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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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