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Question Number 144791 by imjagoll last updated on 29/Jun/21

$$\mathrm{Express}\sin 5\mathrm{x}\mathrm{as}\mathrm{polynomial}$$$$\mathrm{in}\mathrm{terms}\mathrm{of}\sin \mathrm{x}.$$

Answered by liberty last updated on 29/Jun/21

$$\mathrm{by}{\mathrm{DeMoi}}^{\prime }\mathrm{vre}\mathrm{theorem}$$$$\iff \cos 5\mathrm{x}+i\sin 5\mathrm{x}={(\cos \mathrm{x}+i\sin \mathrm{x})}^5$$$$={\mathrm{c}}^5-{10\mathrm{c}}^3{\mathrm{s}}^2+{5\mathrm{cs}}^4+i({5\mathrm{c}}^4\mathrm{s}-{10\mathrm{c}}^2{\mathrm{s}}^3+{\mathrm{s}}^5)$$$$\mathrm{with}\{\begin{array}{l}\mathrm{c}=\cos \mathrm{x}\\ \mathrm{s}=\sin \mathrm{x}\end{array}$$$$\mathrm{compare}\mathrm{imaginary}\mathrm{parts}\mathrm{we}$$$$\mathrm{have}\sin 5\mathrm{x}={5\mathrm{c}}^4\mathrm{s}-{10\mathrm{c}}^2{\mathrm{s}}^3+{\mathrm{s}}^5$$$$={16\mathrm{s}}^5-{20\mathrm{s}}^3+5\mathrm{s}$$$$=16\sin {}^5\mathrm{x}-20\sin {}^3\mathrm{x}+5\sin \mathrm{x}$$

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