Given sin x=-1/8, what is tan 2x if x is in the interval (pi,3pi/2)?

Since x belongs to the range (pi,3pi/2), then x is located
in the 3rd quadrant and the values of tangent function are
positive.

Since the tan function is a ratio between sine
and cosine functions,we need to calculate the cosine function, using the fundamental
formula of trigonometry.

(sin x)^2 + (cos x)^2 =
1

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

We
know that (sin x)= -1/8

(cos x)^2 = 1 -
1/64

(cos x)^2 = 63/64

cos x =
- 3*sqrt7/8

We'll write tangent function as a
ratio:

tan x = sin x / cos
x

tan x =  (- 1/8)/(-
3*sqrt7/8)

tan x
=sqrt7/21

We'll apply double angle identity to determine
tan (2x):

tan (2x) = 2*tan x/[1 - (tan
x)^2]

tan (2x) = (2sqrt7/21)/(1 -
7/441)

tan (2x) =
(2sqrt7/21)/434/441

tan (2x)
=42*sqrt7/434