A.Math General Solution

SOLUTION
2sinxsin3x=1

2{-1/2[cos(x+3x)-cos(x-3x)]}=1 [By product to sum formula]

-(cos4x)+cos(2x)=1

-(2cos^2 2x -1)+cos 2x = 1

-2cos^2 2x + cos 2x = 0

cos2x(-2cos2x + 1) = 0

cos 2x = 0 or (-2cos2x + 1) = 0

when cos 2x = 0
2x=360n+90 or 360n-90
x= 180n+45 or 180n-45
when (-2cos2x + 1) = 0
cos2x = 1/2
2x=360n+60 or 360n-60
x= 180n+30 or 180n-30

利用sinAsinB=-1/2[cos(A+B)-cos(A-B),即2sinAsinB=-cos(A+B)+cos(A-B),
cos(-A)=cosA及cos2A=2cos^2A-1
2sin x sin 3x= 1
-cos4x+cos(-2x)=1
-cos4x+cos2x-1=0
cos2*(2x)-cos2x+1=0
2cos^2(2x)-1-cos2x+1=0
2cos^2(2x)-cos2x=0
cos2x(2cos2x-1)=0
cos2x=0 or cos2x=1/2
2x=2nπ±π/2 or 2x=2nπ±π/3
x=nπ±π/4 or x=nπ±π/6

2 sin x sin 3x = 1

2 sin x sin ( x + 2x ) = 1

2 sin x ( sin x cos 2x + sin 2x cos x ) = 1

2 sin^2 x cos 2x + sin^2 2x = 1

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

2 sin^2 x - 4 sin^4 x + 4 sin^2 x ( 1 - sin^2 x ) - 1 = 0

8 sin^4 x - 6 sin^2 x + 1 = 0

Let y be sin^2 x, we have

8y^2 - 6y + 1 = 0

( 1 - 4y) ( 1 - 2y ) = 0

y = 1/4 or y = 1/2

Therefore,

sin^2 x = 1/4 or sin^2 x = 1/2

sin x = +- 1/2 or sin x = +- 1/(開方2)

x = 180n +- 30 or x = 180n +- 45

2007-04-14 00:28:03 補充：
Presentation of answer:x = 180n 30 or x = 180n - 30x = 180n 45 or x = 180n - 45