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Question Number 29848 by abdo imad last updated on 12/Feb/18

find  Σ_(k=0) ^n cos(kx) and Σ_(k=0) ^n  sin(kx) .

$${find}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} {cos}\left({kx}\right)\:{and}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{sin}\left({kx}\right)\:. \\ $$

Commented by Tinkutara last updated on 13/Feb/18

1+cos x+cos 2x+...+cos nx  =((sin (((n+1)x)/2))/(sin (x/2)))cos (((nx)/2))  sin x+sin 2x+sin 3x+...+sin nx  =((sin ((nx)/2))/(sin (x/2)))sin (x+(((n−1)x)/2))  =((sin ((nx)/2))/(sin (x/2)))sin ((((n+1)x)/2))

$$\mathrm{1}+\mathrm{cos}\:{x}+\mathrm{cos}\:\mathrm{2}{x}+...+\mathrm{cos}\:{nx} \\ $$$$=\frac{\mathrm{sin}\:\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}}{\mathrm{sin}\:\frac{{x}}{\mathrm{2}}}\mathrm{cos}\:\left(\frac{{nx}}{\mathrm{2}}\right) \\ $$$$\mathrm{sin}\:{x}+\mathrm{sin}\:\mathrm{2}{x}+\mathrm{sin}\:\mathrm{3}{x}+...+\mathrm{sin}\:{nx} \\ $$$$=\frac{\mathrm{sin}\:\frac{{nx}}{\mathrm{2}}}{\mathrm{sin}\:\frac{{x}}{\mathrm{2}}}\mathrm{sin}\:\left({x}+\frac{\left({n}−\mathrm{1}\right){x}}{\mathrm{2}}\right) \\ $$$$=\frac{\mathrm{sin}\:\frac{{nx}}{\mathrm{2}}}{\mathrm{sin}\:\frac{{x}}{\mathrm{2}}}\mathrm{sin}\:\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right) \\ $$

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