Question-197982

Question Number 197982 by hardmath last updated on 07/Oct/23

Answered by witcher3 last updated on 11/Oct/23

sin(((12π)/(180)))sin(((15π)/(180)))sin(((51π)/(180)))sin(((57π)/(180)))sin(((63π)/(180))) sin(((2π)/(60)))sin(((5π)/(60)))sin(((17π)/(60)))sin(((19π)/(60)))sin(((21π)/(60))) sin(((2π)/(60)))=sin((π/(30)))=sin((π/5)−(π/6)) sin(((5π)/(60)))=sin((π/(12)))=sin((π/3)−(π/4)) sin(((17π)/(60)))=sin((π/5)+(π/(12))) sin(((19π)/(60)))sin(((21π)/(60))).=(1/2)(cos((π/(30)))−cos(((2π)/3))) sin((π/5))=((√(5−(√5)))/(2(√2))),cos((π/5))=((1+(√5))/4) cos((π/(30)))=(√(1−sin^2 ((π/5)−(π/(30)))))... just compute all value

$$\mathrm{sin}\left(\frac{\mathrm{12}\pi}{\mathrm{180}}\right)\mathrm{sin}\left(\frac{\mathrm{15}\pi}{\mathrm{180}}\right)\mathrm{sin}\left(\frac{\mathrm{51}\pi}{\mathrm{180}}\right)\mathrm{sin}\left(\frac{\mathrm{57}\pi}{\mathrm{180}}\right)\mathrm{sin}\left(\frac{\mathrm{63}\pi}{\mathrm{180}}\right) \\ $$$$\mathrm{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{60}}\right)\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{60}}\right)\mathrm{sin}\left(\frac{\mathrm{17}\pi}{\mathrm{60}}\right)\mathrm{sin}\left(\frac{\mathrm{19}\pi}{\mathrm{60}}\right)\mathrm{sin}\left(\frac{\mathrm{21}\pi}{\mathrm{60}}\right) \\ $$$$\mathrm{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{60}}\right)=\mathrm{sin}\left(\frac{\pi}{\mathrm{30}}\right)=\mathrm{sin}\left(\frac{\pi}{\mathrm{5}}−\frac{\pi}{\mathrm{6}}\right) \\ $$$$\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{60}}\right)=\mathrm{sin}\left(\frac{\pi}{\mathrm{12}}\right)=\mathrm{sin}\left(\frac{\pi}{\mathrm{3}}−\frac{\pi}{\mathrm{4}}\right) \\ $$$$\mathrm{sin}\left(\frac{\mathrm{17}\pi}{\mathrm{60}}\right)=\mathrm{sin}\left(\frac{\pi}{\mathrm{5}}+\frac{\pi}{\mathrm{12}}\right) \\ $$$$\mathrm{sin}\left(\frac{\mathrm{19}\pi}{\mathrm{60}}\right)\mathrm{sin}\left(\frac{\mathrm{21}\pi}{\mathrm{60}}\right).=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\left(\frac{\pi}{\mathrm{30}}\right)−\mathrm{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\right) \\ $$$$\mathrm{sin}\left(\frac{\pi}{\mathrm{5}}\right)=\frac{\sqrt{\mathrm{5}−\sqrt{\mathrm{5}}}}{\mathrm{2}\sqrt{\mathrm{2}}},\mathrm{cos}\left(\frac{\pi}{\mathrm{5}}\right)=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\mathrm{cos}\left(\frac{\pi}{\mathrm{30}}\right)=\sqrt{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{5}}−\frac{\pi}{\mathrm{30}}\right)}… \\ $$$$\mathrm{just}\:\mathrm{compute}\:\mathrm{all}\:\mathrm{value} \\ $$$$ \\ $$$$ \\ $$

Answered by Sutrisno last updated on 14/Oct/23

((2.sin12.cos12.sin15.sin51.2.sin57.cos57sin63)/(2cos12.2cos57)) ((sin24.sin15.sin51.sin114.sin63)/(4sin102.cos57)) ((sin24.sin15.sin51.cos24.sin63)/(4.2.sin51.cos51.cos57)) ((2.sin24.cos24.sin63)/(2.4.2cos51.cos57)) ((sin48.sin15.sin63)/(16cos51.cos57)) ((sin48(cos48−cos78))/(16(cos108+cos6))) ((sin48.cos48−cos78sin48)/(16(cos108+cos6))) ((sin96−(sin126−sin30))/(32(cos108+cos6))) ((cos6−(cos36−(1/2)))/(32(cos108+cos6))) (cos36−(1/2)=sin18=cos72) ((cos6−cos72)/(32(cos6−cos72))) =(1/(32))

$$\frac{\mathrm{2}.{sin}\mathrm{12}.{cos}\mathrm{12}.{sin}\mathrm{15}.{sin}\mathrm{51}.\mathrm{2}.{sin}\mathrm{57}.{cos}\mathrm{57}{sin}\mathrm{63}}{\mathrm{2}{cos}\mathrm{12}.\mathrm{2}{cos}\mathrm{57}} \\ $$$$\frac{{sin}\mathrm{24}.{sin}\mathrm{15}.{sin}\mathrm{51}.{sin}\mathrm{114}.{sin}\mathrm{63}}{\mathrm{4}{sin}\mathrm{102}.{cos}\mathrm{57}} \\ $$$$\frac{{sin}\mathrm{24}.{sin}\mathrm{15}.{sin}\mathrm{51}.{cos}\mathrm{24}.{sin}\mathrm{63}}{\mathrm{4}.\mathrm{2}.{sin}\mathrm{51}.{cos}\mathrm{51}.{cos}\mathrm{57}} \\ $$$$\frac{\mathrm{2}.{sin}\mathrm{24}.{cos}\mathrm{24}.{sin}\mathrm{63}}{\mathrm{2}.\mathrm{4}.\mathrm{2}{cos}\mathrm{51}.{cos}\mathrm{57}} \\ $$$$\frac{{sin}\mathrm{48}.{sin}\mathrm{15}.{sin}\mathrm{63}}{\mathrm{16}{cos}\mathrm{51}.{cos}\mathrm{57}} \\ $$$$\frac{{sin}\mathrm{48}\left({cos}\mathrm{48}−{cos}\mathrm{78}\right)}{\mathrm{16}\left({cos}\mathrm{108}+{cos}\mathrm{6}\right)} \\ $$$$\frac{{sin}\mathrm{48}.{cos}\mathrm{48}−{cos}\mathrm{78}{sin}\mathrm{48}}{\mathrm{16}\left({cos}\mathrm{108}+{cos}\mathrm{6}\right)} \\ $$$$\frac{{sin}\mathrm{96}−\left({sin}\mathrm{126}−{sin}\mathrm{30}\right)}{\mathrm{32}\left({cos}\mathrm{108}+{cos}\mathrm{6}\right)} \\ $$$$\frac{{cos}\mathrm{6}−\left({cos}\mathrm{36}−\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{32}\left({cos}\mathrm{108}+{cos}\mathrm{6}\right)}\:\:\:\left({cos}\mathrm{36}−\frac{\mathrm{1}}{\mathrm{2}}={sin}\mathrm{18}={cos}\mathrm{72}\right) \\ $$$$\frac{{cos}\mathrm{6}−{cos}\mathrm{72}}{\mathrm{32}\left({cos}\mathrm{6}−{cos}\mathrm{72}\right)}\:\:\: \\ $$$$=\frac{\mathrm{1}}{\mathrm{32}} \\ $$

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