If sin x = 3/4 and x is in the set (90,180), what is cot x?

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In the 2nd quadrant, the values of cotangent function are negative.

To determine the value of cotangent function, in the given conditions, we'll apply Pythagorean identity:

1 + (cot x)^2 = 1/(sin x)^2

(cot x)^2 = 1/(sin x)^2 - 1

cot x = - sqrt [1/(sin x)^2 - 1]

cot x = - sqrt (16/9 - 1)

cot x = - (sqrt 7)/3

We'll keep only the negative value for cotangent function

You need to use the following trigonometric identity, such that:

`cot x = cos x/sin x`

The problem provides `sin x = 3/4` , hence, using the fundamental formula of trigonometry, you may find `cos x` , such that:

`cos x = +-sqrt(1 - sin^2 x)`

`cos x = +- sqrt(1 - 9/16) => cos x = +-sqrt(7/16)`

You should remember that for any angle `x in (90^o, 180^o)` , the cosine is negative and sine is positive, hence cotangent is negative, such that:

`cos x = -sqrt7/4`

`cot x = -(sqrt7/4)/(3/4) => cot x = -sqrt7/3`

**Hence, evaluating cot x, under the given conditions, yields **`cot x = -sqrt7/3.`

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