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

You need to remember that the tangent function is
rational, hence `tan 2x = (sin 2x)/(cos
2x).`

The problem provides the information
that the angle x is in quadrant 3, hence `sin xlt0 ; cos xlt
0` .

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

`sin x =
sqrt((1-cos 2x)/2)`

Substituting class="AM">`-1/3`  for sin x
yields:

`-1/3 = sqrt((1-cos
2x)/2)`

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

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

class="AM">`cos 2x = 1 - 2/9 =gt cos 2x =
7/9`

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

class="AM">`sin 2x = sqrt(1 - cos^2
2x)`

`sin 2x = sqrt(1 -
49/81) =gt sin 2x = sqrt(32/81)`

class="AM">`sin 2x = sqrt32/9`

You need to
substitute `sqrt32/9`  for class="AM">`sin 2x`  and `7/9`  for
`cos 2x`  in `tan 2x = (sin
2x)/(cos 2x) ` such that:

class="AM">`tan 2x = (sqrt32/9)/(7/9) =gt tan 2x =
(sqrt32/9)*(9/7)`

`tan
2x = sqrt32/7`

Hence, evaluating
the tangent of double of angle x yields `tan 2x =
sqrt32/7.`