Question

If $\alpha \in R$, then find the maximum value of $|e^{2\alpha \pi i}-1|$.

• A. $2\pi \alpha$
• B. $2\pi$
• C. $2\pi |\alpha |$
• D. $2\alpha$
• E. $2|\alpha |$

Answer 1

The correct option is C $2\pi |\alpha |$

$|e^{2\alpha \pi \iota }-1|=|(\cos 2\alpha \pi -1)+\iota \sin 2\alpha \pi |$

$=|-2{\sin }^2\alpha \pi +\iota 2\sin \alpha \pi \cos \alpha \pi |$

$=|\iota \left(2\sin \alpha \pi \right)+\cos \alpha \pi +\iota \sin \alpha \pi |$

$=|2\sin \alpha \pi |$

Maximum value of $\sin \alpha \pi \le \alpha \pi$

$|e^{2\alpha \pi \iota }-1|\le |2\alpha \pi |$