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Question Number 123248 by Dwaipayan Shikari last updated on 24/Nov/20

Find the value of sin((π/9)) in exact form

$${Find}\:{the}\:{value}\:{of}\:{sin}\left(\frac{\pi}{\mathrm{9}}\right)\:\:{in}\:{exact}\:{form} \\ $$

Commented by MJS_new last updated on 24/Nov/20

sin (π/9) =sin 20° I don′t think it′s possible. the smallest angle α for which we can exactly express sin α is (π/(60))=3° ⇒ for β with β≠((nπ)/(60)) or β≠(3n)° this is impossible. I saw an “expression” for sin 1° but it uses Cardano′s Solution for x^3 +px+q=0 while (p^3 /(27))+(q^2 /4)<0 but in this case it′s not defined. you can write it down but it makes no sense at all.

$$\mathrm{sin}\:\frac{\pi}{\mathrm{9}}\:=\mathrm{sin}\:\mathrm{20}° \\ $$$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{it}'\mathrm{s}\:\mathrm{possible}.\:\mathrm{the}\:\mathrm{smallest} \\ $$$$\mathrm{angle}\:\alpha\:\mathrm{for}\:\mathrm{which}\:\mathrm{we}\:\mathrm{can}\:\mathrm{exactly}\:\mathrm{express} \\ $$$$\mathrm{sin}\:\alpha\:\mathrm{is}\:\frac{\pi}{\mathrm{60}}=\mathrm{3}°\:\Rightarrow\:\mathrm{for}\:\beta\:\mathrm{with}\:\beta\neq\frac{{n}\pi}{\mathrm{60}}\:\mathrm{or}\:\beta\neq\left(\mathrm{3}{n}\right)° \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{impossible}.\:\mathrm{I}\:\mathrm{saw}\:\mathrm{an}\:``\mathrm{expression}''\:\mathrm{for} \\ $$$$\mathrm{sin}\:\mathrm{1}°\:\mathrm{but}\:\mathrm{it}\:\mathrm{uses}\:\mathrm{Cardano}'\mathrm{s}\:\mathrm{Solution}\:\mathrm{for} \\ $$$${x}^{\mathrm{3}} +{px}+{q}=\mathrm{0}\:\mathrm{while}\:\frac{{p}^{\mathrm{3}} }{\mathrm{27}}+\frac{{q}^{\mathrm{2}} }{\mathrm{4}}<\mathrm{0}\:\mathrm{but}\:\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{defined}.\:\mathrm{you}\:\mathrm{can}\:\mathrm{write}\:\mathrm{it}\:\mathrm{down}\:\mathrm{but}\:\mathrm{it} \\ $$$$\mathrm{makes}\:\mathrm{no}\:\mathrm{sense}\:\mathrm{at}\:\mathrm{all}. \\ $$

Commented by Dwaipayan Shikari last updated on 24/Nov/20

sin((π/9))=sinθ 9θ=π ⇒sin6θ =sin3θ ⇒ 2cos3θ =1 (sinθ=(√(1−t^2 )) 4t^3 −3t=(1/2) ⇒ t^3 −(3/4)t−(1/8)=0 (1/(27))(((−3)/4))^3 +(1/4)(−(1/8))^2 =−(1/(64))+(1/(4.64))<0 (Invalid) But any other way sir?

$${sin}\left(\frac{\pi}{\mathrm{9}}\right)={sin}\theta\:\: \\ $$$$\mathrm{9}\theta=\pi\:\Rightarrow{sin}\mathrm{6}\theta\:={sin}\mathrm{3}\theta\:\Rightarrow\:\mathrm{2}{cos}\mathrm{3}\theta\:=\mathrm{1}\:\:\:\:\:\:\:\:\left({sin}\theta=\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }\right. \\ $$$$\mathrm{4}{t}^{\mathrm{3}} −\mathrm{3}{t}=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\:{t}^{\mathrm{3}} −\frac{\mathrm{3}}{\mathrm{4}}{t}−\frac{\mathrm{1}}{\mathrm{8}}=\mathrm{0} \\ $$$$\frac{\mathrm{1}}{\mathrm{27}}\left(\frac{−\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{4}}\left(−\frac{\mathrm{1}}{\mathrm{8}}\right)^{\mathrm{2}} \:=−\frac{\mathrm{1}}{\mathrm{64}}+\frac{\mathrm{1}}{\mathrm{4}.\mathrm{64}}<\mathrm{0}\:\:\left({Invalid}\right) \\ $$$${But}\:{any}\:{other}\:{way}\:{sir}? \\ $$$$ \\ $$

