How do you use the sum and double angle identities to find sin3x?

Question

6061 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-04-26T17:58:57

Since the angle sum formula of sinus:

$sin(alpha+beta)=sinalphacosbeta+cosalphasinbeta$ ,

and the double angle formula of cosine:

$cos2alpha=cos^2alpha-sin^2alpha=2cos^2alpha-1=1-2sin^2alpha$

than:

$sin3x=sin(2x+x)=sin2xcosx+cos2xsinx=$

$=2sinxcosx*cosx+(1-2sin^2x)sinx=$

$=2sinxcos^2x+sinx-2sin^3x=$

$=2sinx(1-sin^2x)+sinx-2sin^3x=$

$=2sinx-2sin^3x+sinx-2sin^3x=$

$=3sinx-4sin^3x$ .