What is sin 3x in terms of sin x?

Answer 1

given the identity sin(x+y)=sinx cosy + siny cosx

sin2x = 2 sinx cosx and

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

using the last two identities gives

sin3x= 2 sinx cosx cosx + sinx cos2x

factoring the sinx we have

sin3x = sinx(2cosx cosx+cos2x)

which satisfies the requirement.

However, we can simplify further since cos 2x = cosx cosx - sinx sinx (a well known identity)

sin3x = sinx (2cosx cosx +cosx cosx - sinx sinx)

so sin3x= sinx(3cosx cosx - sinx sinx)

or sin 3x = 3.cos²x.sinx - sin³x

* * * * *

Good, but not good enough. The answer was required in terms of sin, not a mixture of sinx and cosx. Easily recitified, though, since cos²x = 1 - sin²x

Therefore sin3x = 3*(1-sin²x)*sinx - sin³x

= 3sinx - 3sin³x - sin³x

= 3sinx - 4sin³x