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