Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

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

$$\mathrm{by}\:\mathrm{DeMoi}'\mathrm{vre}\:\mathrm{theorem} \\ $$$$\Leftrightarrow\:\mathrm{cos}\:\mathrm{5x}+{i}\mathrm{sin}\:\mathrm{5x}\:=\:\left(\mathrm{cos}\:\mathrm{x}+{i}\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{5}} \\ $$$$=\:\mathrm{c}^{\mathrm{5}} −\mathrm{10c}^{\mathrm{3}} \mathrm{s}^{\mathrm{2}} +\mathrm{5cs}^{\mathrm{4}} +{i}\left(\mathrm{5c}^{\mathrm{4}} \mathrm{s}−\mathrm{10c}^{\mathrm{2}} \mathrm{s}^{\mathrm{3}} +\mathrm{s}^{\mathrm{5}} \right) \\ $$$$\mathrm{with}\:\begin{cases}{\mathrm{c}=\mathrm{cos}\:\mathrm{x}}\\{\mathrm{s}=\mathrm{sin}\:\mathrm{x}}\end{cases} \\ $$$$\mathrm{compare}\:\mathrm{imaginary}\:\mathrm{parts}\:\mathrm{we} \\ $$$$\mathrm{have}\:\mathrm{sin}\:\mathrm{5x}=\mathrm{5c}^{\mathrm{4}} \mathrm{s}−\mathrm{10c}^{\mathrm{2}} \mathrm{s}^{\mathrm{3}} +\mathrm{s}^{\mathrm{5}} \\ $$$$=\:\mathrm{16s}^{\mathrm{5}} −\mathrm{20s}^{\mathrm{3}} +\mathrm{5s}\: \\ $$$$=\mathrm{16sin}\:^{\mathrm{5}} \mathrm{x}−\mathrm{20sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{5sin}\:\mathrm{x}\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com