三角函数的图象与性质(教案)

§1.4三角函数的图象和性质1(x,y)rP(x,y),(y0)2Prrx2y2x2y则ysinrrxxcosrryytanxxsinMPcosOMtanAT,,MP,,的正弦线,余弦线,yyxO,以O为圆心作单位圆,从这个圆与x轴1112xAn(这里n)等份.把0到nn12x,,,0xxxxx[0,2],xxxRy,xxx重合,则(xR)xOxOAA作x111OA'1起来,再将它们平移,使起点与xxxOM1211111yxcosx22yy1-2-o--5-3-2xyxyxx[0,2]y,3(0,0),((,0),(,(2,0)223xx]y,(,0),(2(,0),(,22(R,)|sinxcosx1,1sinx1cosx1即[.sinx,xRy2k,kZx122k,kZ1x2x,xRy2k,kZ1xx2k,kZ12k)x,x2k)x,(kZ)由xf(x)TxT)f(x)f(x)Tf(x2,4,,2k(kZ,k0)2,4,,f(x)f(x)kZ,k02.由sin(x)x,x)xxxROyyx(xRy(x(xR)k,0kZ,ykkZx;2kkZ,x(xR)y2kkZxxxsinx,x[,]22从y[,]sinx2211当x11[,]sinx22当x2k,2k](kZ)[2k](kZ)221;1[2k,22其值从11.,2k](kZ)[2k1;1](kZ)[2k,2k其值从11.yx,xR和yx,xRkZ)yxyyy=cosx11ooxx,当x2,y1当xx2k当,y2,0kZkkZxkZxkkZ2[2k,2k][2k,2k]22[2k,2k]22tanxy,)22ytanx图像,根据(χ,得到y1o1xxxR,且xkkzπAOπ12-1kkzx2Rxxx且xxR,xk,kzxx2ytanxxR,且xk,kzT2tan(x)xZ(5)对称性:对称中心是k2xx轴的交点,但无对称轴,这(k,k),kZ22例1sinx(xR)yx(xR)y1xx[0,2]y|x|y|x|,2cos(x),x[0,2]y2例3y31sinx2例52sinx116x2cosxysinxcosxy略yf(x)xsin(x)f(x)f(x)32sin(x)1)3cos(x)yxy与xsinx1089107y,,,22sin2x),,,3224242,,,6323,2,4例111ysinx12cosxyylg(2sinx3)3sinxsinxx[,]34yyy222sin(x),x[,]663cosx3cosx3略例2(xRx[,)32若xx1;y2xy略2asin(2x)b[0,],值域为[例3f(x).32求a,b略53sinxacosxa(x])例4y2822略2sinxxxxx]y例5y(sinx2sinxcosxsinxcosx2sinxcosxcosx求f(x)的4322341cosx2sin(x)sinxcosxsin2x)sinx,42略例1f(x)1sinxcosx1sinxcosxsinxcosxcos2x2xxx)f(x)f(x)(4422lg(sinx1sinx)2f(x)xx略例2f(x)b3xa,b为常数)f7f(.f(x)x0f(x)xx2x,x0f(x)f(x)x)略例3y3xy2x1y2sin(5x)2sin(x)y略263AxxRyxxR(其中点评:,函数y0TA、、A0,.2sin2x4sin2xcos2x3cos2x例4yy224sinx)6略例5|x|y|2x1|y|x||x|y略例6f(x)5ff.2f(xf(x)f(x)是Rf略,f.1例7f(x)Rx1x,x[0,],212f(xx)f(x)f(x).121211f(),f();f224f(x)略f(x)xRxaxbba与(例8y(f(x)ba)f(x)xRf(x)f(xa)f(xa)(常数aRy(f(x)6a略例1xRycos(2x)6xycos(2x)yy31sin(x)y2xysin(x)234略例21()ylogcosxy略sin2x212例323174sin(),sin()cos(),cos()18105sin,cos160sin1,sin2,sin3略1sin(x)x[2,2]例4y,23略1x1x例5f(x).12f(x)f(x)略k,kZ}f(x)RT(1){x|x,232,2k],kZ[2k,2k],kZ(2)增区间:[2k222tanx例1y4略x3tan(),x(,)例2y2422yy|x|x,x0,2且x,2212x略例3tan135,tan13817,tantan45tan1,tan2,tan3,tan4略例4tanx3.略例5xxyx)yy略x12tanxtanxx且xk,kZ)例6y22略[,]34x且3(,)tancot例7,.22略xy3例864[,]2xy略logtanx例9ya略31m2nx(n,R)例