Question 737560
<pre>
The trick is to start with sin(3x) and simplify
and it will come out with an equation that you
can solve for  sin³(x).

sin(3x) = sin(2x+x) = sin(2x)cos(x)+cos(2x)sin(x) = 

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

2sin(x)cos²(x)+[1-2sin²(x)]sin(x) =

2sin(x)[1-sin²(x)]+[1-2sin²(x)]sin(x) =
  
2sin(x) - 2sin³(x) + sin(x) - 2sin³(x) =

3sin(x) - 4sin³(x)

So

sin(3x) = 3sin(x) - 4sin³(x)

4sin³(x) = 3sin(x) - sin(3x)

sin³(x) = {{{3/4}}}sin(x) - {{{1/4}}}sin(3x)

Edwin</pre>