To find \(\cot(-\frac{5\pi}{6})\), let's go step-by-step:

### Step 1: Understanding the angle
The given angle is \(-\frac{5\pi}{6}\). In terms of radians, we need to simplify this by understanding its position on the unit circle.

1. Since \(\frac{\pi}{6}\) radians is 30°, we have \(\frac{5\pi}{6} = 150^\circ\).
2. Therefore, \(-\frac{5\pi}{6}\) corresponds to an angle measured clockwise by 150°.

### Step 2: Finding the reference angle
The reference angle is the positive acute angle formed with the x-axis.

- The reference angle for \(-\frac{5\pi}{6}\) is \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\).
- The reference angle is \(\frac{\pi}{6}\), or 30°.

### Step 3: Determine the quadrant
\(-\frac{5\pi}{6}\) falls in the third quadrant since:
- Starting from 0 radians (on the positive x-axis), rotating clockwise by \(150^\circ\) moves us into the third quadrant.

In the third quadrant, both sine and cosine are negative, and therefore the cotangent is positive.

### Step 4: Cotangent of reference angle
We use the reference angle \(\frac{\pi}{6}\) (which corresponds to 30°). The value of \(\cot(\frac{\pi}{6})\) is:

\[
\cot(\frac{\pi}{6}) = \frac{1}{\tan(\frac{\pi}{6})} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}
\]

Since the cotangent is positive in the third quadrant, we have:

\[
\cot\left(-\frac{5\pi}{6}\right) = \sqrt{3}
\]

### Final Answer:
\[
\boxed{\cot\left(-\frac{5\pi}{6}\right) = \sqrt{3}}
\]

Would you like further explanation or details on any of the steps?

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Here are 5 related questions for further exploration:

1. What is \(\cot\left(\frac{5\pi}{6}\right)\)?
2. How do you find the cotangent of any angle in the second quadrant?
3. What is \(\sin\left(-\frac{5\pi}{6}\right)\)?
4. How do you find the reference angle of a negative angle?
5. What is the exact value of \(\tan\left(-\frac{7\pi}{4}\right)\)?

**Tip**: Always verify the quadrant of your angle to determine the sign of trigonometric functions!

This page offers a detailed step-by-step explanation on how to find the cotangent of -5π/6 using trigonometric identities and reference angles. Learn how to determine the sign of cotangent based on the unit circle and quadrant position. Suitable for students in Grades 10-12.

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