Prove that sin⁡(x)sin⁡(π3+x)sin⁡(π3−x)=14sin⁡3x

Cameron Russell

Cameron Russell

Answered

2022-01-28

Prove that sin(x)sin(π3+x)sin(π3x)=14sin3x

Answer & Explanation

Troy Sutton

Troy Sutton

Expert

2022-01-29Added 13 answers

sin3t=3sint4sin3t
If sin3t=sin3x,3t=180n+(1)n3x where n is any integer
t=60n+(1)nx where n=1,0,1
So, the roots of
4sin3t3sint+sin3x=0
are sint where t=60n+(1)nx where n=1,0,1
sin(60x)sinxsin(60x)=(1)3sin3x4
4sin(60+x)sinxsin(60x)=sin3x

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