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To solve \( \cot(570^\circ) \), we need to simplify the angle using the periodicity of trigonometric functions.

### Step 1: Reduce the angle
The cotangent function, like sine and cosine, has a periodicity of \( 360^\circ \). So, we can reduce the angle by subtracting \( 360^\circ \) as many times as needed until the angle is within the standard range of \( 0^\circ \) to \( 360^\circ \).

\[
570^\circ - 360^\circ = 210^\circ
\]

Thus, \( \cot(570^\circ) = \cot(210^\circ) \).

### Step 2: Find \( \cot(210^\circ) \)
The angle \( 210^\circ \) is in the third quadrant, where the cotangent is positive. To find its value, we use the reference angle. The reference angle for \( 210^\circ \) is:
\[
210^\circ - 180^\circ = 30^\circ
\]
Now, \( \cot(210^\circ) \) has the same value as \( \cot(30^\circ) \), but it is positive in the third quadrant. The value of \( \cot(30^\circ) \) is:

\[
\cot(30^\circ) = \frac{\sqrt{3}}{3}
\]

### Conclusion:
\[
\cot(570^\circ) = \frac{\sqrt{3}}{3}
\]

Let me know if you need more details or have any further questions! Here are five related questions to deepen your understanding:

1. What is the periodicity of other trigonometric functions like sine, cosine, and tangent?
2. How do we determine reference angles in each quadrant?
3. Why is \( \cot(210^\circ) \) positive in the third quadrant?
4. Can you find \( \cot(750^\circ) \) using a similar method?
5. What are the values of the other trigonometric functions at \( 30^\circ \)?

**Tip:** Always reduce angles by using the periodicity of the trigonometric function before attempting to evaluate them.

Use the fact that the trigonometric functions are periodic to calculate \(\cot(570^\circ)\).

This solution explains how to compute \(\cot(570^\circ)\) by using the periodic nature of trigonometric functions and simplifying the angle. It covers key trigonometric concepts like cotangent and reference angles, making it suitable for students in Grades 10-12.

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