Prove: \(\displaystyle{\cos{{\left({3}\theta\right)}}}={4}{{\cos}^{{3}}{\left(\theta\right)}}-{3}{\cos{{\left(\theta\right)}}}\)

loraliyeruxi

loraliyeruxi

Answered question

2022-03-28

Prove:
cos(3θ)=4cos3(θ)3cos(θ)

Answer & Explanation

haiguetenteme7zyu

haiguetenteme7zyu

Beginner2022-03-29Added 13 answers

De Moivre's formula reads
(cosθ+isinθ)n=cos(nθ)+isin(nθ)
Of course, this identity implies that equality should also apply to the real component. That is
cos(nθ)=R[(cosθ+isinθ)n]
So we have
cos(3θ)=R[cos3θ+3icos2θsinθ3cosθsin2θisin3θ]
=cos3θ3cosθsin2θ

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