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Question Number 126954 by bramlexs22 last updated on 25/Dec/20

cos (((5π)/(18)))+cos (((7π)/(18)))+cos (((17π)/(18)))=?

$$\:\:\mathrm{cos}\:\left(\frac{\mathrm{5}\pi}{\mathrm{18}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{7}\pi}{\mathrm{18}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{17}\pi}{\mathrm{18}}\right)=? \\ $$

Answered by Dwaipayan Shikari last updated on 25/Dec/20

2cos(((12π)/(36)))cos((π/(18)))+cos(((17π)/(18))) =cos((π/(18)))+cos(((17π)/(18)))=2cos((π/2))cos(((8π)/(18)))=0

$$\mathrm{2}{cos}\left(\frac{\mathrm{12}\pi}{\mathrm{36}}\right){cos}\left(\frac{\pi}{\mathrm{18}}\right)+{cos}\left(\frac{\mathrm{17}\pi}{\mathrm{18}}\right) \\ $$$$={cos}\left(\frac{\pi}{\mathrm{18}}\right)+{cos}\left(\frac{\mathrm{17}\pi}{\mathrm{18}}\right)=\mathrm{2}{cos}\left(\frac{\pi}{\mathrm{2}}\right){cos}\left(\frac{\mathrm{8}\pi}{\mathrm{18}}\right)=\mathrm{0} \\ $$

Commented by bramlexs22 last updated on 25/Dec/20

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Answered by Olaf last updated on 25/Dec/20

X = cos(((5π)/(18)))+cos(((7π)/(18)))+cos(((17π)/(18))) X = cos((π/3)−(π/(18)))+cos((π/3)+(π/(18)))+cos(π−(π/(18))) X = cos((π/3))cos((π/(18)))−sin((π/3))sin((π/(18))) +cos((π/3))cos((π/(18)))+sin((π/3))sin((π/(18)))−cos((π/(18))) X = 2cos((π/3))cos((π/(18)))−cos((π/(18))) X = 2.(1/2).cos((π/(18)))−cos((π/(18))) = 0

$$\mathrm{X}\:=\:\mathrm{cos}\left(\frac{\mathrm{5}\pi}{\mathrm{18}}\right)+\mathrm{cos}\left(\frac{\mathrm{7}\pi}{\mathrm{18}}\right)+\mathrm{cos}\left(\frac{\mathrm{17}\pi}{\mathrm{18}}\right) \\ $$$$\mathrm{X}\:=\:\mathrm{cos}\left(\frac{\pi}{\mathrm{3}}−\frac{\pi}{\mathrm{18}}\right)+\mathrm{cos}\left(\frac{\pi}{\mathrm{3}}+\frac{\pi}{\mathrm{18}}\right)+\mathrm{cos}\left(\pi−\frac{\pi}{\mathrm{18}}\right) \\ $$$$\mathrm{X}\:=\:\mathrm{cos}\left(\frac{\pi}{\mathrm{3}}\right)\mathrm{cos}\left(\frac{\pi}{\mathrm{18}}\right)−\mathrm{sin}\left(\frac{\pi}{\mathrm{3}}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{18}}\right) \\ $$$$+\mathrm{cos}\left(\frac{\pi}{\mathrm{3}}\right)\mathrm{cos}\left(\frac{\pi}{\mathrm{18}}\right)+\mathrm{sin}\left(\frac{\pi}{\mathrm{3}}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{18}}\right)−\mathrm{cos}\left(\frac{\pi}{\mathrm{18}}\right) \\ $$$$\mathrm{X}\:=\:\mathrm{2cos}\left(\frac{\pi}{\mathrm{3}}\right)\mathrm{cos}\left(\frac{\pi}{\mathrm{18}}\right)−\mathrm{cos}\left(\frac{\pi}{\mathrm{18}}\right) \\ $$$$\mathrm{X}\:=\:\mathrm{2}.\frac{\mathrm{1}}{\mathrm{2}}.\mathrm{cos}\left(\frac{\pi}{\mathrm{18}}\right)−\mathrm{cos}\left(\frac{\pi}{\mathrm{18}}\right)\:=\:\mathrm{0} \\ $$

Commented by bramlexs22 last updated on 25/Dec/20

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