Question

If $z=i\log \left(\frac{1-\sqrt{-3}}{2}\right)$, then $\cos z=$

• A. $-1$
• B. $-\frac{1}{2}$
• C. $1$
• D. $\frac{1}{2}$

Answer 1

The correct option is D $\frac{1}{2}$

$z=i\log \left(\frac{1-\sqrt{-3}}{2}\right)$
$=i\log \left(\frac{1-i\sqrt{3}}{2}\right)$
$=i\log \left(e^{-i\frac{\pi }{3}}\right)$
$=i(-i\frac{\pi }{3})$
$=\frac{\pi }{3}$
$\therefore z=\frac{\pi }{3}$
Then, $\cos z=\cos \left(\frac{\pi }{3}\right)$ $=\frac{1}{2}$