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 the approximate value of \( \cot \theta \) given that \( \csc \theta = 3.3272150 \) and \( \theta \) is in quadrant I, we can follow these steps: 1. **Understand the relationship between cosecant and sine:** \[ \csc \theta = \frac{1}{\sin \theta} \] Therefore, we can find \( \sin \theta \): \[ \sin \theta = \frac{1}{\csc \theta} = \frac{1}{3.3272150} \] 2. **Calculate \( \sin \theta \):** \[ \sin \theta = \frac{1}{3.3272150} \] 3. **Find \( \cos \theta \) using the Pythagorean identity:** \[ \sin^2 \theta + \cos^2 \theta = 1 \] First, we calculate \( \sin^2 \theta \): \[ \sin^2 \theta = \left(\frac{1}{3.3272150}\right)^2 \] Then, we can find \( \cos^2 \theta \): \[ \cos^2 \theta = 1 - \sin^2 \theta \] 4. **Calculate \( \cot \theta \):** \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] Now, let's perform the calculations step by step. ### Step 1: Calculate \( \sin \theta \) \[ \sin \theta = \frac{1}{3.3272150} \] ### Step 2: Calculate \( \sin^2 \theta \) \[ \sin^2 \theta = \left(\frac{1}{3.3272150}\right)^2 \] ### Step 3: Calculate \( \cos^2 \theta \) \[ \cos^2 \theta = 1 - \sin^2 \theta \] ### Step 4: Calculate \( \cot \theta \) \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{3.327215}\) - step1: Convert the expressions: \(\frac{1}{\frac{665443}{200000}}\) - step2: Multiply by the reciprocal: \(\frac{200000}{665443}\) Calculate or simplify the expression \( (1/3.3272150)^2 \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{1}{3.327215}\right)^{2}\) - step1: Divide the numbers: \(\left(\frac{200000}{665443}\right)^{2}\) - step2: Use the properties of exponents: \(\frac{200000^{2}}{665443^{2}}\) Calculate or simplify the expression \( 1 - 0.090331 \). Calculate the value by following steps: - step0: Calculate: \(1-0.090331\) - step1: Subtract the numbers: \(0.909669\) Calculate or simplify the expression \( \sqrt(0.909669) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{0.909669}\) - step1: Rewrite the expression: \(\sqrt{\frac{909669}{1000000}}\) - step2: Use the properties of radicals: \(\frac{\sqrt{909669}}{\sqrt{1000000}}\) - step3: Simplify the expression: \(\frac{\sqrt{909669}}{1000}\) Calculate or simplify the expression \( 0.953766/0.300552 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.953766}{0.300552}\) - step1: Convert the expressions: \(\frac{\frac{476883}{500000}}{\frac{37569}{125000}}\) - step2: Multiply by the reciprocal: \(\frac{476883}{500000}\times \frac{125000}{37569}\) - step3: Reduce the numbers: \(\frac{158961}{4}\times \frac{1}{12523}\) - step4: Multiply the fractions: \(\frac{158961}{4\times 12523}\) - step5: Multiply: \(\frac{158961}{50092}\) Now that we have completed the calculations, let's summarize the steps and results: 1. **Calculate \( \sin \theta \):** \[ \sin \theta \approx 0.300552 \] 2. **Calculate \( \sin^2 \theta \):** \[ \sin^2 \theta \approx 0.090331 \] 3. **Calculate \( \cos^2 \theta \):** \[ \cos^2 \theta = 1 - \sin^2 \theta \approx 1 - 0.090331 \approx 0.909669 \] 4. **Calculate \( \cos \theta \):** \[ \cos \theta \approx \sqrt{0.909669} \approx 0.953766 \] 5. **Calculate \( \cot \theta \):** \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \approx \frac{0.953766}{0.300552} \approx 3.173381 \] Thus, the approximate value of \( \cot \theta \) is: \[ \cot \theta \approx 3.173381 \] ### Final Answer: A. \( \cot \theta \approx 3.1733810 \) (rounded to seven decimal places)