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Question Number 111391 by Aina Samuel Temidayo last updated on 03/Sep/20

$$\mathrm{Compute}\cos \frac{\mathrm{\Pi }}{12}$$

Answered by bemath last updated on 03/Sep/20

$$\frac{\pi }{6}=2\times \frac{\pi }{12}\Rightarrow \cos \frac{\pi }{6}=\cos (2.\frac{\pi }{12})$$$$\frac{1}{2}\sqrt{3}=2\cos {}^2\left(\frac{\pi }{12}\right)-1$$$$\frac{\sqrt{3}}{2}+1=2\cos {}^2\left(\frac{\pi }{12}\right)\Rightarrow \cos \frac{\pi }{12}=\sqrt{\frac{2+\sqrt{3}}{4}}$$$$\cos \frac{\pi }{12}=\frac{1}{2}\sqrt{2+\sqrt{3}}$$

Commented by bemath last updated on 03/Sep/20

$$\mathrm{haha}..\mathrm{why}\mathrm{you}\mathrm{think}\mathrm{it}\mathrm{not}\mathrm{equivalent}?$$

Commented by bemath last updated on 03/Sep/20

Commented by bemath last updated on 03/Sep/20

$$\mathrm{look}\mathrm{it}$$

Commented by bobhans last updated on 03/Sep/20

$$\mathrm{creative}\mathrm{sir}.\mathrm{i}\mathrm{like}\mathrm{it}$$

Answered by ajfour last updated on 03/Sep/20

$$\cos \left(\frac{\pi }{12}\right)=\cos (\frac{\pi }{3}-\frac{\pi }{4})$$$$=\cos \frac{\pi }{3}\cos \frac{\pi }{4}+\sin \frac{\pi }{3}\sin \frac{\pi }{4}$$$$=\frac{1}{2}\times \frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{2}}$$$$\cos \frac{\pi }{12}=\frac{\sqrt{2}+\sqrt{6}}{4}.$$

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

$$\mathrm{Thanks}.$$

Commented by bemath last updated on 03/Sep/20

$$\sqrt{{\left(\frac{\sqrt{2}+\sqrt{6}}{4}\right)}^2}=\sqrt{\frac{8+2\sqrt{12}}{16}}=\sqrt{\frac{8+4\sqrt{3}}{16}}$$$$=\sqrt{\frac{2+\sqrt{3}}{4}}=\frac{1}{2}\sqrt{2+\sqrt{3}}$$$$\mathrm{clearr}???$$

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

$$\mathrm{Yes}.\mathrm{Thanks}.$$

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