2015年高考理科数学试题汇编(含答案)三角函数大题

2015年高考理科数学试题汇编(含答案)：三角函数大题
(江苏)15.(本小题满分14分) ,在中，已
知. ,ABCAB,2,AC,3,A,60 (1)求的长; BC (2)求的值. sin2C 43【答案】(1)(2) 77 【解析】考点:余弦定理，二倍角公式fxx,,，sin,,,(10)(安徽)已知函数(，，均为正的常数)的最小正,,，，，，2,fx,,周期为，当x时，函数取得最小值，则下列结论正确的是( ) ，，3 fff220,,,fff022,,, (A) (B) ，，，，，，，，，，，，fff,,,202fff202,,, (C)
(D) ，，，，，，，，，，，，【答案】A of the audit. Fire is a combination of auditing in the field of law enforcement law enforcement job rotation and fire practice of law enforcement, in the promotion changed, retraining retired key timing synchronization in place audit mechanism, and conducive to self-urged self-restraint, the fire law enforcement, to protect themselves. 1.3 preventive fire-fighting urged the current fire law enforcement corruption involving cases of violating law enforcement has an upward trend, judging from national reports of complaints in recent years, reported cases reflect the fire law enforcement is over 50%. In addition, the multiple corruption cases, with a high incidence of leading cadres, a small number of leading cadres ' corruption, corrupt, power-for-money
transaction, especially teams, group reporting increasing complaints involving the supervision of law enforcement jobs, bring great pressure to force honest and serious challenges. Meanwhile, team and Brigade levels bear an administrative license approval, administrative punishment, routine inspections and more than 95% of the amount, some fire staff handled a year go through legal instruments and thousands of contact hundreds of social units, once inside lax, their quality is not high, extremely easy to induce discipline and corruption. In practical operation, the fire law enforcement as 'athletes' and
'umpire' internal inspector, inspection, test and not enough problem touches not found to leading policymakers, deviate from system design, system implementation has to deal with. 2 building fire law 1/6页考点:1.三角函数的图象与应用;2.函数值的大小比较. (福建)19(已知函数的图像是由函数的图像经如下变换得到:先将f()xgxx()cos= 图像上所有点的纵坐标伸长到原来的2倍(横坐标不变)，再将所得到的图像向右平gx() p移个单位长度. 2 (?)求函数的解析式，并求其图像的对称轴方程; f()x (?)已知关于的方程在内有两个不同的解( xf()g()xxm+=[0,2)pab, (1)求实数m的取值范围; 22mcos)1.(ab-=- (2)证明: 5 pxk=+ p(kZ).【答案】(?) ，;(?)(1);(2)详见解f()2sinxx=(5,5)-2 析
( 【解析】试题分析:(?)纵向伸缩或平移: gxkgx()(),或gxgxk()(),，;横向伸缩或平移: 1a,0gxgx()(),,(纵坐标不变，横坐标变为原来的倍)，gxgxa()(),，(时，向, a,0aa左平移f()2sinxx=个单位;时，向右平移个单位);(?) (1)由(?)得，则of the audit. Fire is a combination of auditing in the field of law enforcement law enforcement job rotation and fire practice of law enforcement, in the promotion changed, retraining retired key timing synchronization in place audit mechanism, and conducive to self-urged self-restraint, the fire law enforcement, to protect themselves. 1.3 preventive fire-fighting urged the current fire law enforcement corruption involving cases of violating law enforcement has an upward trend, judging from national reports of complaints in recent years, reported cases reflect the fire law enforcement is over 50%. In addition, the multiple corruption cases, with a high incidence of leading cadres, a small number of leading cadres ' corruption, corrupt, power-for-money transaction, especially teams, group reporting increasing complaints involving the supervision of law enforcement jobs, bring great pressure to force honest and serious challenges. Meanwhile, team and Brigade levels bear an administrative license approval, administrative punishment, routine inspections and more than 95% of the amount, some fire
