Re: [中學] 高中數學

[a016258 (憨)]

※ 引述《semmy214 (陳山河)》之銘言：
: http://imgur.com/YAUm71e
: 第1題不用回答
: 2、3、4煩請版上高手解答

2. θ 省略...   sin^m  = ( sin θ)^m

x^2 + y^2 = cos^6 + (sin^6 + 2sin^4 + sin^2)    (三次方和 拆 前兩項)

          = (cos^4 - cos^2 sin^2 + sin^4) +  2sin^4 + sin^2   (合併二、五項)

          = cos^4 + 4 sin^4

          = (cos^2 + 2sin^2 )^2 - 4sin^2 cos^2

          = sin^4 + 2sin^2 + 1 - 4sin^2 cos^2

          = sin^4 + 2sin^2 + 1 - 4sin^2 (1 - sin^2 )

          = 5sin^4 - 2sin^2 + 1

          = 5( sin^2 - (1/5) )^2 + 4/5

when sin^2 = 1/5 , min = 2/√5

when sin^2 = 1   , max = 2



4. f(x) = sinx cosx + a(sinx + cosx) + a^2   ( t = sinx + cosx ≦ √2 )

        = (t^2 - 1)/2 + at + a^2

        = (1/2) (t^2 + 2at + 2a^2 - 1)    ( ∵ a ≧ 0 )

        ≦(1/2) ( 2 + 2√2 a + 2a^2 - 1)

        = (1/2) (2a^2 + 2√2 a + 1) =   25/2

        => a^2 + √2 a - 12 = 0

             -√2 + √50
        a =  --------------- = 2√2
                   2


有錯還請不吝指正。



--
If people do not believe that mathematics is simple,

it is only because they do not realize how complicated life is.


                                                    － John von Neumann

--
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※ 文章網址: https://www.ptt.cc/bbs/Math/M.1446461803.A.3FC.html
※ 編輯: a016258 (140.114.137.240), 11/02/2015 19:03:32
→ semmy214    : (cos^4 - cos^2 sin^2 + sin^4) 這地方不懂
cos^6 + sin^6 = (cos^2 )^3 + (sin^2 )^3   三次方和化成兩項相乘
※ 編輯: a016258 (140.114.137.240), 11/02/2015 19:06:27
→ semmy214    : a^3+b^3 不是(a+b)(a^2-ab+b^2) 好像漏了(a+b)
(a+b) = ?
→ Eliphalet   : semmy214 你展開就好了，不會費你太多時間
→ semmy214    : E大謝了 我懂了~
→ semmy214    : 第4題.第3行跳第4行不懂...
→ semmy214    : 剛剛看懂了 但為什麼sinx+cosx小於根號2
→ Eliphalet   : 疊加 sin(x)+cos(x) = √2 sin(x+π/4)
※ 編輯: a016258 (218.161.70.138), 11/02/2015 20:34:14