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Express-sin-5x-as-polynomial-in-terms-of-sin-x-




Question Number 144791 by imjagoll last updated on 29/Jun/21
 Express sin 5x as polynomial   in terms of sin x.
$$\:\mathrm{Express}\:\mathrm{sin}\:\mathrm{5x}\:\mathrm{as}\:\mathrm{polynomial} \\ $$$$\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{sin}\:\mathrm{x}.\: \\ $$
Answered by liberty last updated on 29/Jun/21
by DeMoi′vre theorem  ⇔ cos 5x+isin 5x = (cos x+isin x)^5   = c^5 −10c^3 s^2 +5cs^4 +i(5c^4 s−10c^2 s^3 +s^5 )  with  { ((c=cos x)),((s=sin x)) :}  compare imaginary parts we  have sin 5x=5c^4 s−10c^2 s^3 +s^5   = 16s^5 −20s^3 +5s   =16sin^5 x−20sin^3 x+5sin x
$$\mathrm{by}\:\mathrm{DeMoi}'\mathrm{vre}\:\mathrm{theorem} \\ $$$$\Leftrightarrow\:\mathrm{cos}\:\mathrm{5x}+{i}\mathrm{sin}\:\mathrm{5x}\:=\:\left(\mathrm{cos}\:\mathrm{x}+{i}\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{5}} \\ $$$$=\:\mathrm{c}^{\mathrm{5}} −\mathrm{10c}^{\mathrm{3}} \mathrm{s}^{\mathrm{2}} +\mathrm{5cs}^{\mathrm{4}} +{i}\left(\mathrm{5c}^{\mathrm{4}} \mathrm{s}−\mathrm{10c}^{\mathrm{2}} \mathrm{s}^{\mathrm{3}} +\mathrm{s}^{\mathrm{5}} \right) \\ $$$$\mathrm{with}\:\begin{cases}{\mathrm{c}=\mathrm{cos}\:\mathrm{x}}\\{\mathrm{s}=\mathrm{sin}\:\mathrm{x}}\end{cases} \\ $$$$\mathrm{compare}\:\mathrm{imaginary}\:\mathrm{parts}\:\mathrm{we} \\ $$$$\mathrm{have}\:\mathrm{sin}\:\mathrm{5x}=\mathrm{5c}^{\mathrm{4}} \mathrm{s}−\mathrm{10c}^{\mathrm{2}} \mathrm{s}^{\mathrm{3}} +\mathrm{s}^{\mathrm{5}} \\ $$$$=\:\mathrm{16s}^{\mathrm{5}} −\mathrm{20s}^{\mathrm{3}} +\mathrm{5s}\: \\ $$$$=\mathrm{16sin}\:^{\mathrm{5}} \mathrm{x}−\mathrm{20sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{5sin}\:\mathrm{x}\: \\ $$

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