# If sin(x) = 2/3, what is tan(2x)?

May 24, 2016

$4 \sqrt{5}$

#### Explanation:

$\sin x = \frac{2}{3}$ , find cos x.
${\cos}^{2} x = 1 - {\sin}^{2} x = 1 - \frac{4}{9} = \frac{5}{9}$
$\cos x = \pm \frac{\sqrt{5}}{3}$
Since x is in Quadrant I, then, cos x is positive
$\cos x = \frac{\sqrt{5}}{3}$
$\tan x = \sin \frac{x}{\cos x} = \left(\frac{2}{3}\right) \left(\frac{3}{\sqrt{5}}\right) = \frac{2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}$
Apply the trig identity:
$\tan 2 x = \frac{2 \tan x}{1 - {\tan}^{2} x}$
$\tan 2 x = \frac{\frac{4 \sqrt{5}}{5}}{1 - \frac{4}{5}} = \left(4 \frac{\sqrt{5}}{5}\right) \left(\frac{5}{1}\right) = 4 \sqrt{5}$