Quod Erat Demonstrandum

2016/03/21

用D證trigo

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 11:13 上午
Tags: ,

某 M2 題…

Prove the following identity

\cos^2x+\cos^2(x+y)-2\cos y\cos x\cos(x+y)=\sin^2y.

試試"D吓佢"…

思路:因為左邊有 x,而右邊冇,暗示左邊和 x 無關。故,先視 y 為常數,再「用 x D 吓」左邊,應該是零,即

Let f(x)=\cos^2x+\cos^2(x+y)-2\cos y\cos x\cos(x+y)

f'(x)

=-2\cos x\sin x-2\cos(x+y)\sin(x+y)+2\cos y(\sin x\cos(x+y)+\cos x\sin(x+y))

=-(\sin 2x+\sin(2x+2y))+2\cos y\sin(2x+y)

=-2\sin(\frac{1}{2}(2x+2x+2y))\cos(\frac{1}{2}(2x-2x-2y))+2\cos y\sin(2x+y)

=0

所以 f(x) 是常函數(constant function),我們便可以隨便代入,比如 x=0,即

f(x)

\equiv f(0)

=\cos^2(0)+\cos^2(0+y)-2\cos y\cos (0)\cos(0+y)

=1-\cos^2y

=\sin^2y

Q.E.D.

其實運用「D完係零所以 f 是常函數」呢招要小心,我們要考慮可導性(differentiability),見以下例子

https://johnmayhk.wordpress.com/2011/04/05/arcta/

不過,現在只是 M2,咩都唔使理~

習題
1. 試不以微分法,純用 M2 的三角學課程教的技巧證明原題。
2. 試以 core mathematics 課程教的技巧證明原題。(^_^)

發表迴響 »

仍無迴響。

RSS feed for comments on this post. TrackBack URI

發表迴響

在下方填入你的資料或按右方圖示以社群網站登入:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / 變更 )

Twitter picture

You are commenting using your Twitter account. Log Out / 變更 )

Facebook照片

You are commenting using your Facebook account. Log Out / 變更 )

Google+ photo

You are commenting using your Google+ account. Log Out / 變更 )

連結到 %s

在WordPress.com寫網誌.

%d 位部落客按了讚: