Question

The real part of the principle value of $2^{-i}$ is

• A. $\sin \left(\log 2\right)$
• B. $\cos \left(\frac{1}{\log 2}\right)$
• C. $\cos \left[\log \left(\frac{1}{2}\right)\right]$
• D. $\cos \left(\log 2\right)$

Answer 1

The correct option is C $\cos \left[\log \left(\frac{1}{2}\right)\right]$

Let $z=2^{-i}$
Taking log on both sides, we get
$\log z=\log \left(2^{-i}\right)$
$\Rightarrow \log z=-i\log 2$
$\Rightarrow \log z=i\log \left(\frac{1}{2}\right)$
$\Rightarrow z=e^{i\log \left(1/2\right)}(\because e^{i\theta }=\cos \theta +i\sin \theta )$
$\Rightarrow z=\cos \left(\log \frac{1}{2}\right)+i\sin \left(\log \frac{1}{2}\right)$
$\therefore$ The real part of $z=2^{-1}$ is $\cos \left(\log \frac{1}{2}\right)$.