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

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

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Answer 1 · 2016-05-24T15:14:08

$4 \sqrt{5}$ $4sqrt5$

Explanation:

$sin x = \frac{2}{3}$ $sin x = 2/3$ , find cos x.
${cos}^{2} x = 1 - {sin}^{2} x = 1 - \frac{4}{9} = \frac{5}{9}$ $cos^2 x = 1 - sin^2 x = 1 - 4/9 = 5/9$
$cos x = \pm \frac{\sqrt{5}}{3}$ $cos x = +- sqrt5/3$
Since x is in Quadrant I, then, cos x is positive
$cos x = \frac{\sqrt{5}}{3}$ $cos x = sqrt5/3$
$tan x = \frac{sin x}{cos x} = (\frac{2}{3}) (\frac{3}{\sqrt{5}}) = \frac{2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}$ $tan x = sin x/(cos x) = (2/3)(3/sqrt5) = 2/sqrt5 = (2sqrt5)/5$
Apply the trig identity:
$tan 2 x = \frac{2 tan x}{1 - {tan}^{2} x}$ $tan 2x = (2tan x)/(1 - tan^2 x)$
$tan 2 x = \frac{\frac{4 \sqrt{5}}{5}}{1 - \frac{4}{5}} = (4 \frac{\sqrt{5}}{5}) (\frac{5}{1}) = 4 \sqrt{5}$ $tan 2x = ((4sqrt5)/5)/(1 - 4/5) = (4sqrt5/5)(5/1) = 4sqrt5$

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

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