# find the functionIf sin(x)= -1/3 and Pi ≤ x ≤ 3Pi/2, find cot(2x) .

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If Pi ≤ x ≤ 3Pi/2, then x is in the 3rd quadrant and the function cotangent is positive.

Since the cot function is a ratio between cosine and sine 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)= -1/3

(cos x)^2 = 1 - 1/9

(cos x)^2 = 8/9

cos x = - 2sqrt2/3

We'll write cotangent function as a ratio:

cot x = cos x/ sin x

cot x = (- 2sqrt2/3)/(-1/3)

cot x = 2sqrt2