The imaginary part of ^ - 1 ( 5i 3 ) is - Vidyadip

MCQ

The imaginary part of ${\tan ^{ - 1}}\left( {\frac{{5i}}{3}} \right)$ is

A
$0$
B
$\infty $
✓
$\log 2$
D
$\log 4$

✓

Answer

Correct option: C.

$\log 2$

c
(c)${\tan ^{ - 1}}\left( {\frac{{5i}}{3}} \right) = i{\tan ^{ - 1}}\left( {\frac{5}{3}} \right) = \frac{i}{2}\log \left( {\frac{{\frac{5}{3} + 1}}{{\frac{5}{3} - 1}}} \right)$
${\mathop{\rm Im}\nolimits} \left( {{{\tan }^{ - 1}}\left( {\frac{{5i}}{3}} \right)} \right) = \frac{1}{2}\log 4 = \frac{1}{2}.\,2\log 2 = \log 2$.

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