Find the approximate value of \( \cot \theta \), given that \( \csc \theta=3.3272150 \) and \( \theta \) is in quadrant I. Rationalize denominators when applicable. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \cot \theta \approx \square \) (Do not round until the final answer. Then round to seven decimal places as needed.)

To find \( \cot \theta \) given that \( \csc \theta = 3.3272150 \), we start with some trigonometric identities. First, recall the relationship between cosecant, sine, and cotangent: \[ \csc \theta = \frac{1}{\sin \theta} \] Thus, we can find \( \sin \theta \): \[ \sin \theta = \frac{1}{\csc \theta} = \frac{1}{3.3272150} \approx 0.3003821 \] Next, we utilize the identity for cotangent, which is the ratio of cosine to sine: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] We can find \( \cos \theta \) using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \implies \cos^2 \theta = 1 - \sin^2 \theta \] Calculating \( \sin^2 \theta \): \[ \sin^2 \theta \approx (0.3003821)^2 \approx 0.0902296 \] Now find \( \cos^2 \theta \): \[ \cos^2 \theta = 1 - 0.0902296 \approx 0.9097704 \] Taking the square root to find \( \cos \theta \) (and since \( \theta \) is in quadrant I, we take the positive root): \[ \cos \theta \approx \sqrt{0.9097704} \approx 0.9531261 \] Now substituting back to find \( \cot \theta \): \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \approx \frac{0.9531261}{0.3003821} \approx 3.1683578 \] Finally, rounding to seven decimal places, we get: A. \( \cot \theta \approx 3.1683578 \)