Question

如图,在平面直角坐标系xOy中,以Ox轴为始边做两个锐角α,β,它们的终边分别与单位圆相交于A、B两点,已

Answer 1

(1)

∵是单位圆

∴半径r=1

∵yA=(7√2)/10,yB=(√5)/5

∴sinα=yA/r=(7√2)/10,sinβ=yB/r=(√5)/5

∵α和β都是锐角

∴cosα>0,cosβ>0

∴cosα=√[1-(sinα)^2]=√{1-[(7√2)/10]^2}=(√2)/10,cosβ=√[1-(sinβ)^2]=√{1-[(√5)/5]^2}=(2√5)/5

∴tanα=sinα/cosα=[(7√2)/10]/[(√2)/10]=7,tanβ=sinβ/cosβ=[(√5)/5]/[(2√5)/5]=1/2

∴tan(α+β)=(tanα+tanβ)/(1-tanαtanβ)=(7+1/2)/(1-7/2)=-3

(2)

∵tanβ=1/2

∴tan(2β)=(2tanβ)/[1-(tanβ)^2]=(2*1/2)/[1-(1/2)^2]=4/3

∵tanα=7

∴tan(α+2β)=[tanα+tan(2β)]/[1-tanαtan(2β)]=(7+4/3)/(1-7*4/3)=-1

∵β是锐角

∴0