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Question Number 10750 by Joel576 last updated on 24/Feb/17

$$\mathrm{Exact}\mathrm{value}\mathrm{of}$$$$\sin 9°=...$$

Answered by ridwan balatif last updated on 24/Feb/17

$$\mathrm{first}\mathrm{we}\mathrm{must}\mathrm{know}\mathrm{the}\mathrm{value}\mathrm{of}\sin {18}^{\mathrm{o}}$$$$\mathrm{let}\mathrm{x}={18}^{\mathrm{o}}$$$$5\mathrm{x}={90}^{\mathrm{o}}$$$$3\mathrm{x}=90-2\mathrm{x}$$$$\cos \left(3\mathrm{x}\right)=\cos (90-2\mathrm{x})$$$$\cos \left(3\mathrm{x}\right)=\sin \left(2\mathrm{x}\right)$$$$\cos (2\mathrm{x}+\mathrm{x})=\sin \left(2\mathrm{x}\right)$$$$\cos 2\mathrm{x}\times \mathrm{cosx}-\mathrm{sinx}\times \sin 2\mathrm{x}=2\mathrm{sinxcosx}$$$$({\cos }^2\mathrm{x}-{\sin }^2\mathrm{x})\times \mathrm{cosx}-\mathrm{sinx}\times \left(2\mathrm{sinxcosx}\right)=2\mathrm{sinxcosx}$$$$({\cos }^2\mathrm{x}-{\sin }^2\mathrm{x})-{2\sin }^2\mathrm{x}=2\mathrm{sinx}$$$$(1-{2\sin }^2\mathrm{x})-{2\sin }^2\mathrm{x}=2\mathrm{sinx}$$$${4\sin }^2\mathrm{x}+2\mathrm{sinx}-1=0$$$$\mathrm{D}={\mathrm{b}}^2-4\mathrm{ac}=20$$$${\mathrm{sinx}}_{1,2}=\frac{-2\pm \sqrt{20}}{2.4}=\frac{-1\pm \sqrt{5}}{4}$$$$\sin {18}^{\mathrm{o}}\mathrm{must}\mathrm{be}\mathrm{positive}$$$${\sin 18}^{\mathrm{o}}=\frac{\sqrt{5}-1}{4}$$$${\cos 18}^{\mathrm{o}}=\frac{\sqrt{10+2\sqrt{5}}}{4}$$$${\sin 9}^{\mathrm{o}}=\sqrt{\frac{1-{\cos 18}^{\mathrm{o}}}{2}}$$$$=\sqrt{\frac{1-\left(\frac{\sqrt{10+2\sqrt{5}}}{4}\right)}{2}}$$$$=\sqrt{\frac{\frac{4-\left(\sqrt{10+2\sqrt{5}}\right)}{4}}{2}}$$$$=\sqrt{\frac{4-\sqrt{10+2\sqrt{5}}}{8}}$$$$=\frac{1}{2}\sqrt{2-\frac{1}{2}\sqrt{10+2\sqrt{5}}}$$$$$$

Commented by Joel576 last updated on 24/Feb/17

$$thankyouverymuch$$

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