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If \theta is an angle in standard position and its terminal side passes through the point (12,-5), find the exact value of cot\theta in simplest radical form.

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Answer:

Explanation:

To find the exact value of cot(θ), we need to determine the ratio of the adjacent side to the opposite side of the right triangle formed by the given point (12, -5).

Let's label the coordinates of the point as follows: x = 12 and y = -5.

We can calculate the length of the adjacent side and the opposite side using the Pythagorean theorem:

Adjacent side (x-coordinate) = 12

Opposite side (y-coordinate) = -5

Now, we can determine the value of cot(θ) by taking the ratio of the adjacent side to the opposite side:

cot(θ) = adjacent side / opposite side

= x / y

Substituting the values, we get:

cot(θ) = 12 / -5

To simplify the expression, we can multiply the numerator and denominator by -1 to obtain a positive denominator:

cot(θ) = -12 / 5

Therefore, the exact value of cot(θ) in simplest radical form is -12/5.

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