staff handled a year go through legal instruments and thousands of contact hundreds of social units, once inside lax, their quality is not high, extremely easy to induce discipline and corruption. In practical operation, the fire law enforcement as 'athletes' and 'umpire' internal inspector, inspection, test and not enough problem touches not found to leading policymakers, deviate from system design, system implementation has to deal with. 2 building fire law 2/6页，利用辅助角公式变形为(其中
f()g()2sincosxxxx+=+f()g()xx+=+5sin()xj 12)，方程在内有两个不同的解，等价f()g()xxm+=[0,2)pab,sin,cosjj== 55 于直线和函数有两个不同交点，数形结合求实数m的取值范围;ym,yx=+5sin()j p3p(2)结合图像可得和，进而利用诱导公式结合已知条件abj+=2()-abj+=2()-22求解( 试题解析:解法一:(1)将的图像上所有点的纵坐标伸长到原来的2倍(横坐gxx()cos= p标不变)得到的图像，再将的图像向右平移个单位长度后得到y2cos=xy2cos=x2 py2cos()=-x的图像，故，从而函数图像的对称轴方程为f()2sinxx=f()2sinxx=2
pxk=+ p(kZ). 2 21(2)1)
f()g()2sincos5(sincos)xxxxxx+=+=+ 55 12 (其中) sin,cosjj===+5sin()xj 55 mm依题意，在区间内有两个不同的解当且仅当，故msin()=x+j||1[0,2)pab, 55 的取值范围是. (5,5)- 2)因为是方程在区间内有两个不同的
解，ab,[0,2)p5sin()=mx+j mm所以，. sin()=aj+sin()=bj+ 55 p1m?abjabpbj+=2(),2();--=-+当时，2
3p-5abjabpbj+=2(),32();--=-+当时, 2 2mm222所以cos)cos2()2sin()12()11.(abbjbj-=-+=+-=-=-55解法二:(1)同解法一. (2)1) 同解法一. of the audit. Fire is a combination of auditing in the field of law enforcement law enforcement job rotation and fire practice of law enforcement, in the promotion changed, retraining retired key timing synchronization in place audit mechanism, and conducive to self-urged self-restraint, the fire law enforcement, to protect themselves. 1.3 preventive fire-fighting urged the current fire law enforcement corruption involving cases of violating law enforcement has an upward trend, judging from national reports of complaints in recent years, reported cases reflect the fire law enforcement is over 50%. In addition, the multiple corruption cases, with a high incidence of leading cadres, a small number of leading cadres ' corruption, corrupt, power-for-money transaction, especially teams, group reporting increasing complaints involving the supervision of law enforcement jobs, bring great pressure to force honest and serious challenges. Meanwhile, team and Brigade levels bear an administrative license approval, administrative punishment, routine inspections
and more than 95% of the amount, some fire staff handled a year go through legal instruments and thousands of contact hundreds of social units, once inside lax, their quality is not high, extremely easy to induce discipline and corruption. In practical operation, the fire law enforcement as 'athletes' and
'umpire' internal inspector, inspection, test and not enough problem touches not found to leading policymakers, deviate from system design, system implementation has to deal with. 2 building fire law 3/6页2) 因为是方程在区间内有两个不同的解，ab,[0,2)p5sin()=mx+j mm所以，. sin()=aj+sin()=bj+ 55 p当时，1m?abjajpbj+=2(),+();-=-+即2 3p当时, -5abjajpbj+=2(),+3();-=-+即2 所以cos+)cos()(ajbj=-+
于是
cos)cos[()()]cos()cos()sin()sin()(abajbjajbjajbj-=+-+=+++++
2mmm2222 =-++++=--+=-cos()sin()sin()[1()]()1.bjajbj555考点:1、三角函数图像变换和性质;2、辅助角公式和诱导公式( (湖南)17.设的内角A，B，C的对边分别为a，b，c，，且B 为钝,ABCabA,tan角》,(1)证明: ,,BA2 sinsinAC，(2)求的取值范围29【答案】(1)详见解析;(2)(，]. 28 【解析】,试题分析:(1)利用正弦定理，将条件中的式子等价变形为inB=sin(+A)，从而得证;2 sinA，sinC(2)利用(1)中的结论，以及三角恒等变形，将转化为只与有关的表达A
式，再利用三角函数的性质即可求解. sinsinAbB试题解析:(1)由a=btanA及正弦定理，得,,,所以sinB=cosA，即coscosAaB ,sinB=sin(+A). 2 ,,,,,又B为钝角，因此
+A(，A)，故B=+A，即B-A=;(2)由(I)知，
C=-,2222 ,,,,,,,(A+B)=-(2A+)=-2A>0，所以A，于是sinA+sinC=sinA+sin(-2A)= 0,,,,2224,,
