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

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

Answered by Frix last updated on 20/Aug/23

\cos x = \frac{e^{2 i x} + 1}{{2 e}^{i x}}

\cos (a + b i) = \cos a \cosh b - i \sin a \sinh b

\cos \frac{π}{3} \cosh 1 - i \sin \frac{π}{3} \sinh 1 =

= \frac{\cosh 1}{2} - \frac{\sqrt{3} \sinh 1}{2} =

[\cosh x = \frac{e^{2 x} + 1}{{2 e}^{x}} \land \sinh x = \frac{e^{2 x} - 1}{{2 e}^{x}}]

= \frac{e^{2} + 1}{4 e} - \frac{\sqrt{3} (e^{2} - 1)}{4 e} i = \frac{1}{4} (e + \frac{1}{e}) - \frac{\sqrt{3}}{4} (e - \frac{1}{e}) i

Commented by pete last updated on 20/Aug/23

You do all sir, many thanks sir .

Commented by Frix last updated on 20/Aug/23

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