# 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...

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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 `-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`

`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:

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

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

`sin 2x = sqrt32/9`

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

`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.`

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