\( 1 \leftarrow \begin{array}{l}\text { Use the appropriate reciprocal identity to find the exact value of } \cot \theta \text { for the given value of } \tan \theta \\ \tan \theta=6\end{array} \quad \)\begin{tabular}{l} cot \( \theta=\square \) \\ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) 6.67 \\ \hline\end{tabular}

To find \(\cot \theta\) when \(\tan \theta = 6\), we can use the reciprocal identity: \(\cot \theta = \frac{1}{\tan \theta}\). So, substituting the given value of \(\tan \theta\): \[ \cot \theta = \frac{1}{6} \] This means that \(\cot \theta\) simplifies to \(\frac{1}{6}\). Isn't it fun how simple identities can help us swing from one trigonometric function to another? They’re like the backstage pass in the concert of mathematics, allowing you to move seamlessly between different concepts! As you dive deeper into trigonometric identities, you’ll discover just how interconnected the functions are! Pro tip: sketching the unit circle and marking key angles can help you visualize these relationships even better. You’ll be breezing through trigonometry in no time!