1、1 1.3 3.3 3已知三角函数值求角已知三角函数值求角一二三一、已知正弦值,求角【问题思考】一二三一二三二、已知余弦值,求角【问题思考】2.填空:对于余弦函数y=cos x,如果已知函数值y(y-1,1),那么在0,上有唯一的x值和它对应,记作x=arccos y(-1y1,0 x).一二三一二三三、已知正切值,求角【问题思考】1.已知tan x=-1,求x.思考辨析判断下列说法是否正确,正确的打“”,错误的打“”.答案:(1)(2)(3)(4)探究一探究二探究三易错辨析已知正弦值求已知正弦值求角角 分析:借助正弦函数的图象及所给角的范围求解.探究一探究二探究三易错辨析反思感悟给值求角,由
2、于范围不同,所得的角可能不同,一定要注意范围条件的约束作用.对于sin x=a(xR),-1a1,这个方程的解可表示成x=2k+arcsin a或x=2k+-arcsin a(kZ).从而方程的解集为x|x=k+(-1)karcsin a,kZ.探究一探究二探究三易错辨析探究一探究二探究三易错辨析探究一探究二探究三易错辨析已知余弦值求已知余弦值求角角 分析:借助余弦函数的图象及所给角的范围求解即可.探究一探究二探究三易错辨析探究一探究二探究三易错辨析反思感悟cos x=a(-1a1),当x0,时,则x=arccos a,当xR时,可先求得0,2内的所有解,再利用周期性可求得x|x=2karcc
3、os a,kZ.探究一探究二探究三易错辨析变式训练变式训练1已知cos x=-0.345.(1)当x0,时,求x;(2)当xR时,求x的取值集合.解:(1)cos x=-0.345,且x0,x=arccos(-0.345)=-arccos 0.345.(2)当xR时,先求出0,2上的解.cos=-0.345,是第二或第三象限的角,由(1)知x1=-arccos 0.345为第二象限的角,cos(+arccos 0.345)=-0.345且+arccos 0.345 ,x2=+arccos 0.345,当x=2k+x1或2k+x2,kZ时,cos x=-0.345.即所求x的集合为x|x=2ka
4、rccos(-0.345),kZ.探究一探究二探究三易错辨析已知正切值求已知正切值求角角 探究一探究二探究三易错辨析探究一探究二探究三易错辨析反思感悟对于已知正切值求角有如下规律:探究一探究二探究三易错辨析变式训练变式训练2已知tan x=2,且x3,4,求x.(用符号表示)解:3x4,x-3=arctan 2,x=3+arctan 2.探究一探究二探究三易错辨析易错点:因忽视角的范围而致误【典例】求函数y=(arcsin x)2+arcsin x-1的最值.探究一探究二探究三易错辨析纠错心得arcsin x,arccos x,arctan x都是有范围的,忽略它们的范围是求解问题出错的根源.
5、探究一探究二探究三易错辨析答案:C4.满足等式sin(2x+45)=cos(30-x)的最小正角x是.解析:sin(2x+45)=sin(60+x),要使x0,且最小,则2x+45=60+x,所以x=15.答案:155.若arccos(2x-1)有意义,则x的取值范围是.解析:要使arccos(2x-1)有意义,则需-12x-11,即0 x1,故x0,1.答案:0,1dsfdbsy384y982ythb3oibt4oy39y409705923y09y53b2lkboi2y58wy0ehtoibwoify98wy049ywh4b3oiut89u983yf9ivh98y98sv98hv98ys9f
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