F.4 trigonometry---0

用戶36732

2010-03-09 9:05 am

(a) Show that sin 4x = 4sinxcosxcos2x.

(b) By substituting a suitable value into the result of (a), show that cos36cos72 = 1/4

(c) By using the sum and product formulae and the result of (b), find the value of cos36 - cos72.

(d) By using the formula (a+b)^2 = (a-b)^2 + 4ab and the result of (c), find the value of cos36 + cos72.

(e) Hence, find the values of cos36 and cos72.

回答 (1)

Tak Kin

2010-03-09 9:33 am

✔ 最佳答案

(a)
LHS= sin 4x
= 2sin2xcos2x
=4sinxcosxcos2x
=RHS
(b)
let x =36
sin 4x = 4sinxcosxcos2x
sin144 = 4sin36cos36cos72
sin(180-36) = 4sin36cos36cos72
sin36 = 4sin36cos36cos72
∴cos36cos72 = 1/4
(c)
from ans (b)
cos36cos72 = 1/4
1/2 (cos108 + cos36) = 1/4
cos36 - cos72 = 1/2
(d)
( cos36 + cos72 )^2 = ( cos36 - cos72 )^2 +4cos36cos72
( cos36 + cos72 )^2 = ( 1/2)^2 + 4(1/4)
( cos36 + cos72 )^2 = 5/4
cos36 + cos72 = √5/2
(e)
from ans (c) cos36 - cos72 = 1/2 .........(1)
from ans (d) cos36 + cos72 = √5/2........(2)
(1) + (2) :　2cos36 = (1+√5)/2
∴　ｃｏｓ３６　＝　(1+√5)/４
（２）－（１）：　２ｃｏｓ７２＝（√５－１）／２
∴　ｃｏｓ７２　＝　（√５－１）／４

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