If sin(x) = -1/3 and Pi ≤ x ≤ 3Pi/2, then tan(2x) = ?

You need to remember that the tangent function is
rational, hence
2x).

The problem provides the information
that the angle x is in quadrant 3, hence
0 .

You need to remember the formula of half
of angle such that:

sqrt((1-cos 2x)/2)

Substituting class="AM">  for sin x
yields:

2x)/2)

You need to raise to square to remove
the square root such that:

(1-cos 2x)/2 =gt 2/9 = 1 - cos 2x

class="AM">
7/9

You need to use the basic formula of
trigonometry to find sin 2x such that:

class="AM">
2x)

49/81) =gt sin 2x = sqrt(32/81)

class="AM">

You need to
substitute   for class="AM">  and   for
  in
2x)/(cos 2x) such that:

class="AM">
(sqrt32/9)*(9/7)

2x = sqrt32/7

Hence, evaluating
the tangent of double of angle x yields
sqrt32/7.