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Question Number 196211 by pete last updated on 20/Aug/23

$$\mathrm{Show}\mathrm{that}\cos (\frac{\pi }{3}+\mathrm{i})=\frac{1}{4}(\mathrm{e}+\frac{1}{\mathrm{e}})-\frac{\sqrt{3}}{4}(\mathrm{e}-\frac{1}{\mathrm{e}})\mathrm{i}$$

Answered by Frix last updated on 20/Aug/23

$$\cos x=\frac{{\mathrm{e}}^{2\mathrm{i}x}+1}{{2\mathrm{e}}^{\mathrm{i}x}}$$$$\cos (a+b\mathrm{i})=\cos a\cosh b-\mathrm{i}\sin a\sinh b$$$$\cos \frac{\pi }{3}\cosh 1-\mathrm{i}\sin \frac{\pi }{3}\sinh 1=$$$$=\frac{\cosh 1}{2}-\frac{\sqrt{3}\sinh 1}{2}=$$$$[\cosh x=\frac{{\mathrm{e}}^{2x}+1}{{2\mathrm{e}}^x}\wedge \sinh x=\frac{{\mathrm{e}}^{2x}-1}{{2\mathrm{e}}^x}]$$$$=\frac{{\mathrm{e}}^2+1}{4\mathrm{e}}-\frac{\sqrt{3}({\mathrm{e}}^2-1)}{4\mathrm{e}}\mathrm{i}=\frac{1}{4}(\mathrm{e}+\frac{1}{\mathrm{e}})-\frac{\sqrt{3}}{4}(\mathrm{e}-\frac{1}{\mathrm{e}})\mathrm{i}$$

Commented by pete last updated on 20/Aug/23

$$\mathrm{You}\mathrm{do}\mathrm{all}\mathrm{sir},\mathrm{many}\mathrm{thanks}\mathrm{sir}.$$

Commented by Frix last updated on 20/Aug/23

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