本题的关键是，由 利用正余弦函数的有界性得出,f(2cos)0，,,, f10,,，，,，从而，使问题解决，这里正余弦函数的有界性在起了重要作用( f(1)0,,f10,，，,, 【专题训练与高考预测】 一、选择题 22(若1，且1cos1sinsincos,，,,,,,,,，则的取值范围是( ) ,,,,[0,2) ,,3,3,[0,][,][,][,2) A( B( C( D( ,,,2222 1,lgn2(设是锐角，且，，则 ( ) ,lg(1cos),,,mlgsin,,,，1cos 1111mn,()m,(),n A(mn, B( C( D( 22n2m ,,,,,,。0030ab,,ab3(若，与的夹角为，则 ( ) ||2sin15,||4cos15ab,, 13323 A( B( C( D( 22 ,,,,,,,,,,,,,,,,,,,, O,ABC,ABC4(若为的内心，且满足，则的形状为 ()(2)0OBOCOBOCOA,,，,, ( ) A(等腰三角形 B(正三角形 C(直角三角形 D(钝角三角形 abc,ABC,ABC,,5(在中，若，则是 ( ) cosAcosBcosC A(直角三角形 B(等边三角形 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should C(钝角三角形 D(等腰直角三角形 ,,,,,, OAOB6(已知向量、、，则直线与直线 OB,(2，0)OC,(2，2)CA,(2cos,，2sin,) 的夹角的取值范围是 ( ) 55,5,,,,,, A( B( C( D( [，][，][，][0，]12124121224二、填空题 66227(的化简结果是__________( sincos3sincosxxxx，， ,,,,,,,, ,8(若向量与的夹角为，则称为它们的向量积，其长度为，abab，||||||sinabab，,,, ,,,,,, 已知，，且，则_______________( ab,,,4||ab，,||1a,||5b, S15:S209( 一货轮航行到某处，测得灯塔在货轮的北偏东，与灯塔相距海里，随后货轮 30:30按北偏西的方向航行分钟后，又得灯塔在货轮的东北方向，则货轮的速度为每小 时 海里( 三、解答题 ,2sin2()4cos,，,,12,,，,,10( 已知:，( tan()tan()，,,,2310cossin2,,, (1)求的值; tan(),,， (2)求的值( tan, ,,,,2xR,11( 已知函数 ( fxxx3sin22sin(),,，,，，，，,,612,, fx (1)求函数的最小正周期; ，， fx (2)求使函数取得最大值的x的集合( ，， ,,,,2512(已知向量， ， ( ab,,a,(cos,sin),,b,(cos,sin),,5 (1)求cos(),,,的值; ,,5sin,0,,,,,0,,,sin (2)若,， ,， 且， 求( 2213 【参考答案】 sin0,,cos0,,1(解析:B由已知可得，且，故得正确选项B( clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firm ly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 13 22(解析:C 与相加得，lg(1cos)，,,,nlg(1cos),,,mlg(1cos),,,,mn?，故选C( 2lgsin,,,mn ,,。。。3(解析:B ，选B( ab,,,,4sin30cos302sin603 ,,,,,,,,,,,, ，BAC4(解析:A已知即，即边BC与顶角的平分线互相垂直，这表CBABAC,，,()0 ,ABC明是一个以AB、AC为两腰的等腰三角形( sincosAA,sincosBB,sincosCC,5(解析:B依题意，由正弦定理得，且，，故得( ,,226(解析:A由为定值，?A点的轨迹方程为，由图形易知|CA|,2(x,2)，(y,2),2所求角的最大、最小值分别是该圆的切线与轴的夹角，故得( x 2234224227( 解析:1 原式,1( ,，,,，(sincos)3sincos3sincos3sincosxxxxxxxx ,,4333,,,sin,,8(解析: 由夹角公式得，?，?( ||153ab，,，，,cos555 解析:设轮速度为海里/小时，作出示意图，由正弦定理得9( x20(62), 1x202，解得( x,,20(62),sin30sin105:: 11,,，,,,,,10(解析:(1)?tan() ?tan， 33 22sin(2)4cos,,,,，sin24cos,,，tan()，,?, ,,2210cossin2,10cossin2,,,,, 22sincos4cos,,,，sin2costan2,,,，，2cos(sin2cos),,,，,,, ,25cossin5tan,,10cos2sincos,2cos(5cossin),,,,,,,,,, 1,，253,,? ( ，,,tan()116，53 51，31tan()tan,,,，,163,tan,, (2)?tantan[()],,,,,，, ，?( ,51431tan()tan，，,,,,，1163 clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firmly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should ,,,,11(解析:(1)因为 fxxx3sin(2)1cos2,,，,,，，,,612,,，，31,,,,,,,,,,，2sin2cos21xx,,,,,,2626,,,,，， ，，,,,, ,,,，2sin21x,,,,66,,，， ,,,,,，2sin21x,,3,, 2,所以的最小正周期,,( fxT,，，2 ,,,,,22xk,,， (2)当fx取最大值时，，此时 kZ,，即 ,sin21x,,，，，，,,323,, 5,,,5,,，xk kZ,，所以所求的集合为 kZ,( x,xxk,，,，，，，,,1212,, ,, 12(解析:(1)， ， ？a,(cos,sin),,b,(cos,sin),,,, ( ？,,,,abcoscossinsin,,,,，，， ,,252522， ？,，,,,,,,， ？ab,,coscossinsin，，，，55 43,,,,,？,,,,cos22cos即 ， ( ，，，，55 ,,？0,0,0,,,,,？,,, (2)， ,,,,,22 34？,,,,？,,,,sin.cos， ，，，，55 512？,,,？,,sincos， ， 1313 ？,,，,,，,sinsinsincoscossin,,,,,,,,,,，，，，，，，，，， 4123533,,( ,,，,,,,,51351365,, clearly, for maintenance management. 4. install (1) all sorts of switches, sockets and other equipment should be installed firm ly, position accurate, when installing rocker panel, switches, switch should be consistent, generally up to "close" down to "off" switch should cut off the line. Switches the elevation and location of the socket should conform to the drawings, the same interior height difference should not be greater than 5mm. (2) liquid level automatic control device used in a variety of angle brackets, there should be galvanized. (3) fixed lamp hanging boxes are plastic with rotary table round table. (4) light bulb capacity of 100W or less, available resin lamp, 100W and the tide sealed lamps, porcelain lamps. (5) the wiring inside the lamp shall not be less than 0.4mm2 wires. (6) the chandelier with a weight of more than 3kg, the embedded hook-bolts, install lighting in the brick or concrete structures, should use expansion bolts or plastic tube, ban the use of wooden wedges instead. (7) the ceiling mounted light fixtures, border of the shade should cover Panel joints and snapped the ceiling panel, determine the size lamp open position, strictly according to the drawings. (8) the lamps arranged orderly, symmetrical when installing the vertical and horizontal center lines should be on the same line, deviation is less than 5mm. (9) the emergency lighting installation: according to the design and specifications to determine location of evacuation lighting, signs should 15