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

$$Findthevalueofsin\left(\frac{\pi }{9}\right)inexactform$$

Commented by MJS_new last updated on 24/Nov/20

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

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

$$sin\left(\frac{\pi }{9}\right)=sin\theta$$$$9\theta =\pi \Rightarrow sin6\theta =sin3\theta \Rightarrow 2cos3\theta =1(sin\theta =\sqrt{1-t^2}$$$$4t^3-3t=\frac{1}{2}\Rightarrow t^3-\frac{3}{4}t-\frac{1}{8}=0$$$$\frac{1}{27}{\left(\frac{-3}{4}\right)}^3+\frac{1}{4}{(-\frac{1}{8})}^2=-\frac{1}{64}+\frac{1}{4.64}<0\left(Invalid\right)$$$$Butanyotherwaysir?$$$$$$

Commented by MJS_new last updated on 24/Nov/20

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

Answered by zarminaawan last updated on 24/Nov/20

$$sin20=0.342$$

Commented by MJS_new last updated on 24/Nov/20

$$\mathrm{the}\mathrm{question}\mathrm{is}‘‘...inexact{form}^{″}$$

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

$$y=sinx$$$$y+△y=sin(x+△x)$$$$here△x=2^o=\left(\frac{\pi }{180}\times 2\right)=\frac{\pi }{90}$$$$x={18}^o$$$$\frac{△y}{△x}\approx \frac{dy}{dx}$$$$△y=\frac{dy}{dx}△x$$$$△y=cosx\times \left(\frac{\pi }{90}\right)$$$$sin\left({20}^o\right)=sin{18}^o+\frac{\pi }{90}\times cos{18}^o$$$$=\frac{\sqrt{5}-1}{4}+\left(\frac{\pi }{90}\right)\times \sqrt{\frac{10+2\sqrt{5}}{4}}$$$$=0.30901699437+\frac{\pi }{90}\times 0.9510565163\left(usingcalculator\right)$$$$=0.30901699437+0.03321149739$$$$=0.34222849176$$

Commented by MJS_new last updated on 24/Nov/20

$$\mathrm{interesting}!$$$$\mathrm{but}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{also}\mathrm{no}\mathrm{exact}\mathrm{solution}...\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{think}$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{possible}$$

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

$$thankyousir..yesyoucorrect$$

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

$$Thankingforinteractionsirs!$$

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