We are tasked with finding the approximate value of $\cot \theta$, given that $\csc \theta = 3.3272150$ and that $\theta$ is in quadrant I.

Step-by-Step Solution:

1. Relationship between cosecant and sine: $\csc \theta = \frac{1}{\sin \theta}$ Therefore, we can find $\sin \theta$ by taking the reciprocal of $\csc \theta$: $\sin \theta = \frac{1}{3.3272150}$

2. Calculate $\sin \theta$: $\sin \theta \approx \frac{1}{3.3272150} \approx 0.3005546$

3. Use the identity for $\cot \theta$: We know that $\cot \theta = \frac{\cos \theta}{\sin \theta}$. So, we need $\cos \theta$, which we can find using the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ Thus: $\cos^2 \theta = 1 - \sin^2 \theta$ $\cos^2 \theta = 1 - (0.3005546)^2 \approx 1 - 0.0903333 \approx 0.9096667$ $\cos \theta \approx \sqrt{0.9096667} \approx 0.9537675$

4. Calculate $\cot \theta$: Now that we have $\sin \theta \approx 0.3005546$ and $\cos \theta \approx 0.9537675$, we can compute $\cot \theta$: $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{0.9537675}{0.3005546} \approx 3.1736572$

Final Answer:

$\boxed{3.1736572}$ You can round this value to seven decimal places as required.

Would you like further details or have any questions?

Related Questions:

1. How is the reciprocal identity of cosecant related to sine?
2. What is the importance of the Pythagorean identity in trigonometry?
3. How does the sign of the trigonometric functions change across quadrants?
4. Can you explain the process of rationalizing denominators when applicable in trigonometry?
5. How do you compute the value of $\tan \theta$ given $\cot \theta$?

Tip:

Always double-check which quadrant the angle lies in to ensure that the signs of your trigonometric functions are correct.