Imagine

Geometry Level 4

Which of the given options is purely real?

Note: $i$ is the imaginary unit.

None of these choices

\sin{i}

\cos{i}

\tan{i}

1 solution

Digvijay Singh
Jun 20, 2016

Since

$\large \sin{x}=\frac{e^{ix}-e^{-ix}}{2i}$

and

$\large \cos{x}=\frac{e^{ix}+e^{-ix}}{2}$

Hence, for an imaginary value of $x$

$\sin{x}$ is always purely imaginary.
$\cos{x}$ is always purely real.
$\tan{x}$ is always purely imaginary.

For reference:

$\sin{i}=i\bigg(\frac{e^2-1}{2e}\bigg)$

$\cos{i}=\frac{e^2+1}{2e}$

$\tan{i}=i\bigg(\frac{e^2-1}{e^2+1}\bigg)$