maths _28

(28) sin θ/√3 - cos θ = 0
sin θ/√3 = cos θ
tan θ = √3
θ = 210
(30) (I) 90 > 180 - x > 180 - y > 0
Thus sin (180 - x) > sin (180 - y)
sin x > sin y
So (I) is false.
(II) 90 > 180 - x > 180 - y > 0
Thus cos (180 - x) < cos (180 - y)
- cos x < - cos y
cos x > cos y
So (II) is false.
(III) 90 > 180 - x > 180 - y > 0
Thus tan (180 - x) > tan (180 - y)
- tan x > - tan y
tan x < tan y
So (III) is true.
(36) Let x be the distance between the buildings, then
MN = 16 m
x tan 45 - x tan 30 = 16
x = 16/(1 - 1/√3)
x = 37.9 m
(37) ∠DCE is the required angle in which
sin ∠DCE = DE/DC
= 20/36
∠DCE = 33.7

The answers are:
(28) B
(30) C
(36) C
(37) D

2009-06-13 20:57:11 補充：
(28) sin@/sqrt(3)-cos@=0
sin@/sqrt(3)=cos@
sin@/cos@=sqrt(3)
tan@=sqrt(3)
@= 60 or 240

2009-06-13 20:57:22 補充：
(30) for 90<180
sin@ is positive and changes from1 to 0 as @ increases
cos@ is negative and changes from 0 to -1 as @ increases
tan@ is negative and changes from -infinity to 0 as @ increases

2009-06-13 20:57:33 補充：
(36) Let the height of P is x above M, then height of P above N=x+16
x/BQ=tan 30
(x+16)/BQ=tan45
x=21.9
BQ=21.9/tan30=37.9

2009-06-13 20:57:40 補充：
(37)Greatest slope is DC
sinx=DE/DC=20/36=0.556
x=33.7 degree