Use reference angles to find the exact value of the following expression. \[ \begin{array}{l}\cot 210^{\circ} \\ \text { A. } \cot 210^{\circ}= \\ \text { (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or frad } \\ \text { the denominator.) }\end{array} \]

To find \(\cot 210^\circ\), we first need to determine the reference angle. The angle \(210^\circ\) is in the third quadrant, where all trigonometric functions are negative. The reference angle is calculated as: \[ 210^\circ - 180^\circ = 30^\circ \] Now, we can use the definition of cotangent, which is the reciprocal of tangent: \[ \cot 210^\circ = -\cot 30^\circ \] From trigonometry, we know that \(\cot 30^\circ = \frac{\cos 30^\circ}{\sin 30^\circ}\). Since \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\), we have: \[ \cot 30^\circ = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \] Thus, substituting back into our expression, we find: \[ \cot 210^\circ = -\sqrt{3} \] The exact value of \(\cot 210^\circ\) is \(-\sqrt{3}\).