Question

Use your calculator to approximate to four decimal places cot 16°40

Answer 1

To approximate cot 16°40' to four decimal places, one must calculate the tangent of the angle first, take its reciprocal, and then round according to the rules of significant figures, which is determined by the least precise number used in the calculation.

Step-by-step explanation:

When approximating the value of cot 16°40' (cotangent of 16 degrees and 40 minutes) to four decimal places using a calculator, there are a few important considerations regarding significant figures and rounding off. The cot function is the reciprocal of the tangent function, so we need to take the tangent of the angle first and then find its reciprocal to get the cotangent value.

It is crucial to follow the rules of significant figures because the precision of the result can only be as accurate as the least precise number used in the calculation. In this case, the most precise value we are given is to two significant figures (for example, if our angle measurement was taken using an instrument with precision up to 16°40', that indicates two significant figures in the minutes portion).

To find the cotangent using a calculator:

1. Enter the angle 16°40' into the calculator and convert it to decimal form if necessary.
2. Calculate tan(16.6667) - ensuring that the calculator is set to the correct angle measure mode (degrees in this case).
3. Since cot(x) = 1/tan(x), calculate the reciprocal of the tangent value.
4. Finally, round the result to four decimal places, being careful to only report as many significant figures as the least precise number used in the calculation.

Remember that while calculators provide a raw output, the user must apply the rules of significant figures to achieve an accurate result. Therefore, when you get the answer on the calculator, you must round it according to the two significant figures of the minutes in your angle measurement.

For example, if the calculator gives an answer of 2.0855688>, we cannot simply take all the digits because our angle is only precise to two significant figures in the minutes portion. Thus, your final reported answer should be rounded accordingly, in this case to 2.086, assuming the third decimal digit after rounding is a 6 or more, which would require rounding up.