Question

How do you write sin(4x) in terms of sin(x)? Thank you!

Answer 1

`sin4x=4(sinx-2sin^3x)sqrt(1-sin^2x)`

Explanation:

`sin4x=2sin2xcos2x`

= `2xx2sinxcosx xx(1-2sin^2x`

= `4sinxcosx-8sin^3xcosx`

This is generally used expansion for `sin4x`, however to write in terms of pure `sinx`, one can substitute `cosx` with `sqrt(1-sin^2x)`, although there could be issues regarding choosing appropriate sign based on which quadrant `x` falls.

In such a case `sin4x=4(sinx-2sin^3x)sqrt(1-sin^2x)`