Question

If $ z=i{{\log }_{e}}(2-\sqrt{3}) $ , then find the value of $ \cos z $ .

• A. $ 2 $
• B. $ -2 $
• C. $ 2i $
• D. $ -2i $

Answer 1

Answer: (A)

Given, $z=i \log _{e}(2-\sqrt{3}) \because \cos z=\frac{e^{j z}+e^{-i z}}{2}=\frac{e^{i^{2} \log _{e}(2-\sqrt{3})}+e^{\log _{e}(2-\sqrt{3})}}{2}$
$=\frac{e^{\log _{e}(2-\sqrt{3})^{-1}+e^{\log _{e}(2-\sqrt{3})}}}{2}=\frac{(2-\sqrt{3})^{-1}+(2-\sqrt{3})}{2}=\frac{1}{2}\left[\frac{1}{2-\sqrt{3}}+2-\sqrt{3}\right]$
$=\frac{1}{2}[2+\sqrt{3}+2-\sqrt{3}]=2$