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

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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#.