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Question Number 12028 by tawa last updated on 09/Apr/17

$$\mathrm{Express}:\sin \left(33\right)\mathrm{in}\mathrm{surd}\mathrm{form}.$$

Answered by sandy_suhendra last updated on 10/Apr/17

$$\mathrm{let}\mathrm{A}=18°$$$$5\mathrm{A}=90°$$$$3\mathrm{A}=90°-2\mathrm{A}$$$$\cos 3\mathrm{A}=\cos (90°-2\mathrm{A})$$$${4\cos }^3\mathrm{A}-3\mathrm{cosA}=\sin 2\mathrm{A}$$$${4\cos }^3\mathrm{A}-3\mathrm{cosA}=2\mathrm{sinAcosA}$$$${4\cos }^2\mathrm{A}-3=2\mathrm{sinA}$$$$4-{4\sin }^2\mathrm{A}-3=2\mathrm{sinA}$$$${4\sin }^2\mathrm{A}+2\mathrm{sinA}-1=0$$$$\mathrm{sinA}=\frac{-2+\sqrt{4+4.4(-1)}}{2.4}$$$$\sin 18°=\frac{-2+2\sqrt{5}}{8}=\frac{\sqrt{5}-1}{4}$$$$\cos 18°=\sqrt{1-{\sin }^{2}18°}$$$$=\sqrt{1-{\left(\frac{\sqrt{5}-1}{4}\right)}^2}$$$$=\sqrt{\frac{10+2\sqrt{5}}{16}}=\frac{1}{4}\sqrt{10+2\sqrt{5}}$$$$\sin 15°=\sqrt{\frac{1-\cos 30°}{2}}$$$$=\sqrt{\frac{1-\frac{1}{2}\sqrt{3}}{2}}$$$$=\sqrt{\frac{4-2\sqrt{3}}{8}}=\sqrt{\frac{{(\sqrt{3}-1)}^2}{8}}$$$$=\frac{\sqrt{3}-1}{2\sqrt{2}}$$$$\cos 15°=\sqrt{\frac{1+\cos 30°}{2}}=\frac{\sqrt{3}+1}{2\sqrt{2}}$$$$\sin 33°$$$$=\sin (18°+15°)$$$$=\sin 18°.\cos 15°+\cos 18°.\sin 15°$$$$=\frac{\sqrt{5}-1}{4}\times \frac{\sqrt{3}+1}{2\sqrt{2}}+\frac{1}{4}\sqrt{10+2\sqrt{5}}\times \frac{\sqrt{3}-1}{2\sqrt{2}}$$$$=\frac{1}{8\sqrt{2}}[(\sqrt{5}-1)(\sqrt{3}+1)+\sqrt{10+2\sqrt{5}}.(\sqrt{3}-1)]$$

Commented by tawa last updated on 10/Apr/17

$$\mathrm{Wow},\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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