To find the decimal approximation of $\cot(247^\circ 48')$, we can break it down into two steps:

1. Convert the angle into decimal degrees: The angle is given as $247^\circ 48'$. First, convert the minutes (48') into a decimal:

   $48' = \frac{48}{60} = 0.8^\circ$

   So, the angle in decimal degrees is:

   $247^\circ 48' = 247 + 0.8 = 247.8^\circ$

2. Calculate the cotangent: The cotangent of an angle is the reciprocal of the tangent. Therefore,

   $\cot(\theta) = \frac{1}{\tan(\theta)}$

   Using a calculator, we find:

   $\cot(247.8^\circ) \approx \frac{1}{\tan(247.8^\circ)} \approx -0.26794919$

Thus, the decimal approximation of $\cot(247^\circ 48')$ is:

$\boxed{-0.26794919}$

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Would you like more details on the steps, or do you have any other questions? Here are some related questions:

1. How can I convert an angle from degrees, minutes, and seconds to decimal degrees?
2. How do you calculate the cotangent of an angle manually?
3. What is the relationship between tangent and cotangent?
4. How can I convert a decimal degree back into degrees, minutes, and seconds?
5. How do you determine the quadrant of an angle and its impact on trigonometric functions?

Tip: Always ensure you are using the correct mode (degree or radian) on your calculator when performing trigonometric calculations.