半角的正弦、余弦和正切1.tan15°+cot15°等于()A.2B.23C.4D.4332.设α∈(π,2π),则1cosπ2等于()A.sin2B.cos2C.sin2D.cos23.若sin11cos2,则sinα+cosα的值是()A.75B.85C.1D.29154.若sin2α=14,且α∈ππ,42,则cosα-sinα的值是()A.32B.34C.32D.345.1sin8cos81sin8cos8()A.tan2θB.cot4θC.tan4θD.cot2θ6.已知α为三角形的内角,sinα=35,则cot2________.7.若3π2<α<2π,且cosα=14,则1111cos22222的值是________.8.已知0°<α<β<90°,sinα与sinβ是方程x2-(2cos40°)x+cos240°-12=0的两根,则cos(2α-β)=________.9.已知ππ1sin2sin2444,α∈ππ,42,求2sin2α+tanα-1tan-1的值.10.(2011·北京模拟)已知函数f(x)=3sin2x-2sin2x.(1)求π6f的值;(2)若x∈ππ,63,求f(x)的最大值和最小值.1参考答案1.解析:原式=1cos30sin30sin301cos30=2-3+2+3=4.答案:C2.解析:∵α∈(π,2π),∴2∈π,π2,∴sin02.∴1cosπ1cos=sinsin2222.答案:A3.解析:由sin11cos2,①得sin(1cos)1(1cos)(1cos)2,整理得1cos1sin2.②由①得1cos2sin.③②+③得25sin2,解得sinα=45.又由①得cosα=2sinα-1=2×45-1=35.故sinα+cosα=437555.答案:A4.解析:∵(cosα-sinα)2=1-sin2α=1-14=34,∴|cosα-sinα|=32.由α∈ππ,42,知cosα<sinα,∴cosα-sinα=32.答案:C5.解析:由sin1costan21cossin,得tan4θ=sin81cos81cos8sin8,所以1sin8cos81sin8cos8=tan4θ.答案:C6.解析:由条件,得cosα=45,则411cos5cot332sin5或13.2答案:3或137.解析:∵3π2<α<2π,∴3π4<2<π.又cosα=14,∴1+cos10cos224.∴1111cos22222=1110coscoscos22224.答案:1048.解析:由已知得Δ=2cos240°-4cos240°+2=2sin240°,∴x=22cos40°±22sin40°.∴x1=sin45°cos40°+cos45°sin40°=sin85°,x2=sin45°cos40°-cos45°sin40°=sin5°.又由0°<α<β<90°,知β=85°,α=5°,∴cos(2α-β)=cos(-75°)=cos75°=cos(45°+30°)=624.答案:6249.解:∵ππ1sin2sin2444,∴ππ12sin2cos2442,即π1sin422.∴1cos42=.而2sin2α+tanα-1tan-1=-cos2α+22sincossincos=2cos2tan2.∵α∈ππ,42,∴2α∈π,π2.∴cos2α=1cos4322,tan2α=1cos431cos43.∴23253cos2tan22233,即2sin2α+tanα-1tan-1的值为532.310.解:(1)π6f=2ππ313sin2sin213624.(2)f(x)=3sin2x+cos2x-1=2πsin26x-1.因为x∈ππ,62,所以ππ5π2666x,所以12≤πsin26x≤1,所以f(x)的最大值为1,最小值为-2.4