Law of Cosines w/ paraellogram

2009-03-08 4:23 pm
我不知道如何找出那兩個角和b
請你們幫忙一下^^

http://img228.imageshack.us/img228/959/math.jpg
更新1:

還有一題 http://img3.imageshack.us/img3/8271/math2i.jpg 謝謝!!

回答 (1)

2009-03-08 8:52 pm
✔ 最佳答案
設左上角是 y ,則左下角便是 180 - y

由 law of cos :

cos y = (a^2 + b^2 - c^2) / 2ab

cos y = (15^2 + b^2 - 25^2) / 2*15b

cos y = (b^2 - 400) / 30b.......(1)


cos (180-y) = (a^2 + b^2 - d^2) / 2ab

cos (180-y) = (15^2 + b^2 - 20^2) / 2*15b

cos (180-y) = (b^2 - 175) / 30b.....(2)

(1) + (2) :

cos y + cos (180-y) = (b^2 - 400) / 30b + (b^2 - 175) / 30b

cos y + (- cosy) = (2b^2 - 575) / 30b

0 = (2b^2 - 575) / 30b

0 = 2b^2 - 575

b^2 = 575/2

b = 16.955825(六位小數)

把b代入(1):
cos y =( 575/2 - 400) / 30 * 16.955825

cosy = - 0.2211629...

y = 102.77734...

y = 102.777度(三位小數)

180 - y = 180 - 102.77734 = 77.223度;(三位小數)

*************************************************
設左上角是 y ,左下角 是 180 - y

(a^2 + b^2 - c^2) / 2ab = cos y

(a^2 + 625 - 2500) / 50a = cos y

(a^2 - 1875) / 50a = cosy...(1)


(a^2 + b^2 - d^2) / 2ab = cos(180-y)

(a^2 + 625 - 1225) / 50a = cos(180-y)

(a^2 - 600) / 50a = cos(180-y)....(2)

(1) + (2) :

(a^2 - 1875) / 50a + (a^2 - 600) / 50a = cos y + cos(180-y)

2a^2 -2475 = 0

a^2 = 2475/2

a = 35.1781182...

代入(1) :

(2475/2 - 1875) / 50 * 35.1781182 = cosy

cos y = - 0.362441217...

y = 111.250195... = 111.2502度(四位小數)

180 - y = 180 - 111.250195... = 68.7498度(四位小數)


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