作业帮 > 数学 > 作业

已知0<α<π/2,sinα=4/5

来源:学生作业帮 编辑:拍题作业网作业帮 分类:数学作业 时间:2024/03/29 08:52:16
已知0<α<π/2,sinα=4/5
(1)求(sin²α+sin2α)/(cos²α+cos2α)的值
(2)求tan(α-5π/4)的值
∵0<α<π/2,sinα=4/5
∴cosα>0
∵(sinα)^2+(cosα)^2=1
∴(cosα)^2=1-(sinα)^2=1-(4/5)^2=9/25
从而 cosα=3/5
又 sin2α=2sinα*cosα=2*4/5*3/5=24/25
cos2α=1-2(sinα)^2=1-2*(4/5)^2=1-2*16/25=-7/25
∴(sin²α+sin2α)/(cos²α+cos2α)=((4/5)^2+24/25)/((3/5)^2-7/25)
=(16/25+24/25)/(9/25-7/25)
=40/25/(2/25)
=20.
(2)
∵tanα=sinα/cosα=4/5/(3/5)=4/3
tan(-5π/4)=-tan(5π/4)=-tan(π+π/4)=-tanπ/4=-1
∴tan(α-5π/4)=(tanα-tan(-5π/4))/(1+tanα*tan(-5π/4))
=(4/3-(-1))/(1+4/3*(-1))
=(4/3+1)/(1-4/3)
=7/3/(-1/3)
=-7.