Solve sin3x=sinxsin2x

let tanx = t ,

then sinx = t/ sqrt(1+t^2) and cosx = 1/sqrt(1+t^2)

also tan2x=2t/(1-t^2)

then sin2x=2t/(1+t^2) and cos2x=(1-t^2)/(1+t^2)



so sin3x=sinxsin2x

=>sinxcos2x+sin2xcosx=sinxsin2x

=>( t/ sqrt(1+t^2))((1-t^2)/(1+t^2))+(2t/(1+t^2))( 1/sqrt(1+t^2))= (t/ sqrt(1+t^2))(2t/(1+t^2))

=>t-t^3+2t=2t^2

=>t^3-2t^2+t=0

=>t(t-1)^2=0

=>t=0 or t=1

=>tanx=0 or tanx=1

=>x=180n or 180n+45// where n is an integer .

2010-02-18 20:39:20 補充：
wrong step

2010-02-18 20:41:13 補充：
=>t-t^3+2t=2t^2

=>t(t^2+2t-3)=0

=>t=0 or t=1 or t=-3

=>tanx=0 , 1 or 3

=>x=180n , 180n+45 or 180n+arctan(3)

Let t=tan(x/2)
sinx=2t/(1+t²) and cosx=(1-t²)(1+t²) with dx=2dt/(1+t²)
This "t-method" is to solve integration involving trigonometric function.

sorry for stupid question:
what is t-method and where can i find its information?

sin3x = sinx•sin2x

3sinx - 4sin^3x = sinx•2sinxcosx

4sin^3x + 2sin^2xcosx - 3sinx = 0

sinx [4sin^2x + 2sinxcosx - 3] = 0

sinx [4sin^2x + 2sinxcosx - 3(sin^2x + cos^2x)] = 0

sinx (sin^2x + 2sinxcosx - 3cos2x) = 0

sinx (sin^2x + 2sinxcosx - 3cos2x)/cos^2x = 0

sinx (tan^2x + 2tanx - 3) = 0

sinx (tanx - 1) (tanx + 3) = 0

sinx = 0 or tanx = 1 or tanx = -3

Solutions for 0 ≤ x ≤ 360:
x = 0^o, 180^o, 360^o, 45^o, 225^o, 108.43^o,

Common solutions:
x = 180n^o, 180n^o + 45^o, 180n^o - 71.57^o
where n = 1, 2, 3 .....