a) Show that (sin α + sin β) / (cos α + cos β) = tan [(α+β)/2]
b) If 3 sin α - 4 cos α = 4 cos β - 3 sin β, find the value of tan(α+β)
Please do (b).
Thanks.
更新1:
Why is it 2x/(1 - x^2)?
trigo q
2012-05-12 9:18 pm
Given two acute angles α and β.
a) Show that (sin α + sin β) / (cos α + cos β) = tan [(α+β)/2]
b) If 3 sin α - 4 cos α = 4 cos β - 3 sin β, find the value of tan(α+β)
Please do (b).
Thanks.
a) Show that (sin α + sin β) / (cos α + cos β) = tan [(α+β)/2]
b) If 3 sin α - 4 cos α = 4 cos β - 3 sin β, find the value of tan(α+β)
Please do (b).
Thanks.
回答 (2)
2012-05-13 1:36 am
✔ 最佳答案
3 sin α - 4 cos α = 4 cos β - 3 sin β
3 sin α + 3 sin β= 4 cos β+ 4 cos α
3 ( sin α + sin β)/ 4( cos α+ cos β) = 1
tan [(α+β)/2] = 4/3
tan [2 * (α+β)/2] = 2tan [(α+β)/2] / 1- tan ^2 [(α+β)/2]
(Identity of tan 2θ)
tan (α+β) = (2 * 4/3 ) / 1 - (4/3)^2
tan (α+β) = (8/3) / (-7/9)
tan (α+β) = -24/7
2012-05-12 17:37:59 補充：
Here I treat (α+β)/2 as θ
And so when it multiplies by 2, it becomes tan 2θ and it is tan (α+β) the term we want.
2012-05-12 9:31 pm
(a) sin α + sin β
= 2sin[(α + β)/2]cos[(α - β)/2]
cos α + cos β
= 2cos[(α + β)/2]cos[(α - β)/2]
So, (sin α + sin β) / (cos α + cos β) = tan [(α + β)/2]
(b) 3 sin α - 4 cos α = 4 cos β - 3 sin β
3 sin α + 3 sin β = 4 cos α + 4 cos β
3(sin α + sin β) = 4(cos α + cos β)
x = tan[(α+β)/2] = 4/3
tan(α+β) = 2x/(1 - x^2) = (8/3)/(7/9) = 24/7
= 2sin[(α + β)/2]cos[(α - β)/2]
cos α + cos β
= 2cos[(α + β)/2]cos[(α - β)/2]
So, (sin α + sin β) / (cos α + cos β) = tan [(α + β)/2]
(b) 3 sin α - 4 cos α = 4 cos β - 3 sin β
3 sin α + 3 sin β = 4 cos α + 4 cos β
3(sin α + sin β) = 4(cos α + cos β)
x = tan[(α+β)/2] = 4/3
tan(α+β) = 2x/(1 - x^2) = (8/3)/(7/9) = 24/7
收錄日期: 2021-04-27 19:08:12
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20120512000051KK00337