# If sin(x) = -1/3 and Pi ≤ x ≤ 3Pi/2, then 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**

To find x if sin(x) = -1/3 and Pi ≤ x ≤ 3Pi/2, then cot(2x) = ?

Given that sinx = -1/3 and x in 3rd quadrant.

To find cot2x.

We know that if sinx = -1/3 , then tanx = sinx /(sqrt(1-sin^2x) = (1/3)/sqrt(1-(1/3)^2) = 1/sqrt 8

We know that cot(2x) = 1/tan2x

cot(2x) = 1/{2tanx/1-tan^2x)}

cot(2x) = {1-(tan^2x)}/2tanx

cot(2x) = (1- 1/8)/2sqrt(1/8)

cot(2x) = 7/16(1/sqrt8)

cot(2x) = 7sqrt8/16.

Therefore cot2x = 7sqrt8/16.