1. **State the problem:** Find the value of $\cot 231^\circ 48'$ and simplify it to a decimal rounded to eight decimal places. 2. **Convert the angle to decimal degrees:** $$231^\circ 48' = 231 + \frac{48}{60} = 231 + 0.8 = 231.8^\circ$$ 3. **Use the cotangent identity:** $$\cot \theta = \frac{1}{\tan \theta}$$ 4. **Find the reference angle:** Since $231.8^\circ$ is in the third quadrant (between $180^\circ$ and $270^\circ$), the reference angle is: $$231.8^\circ - 180^\circ = 51.8^\circ$$ 5. **Evaluate $\tan 231.8^\circ$ using the reference angle:** In the third quadrant, tangent is positive, so: $$\tan 231.8^\circ = \tan 51.8^\circ$$ 6. **Calculate $\tan 51.8^\circ$ (approximate):** $$\tan 51.8^\circ \approx 1.25864044$$ 7. **Calculate $\cot 231.8^\circ$:** $$\cot 231.8^\circ = \frac{1}{\tan 231.8^\circ} = \frac{1}{1.25864044} \approx 0.79404039$$ **Final answer:** $$\cot 231^\circ 48' \approx 0.79404039$$