F.4 A maths urgent

👨🏻‍🏫 學術台數學

用戶164611

2009-02-04 3:12 am

If sin ( a+b) = 1/2, sin (a-b) = 1/3, find the value of tan a : tan b

更新1:

漏左題..... Given 0 < x < 90度. show that tan x - sin^2 x 4 ------------------- = --------------------------- - 1 tan x + sin^2 x 2 + sin 2x Hence, or otherwise, find the least value of tan x - sin^2 x ------------------- tan x + sin^2 x

更新2:

Given 0 < x < 90度. show that tan x - sin^2 x 4 ------------------- = --------------------------- - 1 tan x + sin^2 x 2 + sin 2x Hence, or otherwise, find the least value of tan x - sin^2 x ------------------- tan x + sin^2 x

更新3:

.......!!!!!!!!!! 應該係咁!!!!!! Given 0 < x < 90度. show that tan x - sin^2 x 　　　　　4 ------------------- = --------------------------- - 1 tan x + sin^2 x 　　　　2 + sin 2x Hence, or otherwise, find the least value of tan x - sin^2 x ------------------- tan x + sin^2 x

回答 (3)

安怡奶粉

2009-02-04 3:29 am

✔ 最佳答案

sin(a+b) = sin a cos b + cos a sin b =1/2-------------1
sin(a-b) = sin a cos b - cos a sin b =1/3--------------2

1-2 : 2cos a sin b = 1/6
cos a sin b = 1/12------------------------3

1+2 : 2sin a cos b = 5/6
sin a cos b = 5/12-------------------------4

4/3 : (sin a cos b) / (cos a sin b) = 5
(sin a / cos a)x(cos b sin b) =5
tan a / tan b = 5
SO, tan a: tan b = 5:1

參考: ME

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Gabriella Montez

2009-02-05 12:37 am

L.H.S. = ( tan x - sin^2 x ) / ( tan x + sin^2 x )
= [ ( sin x / cos x ) - sin^2 x ] / [ ( sin x / cos x ) + sin^2 x ]
= ( sin x - sin^2 x cos x ) / ( sin x + sin^2 x cos x )
= ( 1 - sin x cos x ) / ( 1 + sin x cos x )

2009-02-04 16:37:46 補充：
R.H.S. = [4 / ( 2 + sin 2x )] - 1
= [ 4 - ( 2 + sin 2x ) ] / ( 2 + sin 2x )
= 2 ( 1 - sin x cos x ) / 2 ( 1 + sin x cos x )
= ( 1 - sin x cos x ) / ( 1 + sin x cos x ) = L.H.S.

2009-02-04 16:40:57 補充：
0 < x < 90 deg and so 0 < 2x < 180 deg
When sin 2x is a max., ( tan x - sin^2 x ) / ( tan x + sin^2 x ) is a min.
So take sin 2x = 1, then [4 / ( 2 + 1 )] - 1 = 1 / 3

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用戶10012

2009-02-04 3:29 am

sin(a + b) + sin (a-b) = 2 sin a cos b = 1/2 + 1/3 = 5/6
so sin a cos b = 5/12................(1)
sin (a + b) - sin (a - b) = 2 cos a sin b = 1/2 - 1/3 = 1/6
so cos a sin b = 1/12...........(2)
(1)/(2), we get
sin a cos b / cos a sin b = 5
tan a/tan b = 5
so tan a : tan b = 5: 1.

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