Final answer:
To find tan x when sin x = 1/4 and x is in the third quadrant, use the Pythagorean theorem to find cos x, and then divide sin x by cos x. The result is tan x = -√15/15.
Step-by-step explanation:
If sin x = 1/4 and π < x < 3π/2, we can find tan x by using Pythagoras' theorem and the properties of the unit circle. Since x is in the third quadrant (where both sine and cosine are negative), we can write:
√(1 - {sin² x) = √(1 - (1/4)²) = √(1 - 1/16) = √(15/16) = ∕√15 / 4.
Therefore, cos x is -∕√15 / 4 (negative in the third quadrant). The tangent is given by sin x / cos x, so:
tan x = (1/4) / (-∕√15 / 4) = -1/∕√15.
To simplify the tan x expression further, multiply the numerator and the denominator by √15:
tan x = -1/∕√15 × (√15 / √15) = -√15/15.