A.math....緊急.....20分

2008-01-18 7:11 am
解方程:
2sin^2x- sin2x=2

回答 (3)

2008-01-18 7:20 am
✔ 最佳答案
2(sinx)^2- sin2x=2
2(sinx)^2 - 2 sinx cos x = 2
Dividing both sides by (cosx)^2,
2(tanx)^2-2tanx-2(secx)^2=0
(tanx)^2-tanx-[1+(tanx)^2]=0
tanx = -1
x = 135* or 315* for 0 < x < 360*

2008-01-17 23:21:42 補充:
Remark: 1 + (tanx)^2 = (secx)^2

2008-01-18 13:32:41 補充:
抱歉, 這個方法是會解漏了兩個根的, 正確部驟如下:2(sinx)^2- sin2x = 22[ 1 - (cosx)^2 ] - 2 sin x cos x = 21 - (cosx)^2 - sinxcosx = 1-cos x ( cos x + sin x ) = 0 cos x = 0 or tan x = -1 x = 90*, 270*, 135*, 315* for 0 &lt; x &lt; 360*

2008-01-18 13:36:40 補充:
下面那位有兩處錯了:( 1 ) sin^2 x = ( sin x )^2 這樣表示是為免與sin 2x混淆( 2 ) 0*, 180* 不是cos x = 0 的根, 代回原式便可得知:2(sin0)^2 - sin2(0) = 0 - 0 = 0 不等於22(sin180*)^2 - sin2(180*) = 0 - 0 = 0 不等於2

2008-01-18 13:42:18 補充:
To下面再下面那位:題目是要解方程而不是求方程的通解呀!
參考: My Maths Knowledge
2008-01-19 6:03 pm
2sin^2x- sin2x=2
2(1-cos^2x)-2sinxcosx=2
2-2cos^2x-2sinxcosx=2
-2cosx(cosx+sinx)=0
cosx(cosx+sinx)=0
cosx=0 or cosx+sinx=0
x=90,270 or (cosx+sinx)/cosx=0
x=90,270 or 1+tanx=0
x=90,270 or tanx=-1
x=90,270,135,315 for 0&lt;360//
2008-01-18 7:36 am
解方程:
2sin^2x- sin2x=2 *不是2(sinx)^2- sin2x=2

2sin^2x- sin2x =2
2(1-cos^2x)- sin2x =2
2- 2cos^2x- (2cosxsinx) =2
2- 2cos^2x- 2cosxsinx =2
2cosx(cosx+sinx)=0
cosx =0 or cosx=-sinx
cosx=0 or tanx=-1
x=0, 135, 180 or 315 for 0&lt;360
參考: calculation for leisure


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