Commented by MJS_new last updated on 24/Nov/20

no other way

$$\mathrm{no}\:\mathrm{other}\:\mathrm{way} \\ $$

Answered by zarminaawan last updated on 24/Nov/20

sin20=0.342

$${sin}\mathrm{20}=\mathrm{0}.\mathrm{342} \\ $$

Commented by MJS_new last updated on 24/Nov/20

the question is “...in exact form”

$$\mathrm{the}\:\mathrm{question}\:\mathrm{is}\:``...{in}\:{exact}\:{form}'' \\ $$

Answered by TANMAY PANACEA last updated on 24/Nov/20

y=sinx y+△y=sin(x+△x) here △x=2^o =((π/(180))×2)=(π/(90)) x=18^o ((△y)/(△x))≈(dy/dx) △y=(dy/dx)△x △y=cosx×((π/(90))) sin(20^o )=sin18^o +(π/(90))×cos18^o =(((√5) −1)/4)+((π/(90)))×(√((10+2(√5))/4)) =0.30901699437+(π/(90))×0.9510565163 (using calculator) =0.30901699437+0.03321149739 =0.34222849176

$${y}={sinx} \\ $$$${y}+\bigtriangleup{y}={sin}\left({x}+\bigtriangleup{x}\right) \\ $$$${here}\:\bigtriangleup{x}=\mathrm{2}^{{o}} =\left(\frac{\pi}{\mathrm{180}}×\mathrm{2}\right)=\frac{\pi}{\mathrm{90}} \\ $$$${x}=\mathrm{18}^{{o}} \\ $$$$\frac{\bigtriangleup{y}}{\bigtriangleup{x}}\approx\frac{{dy}}{{dx}} \\ $$$$\bigtriangleup{y}=\frac{{dy}}{{dx}}\bigtriangleup{x} \\ $$$$\bigtriangleup{y}={cosx}×\left(\frac{\pi}{\mathrm{90}}\right) \\ $$$${sin}\left(\mathrm{20}^{{o}} \right)={sin}\mathrm{18}^{{o}} +\frac{\pi}{\mathrm{90}}×{cos}\mathrm{18}^{{o}} \\ $$$$=\frac{\sqrt{\mathrm{5}}\:−\mathrm{1}}{\mathrm{4}}+\left(\frac{\pi}{\mathrm{90}}\right)×\sqrt{\frac{\mathrm{10}+\mathrm{2}\sqrt{\mathrm{5}}}{\mathrm{4}}}\: \\ $$$$=\mathrm{0}.\mathrm{30901699437}+\frac{\pi}{\mathrm{90}}×\mathrm{0}.\mathrm{9510565163}\:\left({using}\:{calculator}\right) \\ $$$$=\mathrm{0}.\mathrm{30901699437}+\mathrm{0}.\mathrm{03321149739} \\ $$$$=\mathrm{0}.\mathrm{34222849176} \\ $$

Commented by MJS_new last updated on 24/Nov/20

interesting! but it′s also no exact solution... I don′t think it′s possible

$$\mathrm{interesting}! \\ $$$$\mathrm{but}\:\mathrm{it}'\mathrm{s}\:\mathrm{also}\:\mathrm{no}\:\mathrm{exact}\:\mathrm{solution}...\:\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{possible} \\ $$

Commented by TANMAY PANACEA last updated on 24/Nov/20

thank you sir..yes you correct

$${thank}\:{you}\:{sir}..{yes}\:{you}\:{correct} \\ $$

Commented by Dwaipayan Shikari last updated on 24/Nov/20

Thanking for interaction sirs !

$${Thanking}\:{for}\:{interaction}\:{sirs}\:! \\ $$

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