19,222sinsinA+cos2A=-2A+sinA+1 =-2(sinA-)+，因为0,所以0,因此4428 of the audit. Fire is a combination of auditing in the field of law enforcement law enforcement job rotation and fire practice of law enforcement, in the promotion changed, retraining retired key timing synchronization in place audit mechanism, and conducive to self-urged self-restraint, the fire law enforcement, to protect themselves. 1.3 preventive
fire-fighting urged the current fire law enforcement corruption involving cases of violating law enforcement has an upward trend, judging from national reports of complaints in recent years, reported cases reflect the fire law enforcement is over 50%. In addition, the multiple corruption cases, with a high incidence of leading cadres, a small number of leading cadres ' corruption, corrupt, power-for-money transaction, especially teams, group reporting increasing complaints involving the supervision of law enforcement jobs, bring great pressure to
force honest and serious challenges. Meanwhile, team and Brigade levels bear an administrative license approval, administrative punishment, routine inspections and more than 95% of the amount, some fire staff handled a year go through legal instruments and thousands of contact hundreds of social units, once inside lax, their quality is not high, extremely easy to induce discipline and corruption. In practical operation, the fire law enforcement as 'athletes' and 'umpire' internal inspector, inspection, test and not enough problem touches not found to leading policymakers, deviate from system design, system implementation has to deal with. 2 building fire law 4/6页22199,, sinA,，,,,2488,, 29由此可知sinA+sinC的取值范围是(，]. 28 考点:1.正弦定理;2.三角恒等变形;3.三角函数的性质. (四川)19.如图，A,B,C,D为平面四边形ABCD 的四个内角. AA1cos,(1)证明:tan;, 2sinA oACABBCCDAD，,,,,,180,6,3,4,5,(2)若求ABCDtantantantan，，，的值. 2222 410【答案】(1)详见解析;(2). 3 【解析】AsinA2tan,试题分析:(1)首先切化弦得，为了将半角变为单角，可在分子分母同时A2cos2 A2sin乘以，然后逆用正弦与余弦的二倍角公式即可.(2)由题设知，该四边形的两对角2 22ABCDtantantantan，，，,，互补.再结合(1)的结果，有，所以只需2222sinsinAB
coscosCA,,coscosDB,,sin,sinAB求出即可.由于已知四边，且，，故考虑用余cos,cosABsin,sinAB弦定理列方程组求，从而求出. AA2sin2sinAA1cos,22tan,,,试题解析:(1). AAA2sinAcos2sincos222 ,,,AC，,180(2)由CADB,,,,180,180，得. of the audit. Fire is a combination of auditing in the field of law enforcement law enforcement job rotation and fire practice of law enforcement, in the promotion changed, retraining retired key timing synchronization in place audit mechanism, and conducive to self-urged self-restraint, the fire law enforcement, to protect themselves. 1.3 preventive
fire-fighting urged the current fire law enforcement corruption involving cases of violating law enforcement has an upward trend, judging from national reports of complaints in recent years, reported cases reflect the fire law enforcement is over 50%. In addition, the multiple corruption cases, with a high incidence of leading cadres, a small number of leading cadres ' corruption, corrupt, power-for-money transaction, especially teams, group reporting increasing complaints involving the supervision of law enforcement jobs, bring great pressure to force honest and serious challenges. Meanwhile, team and Brigade levels bear an administrative license approval, administrative punishment, routine inspections and more than
95% of the amount, some fire staff handled a year go through legal instruments and thousands of contact hundreds of social units, once inside lax, their quality is not high, extremely easy to induce discipline and corruption. In practical operation, the fire law enforcement as 'athletes' and 'umpire' internal inspector, inspection, test and not enough problem touches not found to leading policymakers, deviate from system design, system implementation has to deal with. 2 building fire law 5/6页。