F.4 AMATHS(trigonometry)

(1)
sinα and secα are the roots of the equation 2X^2 -kx+3=0
∴sinα + secα = k/2-------------------(1)

sinα secα = 3/2
∴sinα /cosα = 3/2
tanα = 3/2
∴sinα = 3/√13---------------------(2)
cosα = 2/√13
secα = √13 /2-----------------------(3)
sub (2),(3) into (1):
3/√13 + √13 /2= k/2
k = 6/√13 + √13

f(α)
= sin^2 α - cosα + 1
=1-cos^2 α - cosα +1
=-cos^2 α - cosα +2
=-(cosα +1/2)^2 +9/4
∴maximum of f(α) = 9/4

maximum of (cosα +1/2) =3/2
∴ minimum of [-(cosα +1/2)^2] = - 9/4 [-(3/2)^2]
∴minimum of f(α) = 0[因為9/4 - 9/4 = 0]
∴0＜f(α)＜9/4

2007-02-17 01:13:44 補充：
f(α)=－(cosα ＋1/2)^2 ＋9/4要f(α)是最小,9/4是不變的,所以只是需要[－(cosα ＋1/2)^2]是最小但因為已經有"－",所以需要(cosα ＋1/2)^2 是最大就行了由於cos一個數最大是1,最小是－1我們需要(cosα ＋1/2)^2 是最大,所以需要cosα = 1∴最小的f(α)=－(1 ＋1/2)^2 ＋9/4 = 0留意:即使f(α)=－(cosα － 1/2)^2 ＋9/4,最小仍然是0,你知道為什麼嗎?