Find \( \cot (\theta) \) if \( \sin (\theta)=\frac{\sigma}{13} \) and \( \theta \) terminates in QI. Answer exactly. \( \cot (\theta)=\square \)

To find \( \cot(\theta) \) given that \( \sin(\theta) = \frac{\sigma}{13} \) and that \( \theta \) is in the first quadrant (QI), we can follow these steps: 1. **Use the Pythagorean identity**: We know that \[ \sin^2(\theta) + \cos^2(\theta) = 1. \] Substituting \( \sin(\theta) \) into the identity gives us: \[ \left(\frac{\sigma}{13}\right)^2 + \cos^2(\theta) = 1. \] 2. **Calculate \( \cos(\theta) \)**: \[ \cos^2(\theta) = 1 - \left(\frac{\sigma}{13}\right)^2. \] Simplifying this, we have: \[ \cos^2(\theta) = 1 - \frac{\sigma^2}{169} = \frac{169 - \sigma^2}{169}. \] Therefore, \[ \cos(\theta) = \sqrt{\frac{169 - \sigma^2}{169}} = \frac{\sqrt{169 - \sigma^2}}{13}. \] Since \( \theta \) is in the first quadrant, \( \cos(\theta) \) is positive. 3. **Calculate \( \cot(\theta) \)**: The cotangent function is defined as: \[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}. \] Substituting the values we found: \[ \cot(\theta) = \frac{\frac{\sqrt{169 - \sigma^2}}{13}}{\frac{\sigma}{13}} = \frac{\sqrt{169 - \sigma^2}}{\sigma}. \] Thus, the exact value of \( \cot(\theta) \) is: \[ \cot(\theta) = \frac{\sqrt{169 - \sigma^2}}{\sigma}. \] So, the final answer is: \[ \cot(\theta) = \frac{\sqrt{169 - \sigma^2}}{\sigma}. \]