"Is my evaluation of complex trigonometric expression sin⁡(π4+2i)−sin⁡(2i)..." was answered by our experts

zeotropojd

Answered question

2022-01-16

Is my evaluation of complex trigonometric expression

\sin (\frac{π}{4} + 2 i) - \sin (2 i)

correct?

Thomas Nickerson

Beginner2022-01-17Added 32 answers

\sin (\frac{π}{4} + 2 i) - \sin (2 i)

= \frac{1}{2 i} (e^{- 2} (\cos \frac{π}{4} + i \sin \frac{π}{4}) - e^{2} (\cos \frac{π}{4} - i \sin \frac{π}{4})) - \frac{1}{2 i} (e^{- 2} - e^{2})

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

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

= \frac{\sqrt{2}}{2} cosh (2) - i \frac{2 - \sqrt{2}}{2} sinh (2)

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